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Graduate Texts in Mathematics

Konrad Schmüdgen

An Invitation
to Unbounded
Representations
of ∗-Algebras
on Hilbert Space
Graduate Texts in Mathematics 285
Graduate Texts in Mathematics

Series Editors

Sheldon Axler, San Francisco State University

Kenneth Ribet, University of California, Berkeley

Advisory Board

Alejandro Adem, University of British Columbia

David Eisenbud, University of California, Berkeley & MSRI
Brian C. Hall, University of Notre Dame
Patricia Hersh, University of Oregon
J. F. Jardine, University of Western Ontario
Jeffrey C. Lagarias, University of Michigan
Eugenia Malinnikova, Stanford University
Ken Ono, University of Virginia
Jeremy Quastel, University of Toronto
Barry Simon, California Institute of Technology
Ravi Vakil, Stanford University
Steven H. Weintraub, Lehigh University
Melanie Matchett Wood, University of California, Berkeley

Graduate Texts in Mathematics bridge the gap between passive study and
creative understanding, offering graduate-level introductions to advanced topics in
mathematics. The volumes are carefully written as teaching aids and highlight
characteristic features of the theory. Although these books are frequently used as
textbooks in graduate courses, they are also suitable for individual study.

More information about this series at http://www.springer.com/series/136

Konrad Schmüdgen

An Invitation to Unbounded
Representations of

-Algebras on Hilbert Space

123
Konrad Schmüdgen
Fakultät für Mathematik und Informatik
Universität Leipzig
Leipzig, Germany

ISSN 0072-5285 ISSN 2197-5612 (electronic)

Graduate Texts in Mathematics
ISBN 978-3-030-46365-6 ISBN 978-3-030-46366-3 (eBook)
https://doi.org/10.1007/978-3-030-46366-3

Mathematics Subject Classification: 47L60, 16G99, 16W10, 81S05

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Preface and Overview

Everything Should Be Made as Simple as Possible, But Not

Simpler1

The purpose of this book is to give an introduction to the unbounded representation

theory of
-algebras on Hilbert space. As the title indicates, the book should be
considered as an invitation to this subject rather than a monograph or a compre-
hensive presentation.
Let us briefly explain the two main concepts explored in this book.
A complex
-algebra A is a complex algebra with an involution, denoted by
a 7! a þ . An involution is an antilinear mapping of A into itself which is
antimultiplicative (that is, ðabÞ þ ¼ b þ a þ ) and involutive (that is, ða þ Þ þ ¼ a).
The complex conjugation of functions and the Hilbert space adjoint of operators are
standard examples of involutions.
Just as rings are studied in terms of their modules in algebra, it is natural to
investigate
-representations of
-algebras. Let D be a complex inner product space,
that is, D is a complex vector space equipped with an inner product h; i, and let H
be the corresponding Hilbert space completion. A
-representation of a
-algebra A
on D is an algebra homomorphism … of A into the algebra of linear operators on D
such that

h…ðaÞu; wi ¼ hu; …ða þ Þwi; u; w 2 D; ð1Þ

for all a 2 A. In general, the operators …ðaÞ are unbounded. Equation (1) is crucial,
because it translates algebraic properties of elements of A into operator-theoretic
properties of their images under …. For instance, if a 2 A is hermitian (that is,
a þ ¼ a), then the operator …ðaÞ is symmetric, or if a is normal (that is,
a þ a ¼ aa þ ), then …ðaÞ is formally normal (that is, k…ðaÞuk ¼ k…ða þ Þuk;
u 2 D). Since the closure of the symmetric operator …ðaÞ for a ¼ a þ on the Hilbert
space H is not necessarily self-adjoint, we are confronted with all the difficulties of
unbounded operator theory.

1
Attributed to Albert Einstein.

v
vi Preface and Overview

In quantum mechanics the canonical commutation relation

PQ  QP ¼ i
hI ð2Þ

plays a fundamental role. Here P is the momentum operator, Q is the position

operator, and
h ¼ 2…
h
is the reduced Planck’s constant. Historically, relation (2) is
attributed to Max Born (1925)2. It implies Werner Heisenberg’s uncertainty prin-
ciple [Hg27]. Born and Jordan [BJ26] found a representation of (2) by infinite
matrices. Schrödinger [Schr26] discovered that the commutation relation (2) can be
represented by the unbounded operators P and Q, given by

du
ðQuÞðxÞ ¼ xuðxÞ and ðPuÞðxÞ ¼ i

h ; ð3Þ
dx

acting on the Hilbert space L2 ðRÞ. It was shown later by Wielandt [Wie49] and
Wintner [Wi47] that (2) cannot be realized by bounded operators. For the mathe-
matical treatment of the canonical commutation relation (2), there is no loss of
h1 P.
generality in setting
h ¼ 1, upon replacing P by

The unital
-algebra W with hermitian generators p and q satisfying the relation
pq  qp ¼ i  1 is called the Weyl algebra. Since relation (2) cannot hold for
bounded operators, W has no
-representation by bounded operators, but it has
many unbounded
-representations. Among them there is one distinguished
“well-behaved” representation, the Schrödinger representation …S , or its unitarily
equivalent version, the Bargmann–Fock representation. The
-representation …S
acts on the Schwartz space SðRÞ, considered as a subspace of the Hilbert space
L2 ðRÞ, by

…S ðpÞu ¼ Pu and …S ðqÞu ¼ Qu; u 2 SðRÞ;

where P and Q are given by (3) with
h ¼ 1. The Weyl algebra has a rich algebraic
structure and an interesting representation theory. This
-algebra will be our main
guiding example through the whole book; it is treated in detail in Chap. 8.

Aims of the Book

For decades, operator theory on Hilbert space and operator algebras have provided
powerful methods for quantum theory and mathematical physics. Among the many
books on these topics, two can be recognized as standard textbooks for graduate
students and researchers. These are the four volumes [RS72]–[RS78] by

2
In a letter to Pauli [Pa79, pp. 236–241], dated September 18, 1925, Heisenberg called the
commutation relation (2) “eine sehr gescheite Idee von Born” (“a very clever idea of Born”). In the
literature the relation (2) was first formulated by Born and Jordan [BJ26] and by Dirac [D25].
Preface and Overview vii

Reed–Simon covering operator theory and the two volumes [BR87]–[BR97] by

Bratteli-Robinson for C
- and W
-algebras. The present book might be considered a
supplement covering unbounded representations of general
-algebras.
The aims and features of this book are the following:
• The main aim is to provide a careful and rigorous treatment of the basic
concepts and results of unbounded representation theory on Hilbert space.
Our emphasis is on representations of important nonnormed
-algebras. In
general, representations of
-algebras on Hilbert space act by unbounded operators.
It is well known that algebraic operations involving unbounded operators are del-
icate matters, so it is not surprising that unbounded representations lead to new and
unexpected difficulties and pathologies. Some of these are collected in Sect 4.7. In
fact, these phenomena already occur for very simple algebras such as the Weyl
algebra or polynomial algebras.
Compared to bounded Hilbert space representations, many results and devel-
opments require additional assumptions, concepts, and technical arguments. We
point out possible pathologies and propose concepts to circumvent them.
• In the exposition and presentation we try to minimize the use of technicalities
and generalities.
So we treat the representation theory of the Weyl algebra only in dimension one;
positivity only for functionals rather than complete positivity of mappings;
decomposition theory only for functionals and not for representations; and we avoid
details from the theory of quantum groups. Some results with long and technically
involved proofs, such as the trace representation Theorem 3.26 and the integrability
Theorems 9.49 and 9.50 for Lie algebra representations, are stated without proofs.
(The reader can find these topics and complete proofs in the author’s monograph
[Sch90].) We hope to fulfill Einstein’s motto stated above in this manner, at least to
some extent.
• The choice of topics illustrates the broad scope and the usefulness of unbounded
representations.
There are various fields in mathematics and mathematical physics where repre-
sentations of general
-algebras on Hilbert space appear. The canonical commutation
relation of quantum mechanics was already mentioned and is only one example.
Quantum algebras and noncompact quantum groups can be represented by
unbounded operators. Unitary representations of Lie groups lead to in general
unbounded representations of enveloping algebras. Representations of polynomial
algebras play a crucial role in the operator approach to the classical multi-dimensional
moment problem. Noncommutative moment problems are closely related to Hilbert
space representations. Properties of states on general
-algebras are important in
noncommutative probability theory. Dynamical systems appear in the representation
theory of operator relations. Noncomutative real algebraic geometry asks when ele-
ments, which are positive operators in certain representations, are sums of hermitian
viii Preface and Overview

squares, possibly with denominators. These topics will appear in this book; for most
of them we provide introductions to these subjects. Some of them are treated in great
detail, while others are only touched upon.
• Our aim is to present fundamental general concepts and their applications and
basic methods for constructing representations.
The GNS construction is a powerful tool that is useful to reformulate or to solve
problems by means of Hilbert space operators. We carry out this construction in
detail and apply it to the study of positive functionals on
-algebras. Further, we
develop general methods for the construction of classes of representations such as
induced representations, operator relations, and well-behaved representations.
Representations on rigged modules or Hilbert C
-modules is a new topic which
belongs to this list as well. Throughout, our main focus is on basic ideas, concepts,
examples, and results.
• For some selected topics self-contained and deeper presentations are given.
This concerns the representation theory of the Weyl algebra and the theory of
infinitesimal representations of enveloping algebras. Both topics are extensively
developed including a number of advanced and deep results. Also, Archimedean
quadratic modules and the corresponding C
-algebras are explored in detail.

Brief Description of the Contents

Chapter 1 should be considered as a prologue to this book. We give a brief and

informal introduction into the algebraic approach to quantum theories thereby
provided some physical motivation for the study of general
-representations and
states of
-algebras.
Chapter 2 deals with the algebraic structure of general involutive algebras. Basic
constructions (tensor products, crossed products, matrix algebras), examples
(semigroup
-algebras,
-algebras defined by relations), and concepts (characters,
positive functionals, quadratic modules) are introduced and investigated.
Chapter 3 gives a short digression into O
-algebras. These are
-algebras of
linear operators on an invariant dense domain of a Hilbert space. The involution is
the restriction of the Hilbert space adjoint to the domain. We treat three special
topics (graph topology, bounded commutants, and trace functionals) that are used
later in the study of representations.
With Chap. 4 we enter the main topic of this book:
-representations on Hilbert
space. We develop basic concepts (closed, biclosed, self-adjoint, essentially
self-adjoint representations), in analogy to single operator theory, and standard
notions on representations (invariant subspaces, irreducible representations). The
heart of this chapter is the GNS construction which associates a
-representation
with each positive functional. It is probably the most important and useful technical
tool in Hilbert space representation theory.
Preface and Overview ix

Chapter 5 is devoted to a detailed study of positive linear functionals on
-

algebras. The GNS representation allows one to explore the interplay between
properties of Hilbert space representations and positive functionals. Ordering,
orthogonality, transition probability, and a Radon–Nikodym theorem for positive
functionals are treated in this manner. Choquet’s theory is applied to obtain
extremal decompositions of states. Quadratic modules defined by representations
are introduced.
Chapters 6–9 are devoted to the representation theories of some important
special classes of
-algebras.
Chapter 6 deals with tensor algebras and free
-algebras. Positive functionals are
approximated by vector functionals of finite-dimensional representations and
faithful representations are constructed. We define topological tensor algebras such
as the field algebra of quantum field theory and develop continuous representations.
Chapter 7 is about “well-behaved” representations and states of commutative
-
algebras. We characterize these representations by a number of conditions and
express well-behaved representations of finitely generated
-algebras in terms of
spectral measures.
Chapters 8 and 9 are two core chapters that stand almost entirely by themselves.
Chapter 8 gives an extensive treatment of Hilbert space representations of the
canonical commutation relation (2) and the Weyl algebra. After collecting algebraic
properties of this algebra we treat the Bargmann–Fock representation and the
corresponding uniqueness theorem. Then the Schrödinger representation is studied
and the Stone–von Neumann uniqueness theorem is proved. The Bargmann
transform establishes the unitary equivalence of both representations. Kato’s the-
orem on the characterization of Schrödinger pairs in terms of resolvents is derived.
Further, the Heisenberg uncertainty principle and the Groenewold-van Hove
“no-go” theorem for quantization are developed in detail.
Chapter 9 is about infinitesimal representations of universal enveloping algebras
of finite-dimensional Lie algebras. Each unitary representation of a Lie group yields
a
-representation of the corresponding enveloping algebra. Basic properties
of these representations (C 1 -vectors, Gårding domains, graph topologies, essential
self-adjointness of symmetric elements) are studied in detail and elliptic regularity
theory is used to prove a number of advanced results.
Analytic vectors, first for single operators and then for representations, are
investigated. They play a crucial role for the integrability theorems of Lie algebra
representations due to Nelson and to Flato, Simon, Snellman, and Sternheimer.
These results are presented without proof, but with references. Finally, we discuss
K-finite vectors for unitary representations of SLð2; RÞ and the oscillator
representation.
Chapter 10 is concerned with Archimedean quadratic modules and the associ-
ated
-algebras of bounded elements. Two abstract Stellensätze give a glimpse into
noncommutative real algebraic geometry. As an application we derive a strict
Positivstellensatz for the Weyl algebra. Finally, a theorem about the closedness
of the cone of finite sums of hermitian squares in certain
-algebras is proved and
some applications are obtained.
x Preface and Overview

Chapter 11 examines the operator relation XX
¼ FðX
XÞ, where F is Borel

function on ½0; þ 1Þ and X is a densely defined closed operator on a Hilbert space.
The representation theory of this relation is closely linked to properties of the
dynamical system defined by the function F. For instance, finite-dimensional
irreducible representations correspond to cycles of the dynamical system. The
hermitian q-plane and the q-oscillator algebra are treated as important examples.
Chapter 12 presents an introduction to unbounded induced representations of

-algebras. For group graded
-algebras there exists a canonical conditional
expectation which allows one to define induced representations. We develop this
theory for representations which are induced from characters of commutative
subalgebras. The Bargmann–Fock representation of the Weyl algebra is obtained in
this manner.
An important topic of advanced Hilbert space representation theory is to describe
classes of “well-behaved” representations of general
-algebras. In Chap. 13 we
propose some general methods (group graded
-algebras, fraction algebras, com-
patible pairs) and apply them to the Weyl algebra and to enveloping algebras.
Chapter 14 provides a brief introduction to
-representations on rigged modules
and Hilbert C
-modules. This is a new subject of theoretical importance. A rigged
space is a right or left module equipped with an algebra-valued sesquilinear map-
ping which is compatible with the module action. First we explore
-representations
of
-algebras on rigged modules purely algebraically. If the riggings are positive
semi-definite (in particular, in the case of Hilbert C
-modules), induced represen-
tations on “ordinary” Hilbert spaces can be defined and imprimitivity bimodules
yield equivalences between
-representations of the corresponding
-algebras.

Guide to Instructors and Readers

Various courses and advanced seminars can be built on this book. All of them
should probably start with some basics on
-algebras (Sects. 2.1 and 2.2), positive
functionals and states (Sect. 2.4), and
-representations (Sect. 4.1).
One possibility is a graduate course on unbounded representation theory. The
basics should be followed with important notions and tools such as irreducibility
(Sect. 4.3), GNS representations (Sect. 4.4), and bounded commutants (Sects. 3.2
and 5.1). Then there are many ways to continue. One way is to treat representations
of special classes of
-algebras such as tensor algebras (Chap. 6), commutative
algebras (Chap. 7), or the Weyl algebra (Chap. 8). One may also continue with a
detailed study of states (with material taken from Chap. 5) or by developing general
methods such as induced representations (Chap. 12), operator relations (Chap. 11),
and fraction algebras (Sects. 13.2 and 13.3).
Another possible course for graduate students of mathematics and theoretical
physics is on representations of the canonical commutation relation and the Weyl
algebra. Such a course could be based entirely on Chap. 8. Here, after considering
some basics and algebraic properties of the Weyl algebra, the Bargmann–Fock and
Preface and Overview xi

Schrödinger representations, the Fock space, the Bargmann–Segal transform, the

Stone–von Neumann uniqueness theorem should be developed and continued until
the sections on the Heisenberg uncertainty principle and the Groenewald–von Hove
“no-go” theorem.
Chapter 9, which treats integrable representations of enveloping algebras, could
be used in general or advanced courses or as a reference for researchers. Material
from this chapter, for example, the “elementary” parts from Sect. 9.2 on
infinitesimal representations, C1 -vectors and Garding domains, can be integrated
into any general course on infinite-dimensional unitary representation theory of Lie
groups. More complex material such as elliptic elements or analytic vectors (see
e.g. Sects. 9.4 and 9.6) would suit an advanced course. Because Chap. 9 contains a
number of strong results on infinitesimal representations, their domains, and
commutation properties, it might be also useful as a reference for researchers.
Apart from basic concepts and facts, most chapters are more or less
self-contained and could be studied independently of each other. Special topics can
be easily included into courses, treated in seminars or read on their own. Examples
are the noncommutative Positivstellensätze (Chap. 10) or operator relations and
dynamical systems (Chap. 11).
Each chapter is followed by a number of exercises. They vary in difficulty and
serve for different purposes. Most of them are examples or counter-examples
illustrating the theory. Some are slight variations of results stated in the text, while
others contain additional new results or facts that are of interest in themselves.

Prerequisites

The main prerequisite for this book is a good working knowledge of unbounded
Hilbert space operators such as adjoint operators, symmetric operators, self-adjoint
operators, and the spectral theorem. The corresponding chapters of the author’s
Graduate Text [Sch12] contain more material than really needed. The reader should
be also familiar with elementary techniques of algebra, analysis, and bounded
operator algebras. Chapter 9 assumes a familiarity with the theory of Lie groups and
Lie algebras. In three appendices, we have collected some basics on unbounded
operators, C
-algebras and their representations, and locally convex spaces and
separation of convex sets. In addition, we have often restated facts and notions at
the places where they are most relevant.
For parts of the book or for single results, additional facts from other mathe-
matical fields are required, which emphasize the interplay with these fields. There
we have given links to the corresponding literature. In most cases these results are
not needed elsewhere in the book, so the unfamiliar reader may skip these places.

Leipzig, Germany Konrad Schmüdgen

March 2020
xii Preface and Overview

Acknowledgements I am grateful to Prof. J. Cimprič and Prof. V. L. Ostrovskyi for careful

reading of some chapters and for many useful comments. Also, I would like to thank Dr. R. Lodh
from Springer-Verlag for his indispensable help getting this book published.
Contents

1 Prologue: The Algebraic Approach to Quantum

Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2
-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1
-Algebras: Definitions and Examples . . . . . . . . . . . . . . . . . . 7
2.2 Constructions with
-Algebras . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Quadratic Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Positive Functionals and States on Complex
-Algebras . . . . . 20
2.5 Positive Functionals on Real
-Algebras . . . . . . . . . . . . . . . . . 23
2.6 Characters of Unital Algebras . . . . . . . . . . . . . . . . . . . . . . . . 25
2.7 Hermitian Characters of Unital
-Algebras . . . . . . . . . . . . . . . 28
2.8 Hermitian and Symmetric
-Algebras . . . . . . . . . . . . . . . . . . . 33
2.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3 O
-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1 O
-Algebras and Their Graph Topologies . . . . . . . . . . . . . . . . 39
3.2 Bounded Commutants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3 Trace Functionals on O
-Algebras . . . . . . . . . . . . . . . . . . . . . 49
3.4 The Mittag-Leffler Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4
-Representations . . . . . . . . . . . . . . . . . . . . . . . . . ............. 59
4.1 Basic Concepts on
-Representations . . . . . . ............. 59
4.2 Domains of Representations in Terms of
Generators . . . . . . . . . . . . . . . . . . . . . . . . . ............. 67
4.3 Invariant Subspaces and Reducing Subspaces ............. 73

xiii
xiv Contents

4.4 The GNS Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.5 Examples of GNS Representations . . . . . . . . . . . . . . . . . . . . . 82
4.6 Positive Semi-definite Functions on Groups . . . . . . . . . . . . . . 85
4.7 Pathologies with Unbounded Representations . . . . . . . . . . . . . 88
4.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5 Positive Linear Functionals . . . . . . . . . . . . . . . . . . . . . .......... 93
5.1 Ordering of Positive Functionals . . . . . . . . . . . . .......... 93
5.2 Orthogonal Positive Functionals . . . . . . . . . . . . . .......... 97
5.3 The Transition Probability of Positive
Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . .......... 99
5.4 Examples of Transition Probabilities . . . . . . . . . . . . . . . . . . . 104
5.5 A Radon–Nikodym Theorem for Positive
Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.6 Extremal Decomposition of Positive Functionals . . . . . . . . . . . 113
5.7 Quadratic Modules and
-Representations . . . . . . . . . . . . . . . . 117
5.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6 Representations of Tensor Algebras . . . . . . . . . . . . . . . . . . . . . . . . 121
6.1 Tensor Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.2 Positive Functionals on Tensor Algebras . . . . . . . . . . . . . . . . 123
6.3 Operations with Positive Functionals . . . . . . . . . . . . . . . . . . . 125
6.4 Representations of Free Field Type . . . . . . . . . . . . . . . . . . . . 127
6.5 Topological Tensor Algebras . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7 Integrable Representations of Commutative
-Algebras . . . . . . . . . 137
7.1 Some Auxiliary Operator-Theoretic Results . . . . . . . . . . . . . . 137
7.2 “Bad” Representations of the Polynomial Algebra C½x1 ; x2  . . . 139
7.3 Integrable Representations of Commutative
-Algebras . . . . . . 142
7.4 Spectral Measures of Integrable Representations . . . . . . . . . . . 148
7.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
8 The Weyl Algebra and the Canonical Commutation Relation . . . . . 153
8.1 The Weyl Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
8.2 The Operator Equation AA
¼ A
A þ I . . . . . . . . . . . . . . . . . . 157
8.3 The Bargmann–Fock Representation of the Weyl
Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
8.4 The Schrödinger Representation of the Weyl
Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
Contents xv

8.5 The Stone–von Neumann Theorem . . . . . . . . . . . . . . . . . . . . 169

8.6 A Resolvent Approach to Schrödinger Pairs . . . . . . . . . . . . . . 173
8.7 The Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
8.8 The Groenewold–van Hove Theorem . . . . . . . . . . . . . . . . . . . 178
8.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
8.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
9 Integrable Representations of Enveloping Algebras . . . . . . . . . . . . 187
9.1 Preliminaries on Lie Groups and Enveloping Algebras . . . . . . 188
9.2 Infinitesimal Representations of Unitary Representations . . . . . 189
9.3 The Graph Topology of the Infinitesimal Representation . . . . . 196
9.4 Elliptic Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
9.4.1 Preliminaries on Elliptic Operators . . . . . . . . . . . . . . . 199
9.4.2 Main Results on Elliptic Elements . . . . . . . . . . . . . . . . 204
9.4.3 Applications of Elliptic Elements . . . . . . . . . . . . . . . . 205
9.5 Two Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
9.6 Analytic Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
9.6.1 Analytic Vectors for Single Operators . . . . . . . . . . . . . 213
9.6.2 Analytic Vectors for Unitary Representations . . . . . . . . 215
9.6.3 Exponentiation of Representations of Enveloping
Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
9.7 Analytic Vectors and Unitary Representations of SLð2; RÞ . . . . 217
9.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
9.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
10 Archimedean Quadratic Modules and Positivstellensätze . . . . . . . . 225
10.1 Archimedean Quadratic Modules and Bounded Elements . . . . 225
10.2 Representations of
-Algebras with Archimedean
Quadratic Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
. .
10.3 Stellensätze for Archimedean Quadratic Modules . . . . . . . . 233
. .
10.4 Application to Matrix Algebras of Polynomials . . . . . . . . . . 235
. .
10.5 A Bounded
-Algebra Related to the Weyl Algebra . . . . . . 237
. .
10.6 A Positivstellensatz for the Weyl Algebra . . . . . . . . . . . . . . 241
. .
P 2
10.7 A Theorem About the Closedness of the Cone A ... . . . . 244
10.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
10.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
11 The Operator Relation XX
¼ FðX
XÞ . . . .......... . . . . . . . . . 251
11.1 A Prelude: Power Partial Isometries . .......... . . . . . . . . . 252
11.2 The Operator Relation AB ¼ BFðAÞ . .......... . . . . . . . . . 254
11.3 Strong Solutions of the Relation XX
¼ FðX
XÞ . . . . . . . . . . . 258
xvi Contents

11.4 Finite-Dimensional Representations . . . . . . . . . . . . . . . . . . . . 261

11.5 Infinite-Dimensional Representations . . . . . . . . . . . . . . . . . . . 265
11.6 The Hermitian Quantum Plane . . . . . . . . . . . . . . . . . . . . . . . . 270
11.7 The q-Oscillator Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
11.8 The Real Quantum Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
11.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
11.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
12 Induced
-Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
12.1 Conditional Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
12.2 Induced
-Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
12.3 Induced Representations of Group Graded
-Algebras
from Hermitian Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
12.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
12.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
13 Well-Behaved Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
13.1 Well-Behaved Representations of Some Group
Graded
-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
13.2 Representations Associated with
-Algebras of Fractions . . . . . 304
13.3 Application to the Weyl Algebra . . . . . . . . . . . . . . . . . . . . . . 310
13.4 Compatible Pairs of
-Algebras . . . . . . . . . . . . . . . . . . . . . . . 312
13.5 Application to Enveloping Algebras . . . . . . . . . . . . . . . . . . . . 314
13.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
13.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
14 Representations on Rigged Spaces and Hilbert C
-Modules . . . . . . 319
14.1 Rigged Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
14.2 Weak Imprimitivity Bimodules . . . . . . . . . . . . . . . . . . . . . . . 324
14.3 Positive Semi-definite Riggings . . . . . . . . . . . . . . . . . . . . . . . 328
14.4 Imprimitivity Bimodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
14.5 Hilbert C
-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
14.6 Representations on Hilbert C
-modules . . . . . . . . . . . . . . . . . 338
14.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344
14.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

Appendix A: Unbounded Operators on Hilbert Space . . . . . . . . . . . . . . . 347

Appendix B: C*-Algebras and Representations . . . . . . . . . . . . . . . . . . . . 353
Appendix C: Locally Convex Spaces and Separation
of Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
Symbol Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
General Notation

Throughout the book, we use the following notational conventions:

The involution of an abstract
-algebra is denoted by a 7! a þ .
The symbol a
is only used for the adjoint of a Hilbert space operator a.
The symbol K denotes either the real field R or the complex field C.
All algebras or vector spaces are either over R or C.
All inner products of complex inner product spaces or Hilbert spaces are linear
in the first and conjugate linear in the second variables.
Unless stated explicitly otherwise, all inner products and Hilbert spaces are over
the complex field.
We denote
– abstract
-algebras by sanserif letters such as A, B, F, M, W, X,
– unit elements of a unital K-algebras by 1 and write a  1 by a for a 2 K,
– O
-algebras by script letters such as A, B,
– Hilbert spaces by H, H0 , G, K,
– inner products by angle brackets h; i, h; i1
– dense domains or inner product spaces by D, DðTÞ,
– representations by …, …f , q,
– Hilbert space vectors by u, w, g, n.

N0 Set of nonnegative integers,

N Set of positive integers,
Z Set of integers,
R Set of real numbers,
Rþ Set of nonnegative real numbers,
C Set of complex numbers,
T Set of complex numbers of modulus one.
Cd ½x :¼ C½x1 ; . . .; xd , Rd ½x :¼ R½x1 ; . . .; xd .

xvii
xviii General Notation

For a Hilbert space H, we denote by

– BðHÞ the bounded operators on H,
– B1 ðHÞ the trace class operators on H,
– Tr t the trace of a trace class operator t,
– B1 ðHÞ þ the positive trace class operators on H,
– B2 ðHÞ the Hilbert-Schmidt operators on H.
For a
-algebra A we denote by
– A1 the unitization of A,
– Aher the hermitian part of A,
– P(A)* the positive linear functionals on A,
– P e (A)* the extendable positive linear functionals on A,
– S(A) the states of A,
– ^ the hermitian characters of A, if A is commutative and unital,
A
– 1 its unit element, if A is unital.

Cc ðX Þ Compactly supported continuous functions on a topological space X .

C0 ðX Þ Continuous functions on a locally compact space X that vanish at infinity.
L2 ðMÞ L2 -space with respect to the Lebesgue measure if M is a Borel set of Rd .
R
F Fourier transform F ðf ÞðxÞ ¼ ð2…Þd=2 Rd eiðx;yÞ f ðyÞdy.
Chapter 1
Prologue: The Algebraic Approach
to Quantum Theories

Let us begin by recalling some well-known concepts from quantum mechanics.

For details, the reader can consult one of the standard textbooks such as [SN17] or
[Ha13].
The mathematical formulation of quantum mechanics is based on a complex
Hilbert space H, which is called the state space. The two fundamental objects of a
quantum theory, observables and states, are described by the following postulates.
(QM1) Each observable is a self-adjoint operator on the Hilbert space H.
(QM2) Each pure state is given by the unit ray [ϕ] := {λϕ : λ ∈ T} of a unit vector
ϕ ∈ H.
In general, not all self-adjoint operators on H are physical observables and not all
unit vectors of H correspond to physical states. In the subsequent informal discussion
we will ignore this distinction and consider all unit rays as states and all bounded
self-adjoint operators on H as observables.
That each observable A is a self-adjoint operator by axiom (QM1) has impor-
tant consequences. Then the spectral theorem applies, and there exists a unique
projection-valued measure E A (·), called the spectral measure of A, on the Borel
σ-algebra of R such that

A= λ d E A (λ).
R

This spectral measure E A is a fundamental mathematical object in operator theory

and in quantum mechanics as well. All properties of the self-adjoint operator and
the observable A are encoded in E A . First we note that the support of the spectral
measure E A coincides with the spectrum of the operator A.

© The Editor(s) (if applicable) and The Author(s), under exclusive license 1
to Springer Nature Switzerland AG 2020
K. Schmüdgen, An Invitation to Unbounded Representations of *-Algebras
on Hilbert Space, Graduate Texts in Mathematics 285,
https://doi.org/10.1007/978-3-030-46366-3_1
2 1 Prologue: The Algebraic Approach to Quantum Theories

The probabilistic interpretation of quantum mechanics and the measurement the-

ory of observables are essentially based on spectral measures. To explain this, we
consider a unit vector ϕ ∈ H. It is clear that

μ[ϕ] (·) := E A (·)ϕ, ϕ

defines a probability measure μ[ϕ] on R which depends only on the unit ray [ϕ]. The
probabilistic interpretation says that μ[ϕ] (M) is the probability that the measurement
outcome of the observable A in the state [ϕ] lies in the Borel set M of R. Two
observables A1 and A2 are simultaneously measurable if and only if their spectral
measures E A1 and E A2 commute.
Now let ϕ be a unit vector of the domain of A. Then the number
 
Aϕ, ϕ = λ dμ[ϕ] (λ) = λ dE A (λ)ϕ, ϕ
R R

is interpreted as the expectation value and (Δ[ϕ] A)2 := Aϕ2 − Aϕ, ϕ2 as the
variance of the observable A in the state [ϕ]. Finally, the spectral
 measure allows
one to define a function F(A) of an observable A by F(A) = F(λ)d E A (λ) for
any Borel function F on the spectrum of A.
Let [ϕ] and [ψ] be states of H. Then the number

P([ϕ], [ψ]) := |ϕ, ψ|2

depends only on the unit rays, and it is called the transition probability between the
states [ϕ] and [ψ].
A symmetry of the quantum system is a bijection of the set of states [ϕ] which
preserves the transition probabilities between states. By Wigner’s theorem (see, e.g.,
[Em72]), each symmetry θ is implemented by a unitary or an antiunitary operator
U of the Hilbert space H, that is, θ([ϕ]) = U [ϕ]U −1 for all states [ϕ]. (An anti-
unitary operator is an operator U on H such that U (αϕ + βψ) = α U ϕ + β U ψ and
U ϕ, U ψ = ϕ, ψ for ϕ, ψ ∈ H and α, β ∈ C.)
Let U be a unitary or an antiunitary operator on H. If A is an observable, then
the operator θ(A) := U AU −1 is self-adjoint and hence an observable. For arbitrary
A ∈ B(H), we set θ0 (A) = U AU −1 if U is unitary and θ1 (A) = U A∗ U −1 if U is
antiunitary. For self-adjoint operators A, both θ0 (A) and θ1 (A) coincide with θ(A).
Then, θ0 is a ∗-automorphism and θ1 is a ∗-antiautomorphism of the C ∗ -algebra
B(H) of bounded operators on H.
There are also mixed states and states given by density matrices. Assume for a
moment that the observables are bounded operators. Then each positive trace class
operator t on H of trace one defines also a state. The corresponding probability
measure is μt (·) := Tr t E A (·), and the expectation value is Tr t A.
That was the classic approach to quantum mechanics. Let us explain now the
algebraic approach, in which the main objects of study of this book appear.
1 Prologue: The Algebraic Approach to Quantum Theories 3

Here the observable algebra is the central object of the theory. This is an abstract
complex unital algebra A equipped with an algebra involution a → a + , that is, A is
a complex unital ∗-algebra. The key postulates in this approach are the following:
(A1) Each observable is a hermitian element a = a + of the ∗-algebra A.
(A2) Each state is a linear functional f on A such that f (a + a) ≥ 0 for a ∈ A and
f (1) = 1.
If f is a state and a is an observable of A, then the real number f (a) is considered
as the expectation value and the nonnegative number Δ f (a)2 := f (a 2 ) − f (a)2 as
the variance of a in the state f .
Let us motivate this definition of a state. Elements of the form a + a are always
hermitian, and they should be positive, because Hilbert space operators of the form
A∗ A are positive. Then the condition f (a + a) ≥ 0 says that the expectation value of
the “positive” observable a + a is nonnegative. A functional f with this property is
called positive. The requirement f (1) = 1 is a normalization condition for the trivial
observable 1 ∈ A.
Since A is a ∗-algebra, one can form algebraic operations (linear combinations,
products, adjoints) of elements of A. It is easily verified that the product of two
hermitian elements is hermitian if and only if the elements commute. Hence the
product of two observables can be only an observable if they commute in the
algebra A.
To remedy this failure it is convenient to consider the Jordan product

1
a ◦ b := (ab + ba)
2

of elements a, b ∈ A. Obviously, if the elements a and b are hermitian, so is a ◦ b.

Clearly, a ◦ b = 21 ((a + b)2 − a 2 − b2 ). Therefore, if we agree that real linear com-
binations and squares of observables are also observables, then the Jordan product
a ◦ b of observables a, b ∈ A is again an observable. Note that the Jordan product
“◦” is distributive and commutative, but it is not associative in general.
Before we continue our discussion we introduce a few more mathematical notions.
Let θ be a linear map of A into another ∗-algebra B such that θ(a + ) = θ(a)+ for
a ∈ A. Then θ is called a ∗-antihomomorphism if θ(ab) = θ(b)θ(a) for a, b ∈ A
and a Jordan homomorphism if θ(a ◦ b) = θ(a) ◦ θ(b) for a, b ∈ A. In this case, if
A = B and θ is bijective, then θ is said to be a ∗-antiautomorphism and a Jordan auto-
morphism of A, respectively. Clearly, ∗-homomorphisms and ∗-antihomomorphisms
are Jordan homomorphisms.
Roughly speaking, a symmetry of a physical system should be a bijection that
preserves the main structures of the system. In the case of pure states on a Hilbert
space, the transition probability of states was chosen as the relevant concept. In
the algebraic approach, it is natural to require that symmetries preserve the Jordan
product. Thus, we define a symmetry to be a Jordan automorphism of the ∗-algebra
A. Then any symmetry θ preserves observables, and the map f → f ◦ θ preserves
4 1 Prologue: The Algebraic Approach to Quantum Theories

states. Various symmetry concepts for C ∗ -algebras are treated and discussed in [Ln17,
Chap. 5], [Em72, Sect. 2.2.a], [Mo13, Sect. 12.1], and [K65].
In particular, ∗-automorphisms and ∗-antiautomorphisms of A are symmetries.
We say that a group G acts as a symmetry group on the observable algebra A if we
have a homomorphism g → θg of G into the group of ∗-automorphisms of A.
We collect the main concepts introduced so far in the following table:

Quantum mechanics Algebraic approach

State Hilbert space H Observable algebra A
Observable Self-adjoint operator on H Hermitian element of A
State Unit ray [ϕ] of ϕ ∈ H, ϕ = 1 Positive functional f with f (1) = 1
Symmetry Unitary or antiunitary operator on H Jordan automorphism of A

It should be emphasized that for the study of quantum theories usually specific
sets of further axioms and topics are added. Important examples are the Gårding–
Wightman axioms and the Haag–Kastler axioms in algebraic quantum field theory
[Hg55] and the KMS states in quantum statistical mechanics [BR97].
Next we discuss the role of representations of the observable algebra. To avoid
technical difficulties, let us assume throughout the following discussion that the
observable algebra A is a unital C ∗ -algebra. Recall that a ∗-representation of A is
a ∗-homomorphism ρ of A into the ∗-algebra B(H) of bounded operators of some
Hilbert space H. Then the image of each abstract observable a ∈ A is a bounded
self-adjoint operator ρ(a), hence an observable on the Hilbert space H, and each unit
vector ϕ ∈ H defines a state f ρ,ϕ (·) := ρ(·)ϕ, ϕ on A. These states f ρ,ϕ are called
the vector states of the representation ρ. Conversely, if f is a state on A, then the
GNS construction provides a ∗-representation ρ f of A on a Hilbert space H such
that f (·) = ρ f (·)ϕ f , ϕ f  for some unit vector ϕ f ∈ H. Thus, the abstract state f
on A gives a concrete state [ϕ f ] on the Hilbert space H.
Further, two ∗-representations of A are physically equivalent if and only if each
vector state of one is a weak limit of convex combinations of vector states of the
other, or equivalently, if the kernels of both representations coincide [Em72, The-
orem II.1.7]. It is obvious that unitarily equivalent representations are physically
equivalent, but the converse is not true.
Let us turn to symmetries. Suppose ρ is a ∗-representation of A on a Hilbert space
H. A ∗-automorphism θ of A is called unitarily implemented in the representation ρ
if there exists a unitary operator U on H such that ρ(θ(a)) = U ρ(a)U −1 for a ∈ A.
Likewise, an action g → θg of a group G on A is said to be unitarily implemented
in the representation ρ if there is a homomorphism g → U (g) of G into the group
of unitaries on H, called then a unitary representation of G on H, such that

ρ(θg (a)) = U (g)ρ(a)U (g −1 ) for a ∈ A, g ∈ G. (1.1)

It can be shown that (1.1) holds, for instance, for the GNS representation associated
with any state which is invariant under θg . In important cases, G is a Lie group; then
appropriate continuity assumptions on θg and U (g) have to be added.
1 Prologue: The Algebraic Approach to Quantum Theories 5

According to a result of Kadison [K65], [BR97, Proposition 3.2.2], any Jordan

homomorphism into B(H) can be decomposed into a sum of a ∗-homomorphism and
a ∗-antihomomorphism. More precisely, if ρ : A → B is a Jordan homomorphism of
A on a C ∗ -subalgebra B of B(H), then there is a projection P ∈ B ∩ B such that
a → ρ(a)P is a ∗-homomorphism and a → ρ(a)(I − P) is a ∗-antihomomorphism
of A into B(H). Here B and B denote the commutant and bicommutant of B,
respectively. In particular, if the von Neumann algebra B is a factor, then P = 0 or
P = I , so ρ is a ∗-homomorphism or a ∗-antihomomorphism.
Any ∗-representation ρ of the observable algebra allows one to pass from the fixed
abstract observable algebra A to the observable algebra ρ(A) of operators acting on
a Hilbert space. There the power of operator theory on Hilbert spaces can be used
to study the quantum system. The flexibility of choosing the ∗-representation has a
number of advantages. First, various realizations of unitarily equivalent representa-
tions may provide new methods and structural insight. For instance, the Schrödinger
representation and the Bargmann–Fock representation of the Weyl algebra are unitar-
ily equivalent, but their realizations on L 2 (Rd ) and on the Fock space, respectively,
lead to different approaches for the study of the canonical commutation relations.
Second, unitarily or physically inequivalent realizations of quantum systems can be
treated by means of the same abstract observable algebra. Here the canonical commu-
tation relations for infinitely many degrees of freedom form an interesting example.
There exist unitarily inequivalent irreducible representations which are physically
equivalent [BR97, Em72]. Third, let g → θg be an action of a Lie group G as ∗-
automorphisms of A. In “good” cases there exists a ∗-representation ρ of A such that
this action is implemented by a unitary representation g → U (g) of G, as in formula
(1.1). Then the representation theory of Lie groups on Hilbert space can be used to
study the ∗-automorphism group.
The preceding was a brief sketch of some basic general concepts and ideas of
quantum mechanics and the algebraic approach to quantum theories.
In the case of general ∗-algebras a number of additional technical problems appear
in the study of ∗-representations and states. For instance, it may happen that the image
of a hermitian element under a ∗-representation has no self-adjoint extension, so it
cannot be considered as an observable on the representation Hilbert space. An aim
of this book is to lay down a rigorous mathematical foundation of the theory of
representations and states of general ∗-algebras.
The pioneering work for the algebraic approach goes back to Neumann [vN32], Segal [Se47a],
and others. Modern treatments of this approach and various sets of axioms can be found in the
books of Emch [Em72], Moretti [Mo13] and Landsman [Ln17]; see also [K65]. Standard references
are [Hg92, Ak09] for algebraic quantum field theory and [BR87, BR97] for quantum statistical
mechanics.
Chapter 2
∗-Algebras

The aim of this chapter is to develop algebraic properties and structures of ∗-algebras
and of positive functionals and states. Also, we introduce a number of basic concepts,
notations, and facts that will be used later in this book.
Section 2.1 contains basic definitions and examples of ∗-algebras. In Sect. 2.2,
we treat some general constructions of ∗-algebras (tensor products, matrix algebras,
crossed products, group graded algebras). Positivity in ∗-algebras is expressed in
terms of quadratic modules; they are introduced in Sect. 2.3 and studied later in
Sect. 5.7 and Chap. 10.
Sections 2.4–2.8 deal with positive linear functionals. In Sect. 2.4, we develop
basic facts on positive functionals and states on complex ∗-algebras. Positive func-
tionals on real ∗-algebras are briefly considered in Sect. 2.5. In Sect. 2.6 we study
characters of general algebras and prove the Gleason–Kahane–Zelazko characteri-
zation of characters (Theorem 2.56). Section 2.7 is about hermitian characters and
pure states of commutative ∗-algebras (Theorem 2.63). In Sect. 2.8, we give a short
digression into hermitian and symmetric ∗-algebras.
Throughout this chapter, A is an algebra over the field K, where K is R or C.

2.1 ∗-Algebras: Definitions and Examples

The following definitions introduce the first main notions which this book is about.

Definition 2.1 An algebra over K is a vector space A over K, equipped with a

mapping (a, b) → ab of A × A into A, such that for a, b, c ∈ A and α ∈ K:

a(bc) = (ab)c, (αa)b = α(ab) = a(αb), a(b + c) = ab + ac, (b + c)a = ba + ca.

© The Editor(s) (if applicable) and The Author(s), under exclusive license 7
to Springer Nature Switzerland AG 2020
K. Schmüdgen, An Invitation to Unbounded Representations of *-Algebras
on Hilbert Space, Graduate Texts in Mathematics 285,
https://doi.org/10.1007/978-3-030-46366-3_2
8 2 ∗-Algebras

The element ab is called the product of a and b; we also write a · b for ab.
An algebra A is called unital if it has a unit element 1 ∈ A, that is, 1a = a1 = a
for all a ∈ A. An algebra A is commutative if ab = ba for a, b ∈ A.
Definition 2.2 An algebra involution, briefly an involution, of an algebra A over K
is a mapping a → a + from A into A such that for a, b ∈ A and α, β ∈ K:

(αa + βb)+ = α a + + β b+ , (ab)+ = b+ a + , (a + )+ = a. (2.1)

An algebra (over K) equipped with an involution is called a ∗-algebra (over K).

Example 2.3 Let d ∈ N. The polynomial algebra Kd [x] := K[x1 , . . . , xd ] is a unital
∗-algebra with involution defined by
 
f + (x) := a α x α for f (x) = aα x α ∈ Kd [x],
α α

where we set x α := x1α1 · · · xdαd for α = (α1 , . . . , αd ) ∈ Nd0 and x 0j := 1. Note that
the involution on Rd [x] is just the identity mapping. 
Let A be a ∗-algebra over K. It is easily verified that if A has a unit element 1 and
a ∈ A is invertible in A, then 1+ = 1 and (a −1 )+ = (a + )−1 .
Definition 2.4 An element a ∈ A is called hermitian if a = a + and skew-hermitian
if a + = −a.
The hermitian part Aher and the skew-hermitian part Asher of A are

Aher := {a ∈ A : a + = a}, Asher := {a ∈ A : a + = −a}. (2.2)

Clearly, both parts are real vector spaces, Aher is invariant under the Jordan product
a ◦ b := 21 (ab + ba), and Asher is invariant under the commutator [a, b] := ab − ba.
Further, A = Aher + Asher and each a ∈ A can be uniquely written as

a = ah + ash , where ah ∈ Aher , ash ∈ Asher . (2.3)

Indeed, for ah := 21 (a + + a) and ash := 21 (a − a + ) we have (2.3). Conversely, if

ãh ∈ Aher and ãsh ∈ Asher satisfy a = ãh + ãsh , then a + = ãh − ãsh and hence ãh = ah
and ãsh = ash .
Now suppose K = C. Then, obviously, Asher = i Aher , so that A = Aher + iAher .
Therefore, by (2.3), each element a ∈ A can be uniquely represented in the form

a = a1 + ia2 , where a1 , a2 ∈ Aher , (2.4)

and we have a1 = Re a := 21 (a + + a) and a2 = Im a := 2i (a + − a).

If A is a commutative real algebra, the identity map is obviously an involution.
There exist algebras A which admit no algebra involution and others which have
infinitely many involutions making A into a ∗-algebra; see, e.g., [CV59].
2.1 ∗-Algebras: Definitions and Examples 9

Example 2.5 (An algebra which has no involution)

Let A be the K-algebra of 2 × 2 matrices (akl )2k,l=1 , with akl ∈ K, a21 = a22 = 0.
Clearly, the algebra A is isomorphic to the vector space K2 with multiplication

(x1 , x2 )(y1 , y2 ) = (x1 y1 , x1 y2 ). (2.5)

The algebra A has no involution such that A becomes a ∗-algebra.

Indeed, assume to the contrary that a → a + is an algebra involution of A. Set
x := (1, 0) and y := (0, 1). We have x 2 = x, y 2 = 0, x y = y, yx = 0 by (2.5).
Then (x + )2 = x + and (y + )2 = 0. By (2.5), these equations imply x + = (1, x2 ) and
y + = (0, y2 ). Then 0 = (yx)+ = x + y + = (1, x2 )(0, y2 ) = (0, y2 ) = y + and hence
0 = (y + )+ = y, a contradiction. 
We develop different involutions in Example 2.15 below using the next lemma.
Lemma 2.6 Suppose A is an algebra. If ϕ : a → a + is an algebra involution and
θ is an algebra automorphism of A such that

(θ ◦ ϕ) ◦ (θ ◦ ϕ) = Id, that is, θ(θ(a + )+ ) = a for a ∈ A, (2.6)

then ψ := θ ◦ ϕ is also an algebra involution of A.

Conversely, if ϕ and ψ are algebra involutions of A, then θ := ψ ◦ ϕ is an auto-
morphism of the algebra A such that ψ = θ ◦ ϕ and condition (2.6) holds.
The proof of this lemma is given by simple algebraic manipulations based on
(2.1). Equation (2.6) is equivalent to the last condition in (2.1). We omit the details;
see Exercise 1.
Next let us introduce some standard notions.
A map θ of a ∗-algebra A into another ∗-algebra B is called a ∗-homomorphism if θ
is an algebra homomorphism such that θ(a + ) = θ(a)+ for a ∈ A. A ∗-isomorphism
is a bijective ∗-homomorphism of A and B; in this case, A and B are said to be
∗-isomorphic. A ∗-automorphism of A is a ∗-isomorphism of A on itself. A ∗-ideal
of A is a two-sided ideal of A which is invariant under the involution.
Next we consider two useful general constructions.
Unitization of a ∗-algebra
For many considerations it is necessary that the ∗-algebra possesses a unit element.
If a ∗-algebra has no unit, it can be embedded into a unital ∗-algebra by adjoining a
unit. Let A be a ∗-algebra. It is easy to check that the K-vector space B := A ⊕ K is
a unital ∗-algebra with multiplication and involution defined by

(a, α)(b, β) := (ab + αb + βa, αβ) and (a, α)+ := (a + , α) (2.7)

for a, b ∈ A and α, β ∈ K. Obviously, 1 := (0, 1) is the unit element of B. By iden-

tifying a and (a, 0), the ∗-algebra A becomes a ∗-subalgebra of B. For notational
simplicity we write a + α instead of (a, α). Note that if A has a unit element, this
element is no longer a unit element of the larger ∗-algebra B.
10 2 ∗-Algebras

If A is not unital, we denote the unital ∗-algebra B = A ⊕ K by A1 . If A is unital,

we set A1 := A.
Definition 2.7 The unital ∗-algebra A1 is called the unitization of the ∗-algebra A.
For real ∗-algebras we may have Lin Aher = A, as the following example shows.
Example 2.8 On the vector space A := R we define a product by x · y := 0 and an
involution by x + := −x . Then A is a real ∗-algebra and Aher = {0} = A. For the
unitization A1 we have (A1 )her = {(0, α) : α ∈ R} by (2.7). Hence the linear span of
(A1 )her is different from A1 . 
Complexification of a real ∗-algebra

Suppose A is a real ∗-algebra. Let AC be the Cartesian product A × A. It is not

difficult to verify that AC becomes a complex ∗-algebra with addition, multiplication
by complex scalars, multiplication, and involution defined by

(a, b) + (c, d) = (a + c, b + d), (α + iβ)(a, b) = (αa − βb, αb + βa),

(a, b)(c, d) = (ac − bd, bc + ad), (a, b)+ := (a + , −b+ ),

where a, b, c, d ∈ A and α, β ∈ R. The map a → (a, 0) is a ∗-isomorphism of A on

a real ∗-subalgebra of AC . We identify a ∈ A with (a, 0) ∈ AC . Then A becomes a
real ∗-subalgebra of AC , and we have (a, b) = a + ib for a, b ∈ A.
Definition 2.9 The complex ∗-algebra AC is called the complexification of the real
∗-algebra A.
We define θ(a + ib) = a − ib for a, b ∈ A. Then we have

θ(α a + β b) = α θ(a) + β θ(b), θ(x + ) = θ(x)+ , (2.8)

θ(x y) = θ(x)θ(y), (θ ◦ θ)(x) = x (2.9)

for α, β ∈ C, a, b ∈ A, x, y ∈ AC , and A = {x ∈ AC : θ(x) = x}. Conversely, if B

is a complex ∗-algebra and θ : B → B is a map satisfying (2.8) and (2.9), then
A := {x ∈ B : θ(x) = x} is a real ∗-algebra and B is the complexification of A.
Now let A be a commutative real algebra. Then A is a real ∗-algebra with the
identity map as involution and we have (a + ib)+ = a − ib in AC , where a, b ∈ A.
Hence A is the hermitian part (AC )her of its complexification AC . For instance, if
A = R[x1 , . . . , xd ], we obtain AC = C[x1 , . . . , xd ].
Now we turn to examples of ∗-algebras. Large classes of examples of ∗-algebras
are defined by means of generators and defining relations.
1. ∗-Algebras defined by relations
Let K x1 , . . . , xm denote the free unital K-algebra with generators x1 , . . . , xm . The
elements of this algebra can be considered as noncommutative polynomials f in
x1 , . . . , xm ; for instance, f (x1 , x2 ) = 5x1 x27 x13 − 3x1 x2 + x2 x1 + 1.
2.1 ∗-Algebras: Definitions and Examples 11

Let n + k ∈ N, where k, n ∈ N0 . The algebra K x1 , . . . , xn , y1 , . . . , y2k has an

involution determined by (x j )+ = x j for j = 1, . . . , n and (yl )+ = yl+k for l =
1, . . . , k; the corresponding ∗-algebra is denoted by

K x1 , . . . , xn , y1 , . . . , y2k | (x j )+ = x j , j = 1, . . . , n; (yl )+ = yl+k , l = 1, . . . , k .

(2.10)

(If n = 0 or k = 0, we interpret (2.10) by omitting the corresponding variables.)

Now let f 1 , g1 . . . , fr , gr be elements of the ∗-algebra (2.10) and let J be the
∗-ideal of this ∗-algebra generated by the elements f 1 − g1 , . . . , fr − gr . We write

K x1 , . . . , xn , y1 , . . . , y2k | (x j )+ = x j , j = 1, . . . , n; (yl )+ = yl+k , l = 1, . . . , k;

f 1 = g1 , . . . , fr = gr (2.11)

for the quotient ∗-algebra of (2.10) by the ∗-ideal J. Thus, (2.11) is the unital ∗-algebra
with generators (x1 )+ = x1 , . . . , (xn )+ = xn , (y1 )+ = yk+1 , . . . , (yk )+ = y2k and
defining relations f 1 = g1 , . . . , fr = gr .

Example 2.10 (Weyl algebra W(d))

For d ∈ N, the d-dimensional Weyl algebra W(d) is the complex unital ∗-algebra

W(d) := C p1 , . . . , pd , q1 , . . . , qd | ( pk )+ = pk , (qk )+ = qk , pk qk − qk pk = −i;

p j pl = pl p j , q j ql = ql q j , p j ql = ql p j , k, j, l = 1, . . . , d, j = l ,

where i is the complex unit. The one-dimensional Weyl algebra or CCR-algebra is

W := C p, q | p + = p, q + = q, pq − qp = −i . (2.12)

For elements p, q of a complex unital algebra, a := √1 (q

2
+ i p),
+ + +
a := √1 (q
2
− i p) satisfy aa − a a = 1 if and only if pq − qp = −i. From this
fact it follows that the map √12 (q + i p) → a extends to a ∗-isomorphism of W on
the ∗-algebra C a, b | a + = b, ab − ba = 1 . We shall write this ∗-algebra as

C a, a + | aa + − a + a = 1 . (2.13)

Thus, (2.12) and (2.13) are ∗-isomorphic versions of the Weyl algebra; see Sect. 8.1.
Chapter 8 is devoted to the study of representations of the Weyl algebra. 

As angle brackets · denote free algebras, squared brackets [ · ] always refer

to commutative polynomial algebras. In particular, Cd [x] := C[x1 , . . . , xd ] and
Rd [x] := R[x1 , . . . , xd ] are commutative ∗-algebras of polynomials with involu-
tion (x j )+ = x j , j = 1, . . . , d. Commutative algebras with relations are defined
similarly as above and are self-explanatory. For instance, C[x, y | x + x + y + y = 1]
denotes the commutative ∗-algebra of polynomials in x, x + , y, y + satisfying the
equation x + x + y + y = 1 of the unit sphere in C2 .
 Another Random Document on
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Beyer, Dr.: Letter to, II, 259.

Bigot, Librarian of Count Rasoumowsky: II, 73, 125, 146;

his wife, Marie, II, 84, 146.

Bihler, J. N.: III, 156.

Biographers of B.: Early, III, 197 et seq.

“Biographische Notizen.” (See “Wegeler, F. G.” and “Ries,

Ferdinand.”)

Birchall, Robert, English publisher: II, 319, 324, 325;

difficulty in getting a receipt from B., 355, et seq., 345, 350, 346,
359;
death of, 351;
and the overtures bought by the Philharmonic Society, 337.

Birkenstock, Joseph Melchior: II, 178;

Antonie, II, 179.

Blahetka, Leopoldine: III, 50, 138, 157.

Bland, Mrs., English singer: II, 310.

“Blendwerk, Das” (“La fausse Magie”): Opera by Grétry, I, 107.

Blöchlinger, Joseph Karl: III, 7, 23. (See Guardianship under

“Beethoven, Ludwig van,” and “Beethoven, Karl.”)

Boer, S. M. de: Visits B., III, 203.

Bohemian Nobility: Musical culture of the, I, 168.

Böhm, Violinist: Plays the Quartet Op. 127, III, 192, 193;
 torchbearer at B’s funeral, III, 312.

Böhm’s Theatrical Company: I, 86.

Bolla, Signora: B. plays at her concert, I, 191.

Bonaparte, Jerome: I, 190; II, 122;

invites B. to his court, 124, 135 et seq.

Bonaparte, Louis, King of Holland: II, 245, 247.

Bonaparte, Napoleon: Threatens invasion of Vienna, I, 199, 200;

the “Eroica,” 213; II, 24;
B’s remark: “I would conquer him!” 117;
neglects opportunity to hear the “Eroica,” 149;
marches on Moscow, 221;
holds court at Dresden, 221;
effect of his downfall, 295;
and Cherubini, III, 206.

Bonn, City of: Festival in 1838, I, xvii;

selected as Electoral residence, 3;
besieged by Marlborough, 6;
restored to archbishopric of Cologne, 6;
improved by Elector Clemens August, 7;
the Comedy House, 30;
professional and amateur musicians in B’s time, 31;
appearance of the city, 38 et seq.;
Beethovens in before the arrival of the composer’s grandfather,
44;
music in Max Franz’s reign, 88;
theatrical companies, 112;
B’s friends, 117, 125, 126;
B. leaves the city forever, 125;
B’s compositions in, 129 et seq.;
Beethoven Festival of 1845, II, 177.
Boosey, Music publisher in London: III, 111, 128;
makes contract with B. through Ries, 128.

Born, Baroness: III, 42.

Boston Handel and Haydn Society: Commissions B. to write an

oratorio, III, 87.

Botticelli, Singer: III, 169.

Bouilly, J. N.: His opera-texts, II, 36.

Bowater, Mrs.: I, 134, 145;

III, 40.

“Bradamante”: Opera-book by Collin, II, 19.

Brahms, Johannes: Confirms authenticity of Bonn cantatas, I,

131;
comment on the compositions of royal personages, III, 20.

Brauchle: Tutor of Count Erdödy’s children, I, 320;

II, 317.

Braun, Baron: I, 168;

invites the Rombergs to give a concert, 199, 244, 290, 348, 350;
engages Ries at B’s solicitation, 360;
engages Cherubini to compose operas, II, 3;
dismisses Schikaneder, 23, 34, 35;
withdraws “Fidelio,” 63;
ends his management of the Theater-an-der-Wien, 78.
—Baroness, I, 225, 244;
dedication of the Horn Sonata, 290.

Braunhofer, Dr.: Dedication of the “Abendlied,” III, 50, 199;

 canon for, 200, 219, 373;
declines call to B., 272, 274.

Breimann: II, 125.

Breitkopf and Härtel: Acquire publication rights of this biography,

I, xv;
employ Dr. Riemann to revise German edition, xv;
B’s letters to, I, 286, 294, 348, 349, 364, 369;
II, 66, 67, 142, 148, 192, 198, 200, 204, 206;
B. offers them all his works, 67;
attempt to renew association with B., III, 73.

Brentano, Antonie: II, 322;

III, 128.

Brentano, Clemens: II, 196, 222.

Brentano, Elizabeth: (See Arnim, Bettina von)

Brentano, Franz: II, 179, 186;

B. borrows money from, III, 39, 45, 46, 47;
loan repaid, 64, 184.

Brentano, Maximiliane: II, 179, 180, 221.

Brentano, Sophie: II, 179.

Breuning, Christoph von: I, 98, 99;

Christoph (son), I, 198, 303.

Breuning, Eleonore Brigitte von: Wife of Franz Gerhard Wegeler,

I, 99, 118, 119;
lines on B’s birthday, 122;
inscription in his album, 125, 138, 300;
III, 214, 288.
Breuning, Emmanuel Joseph von: I, 98.

Breuning, Georg Joseph von: I, 98.

Breuning, Dr. Gerhard: Visited by Thayer, I, xi 96, 99, 100, 300;

opinion of B’s brother Karl, II, 322, 362;
description of Johann van B., III, 66;
B’s interest in him as a lad, III, 214;
on B’s last illness, 247;
on the medical treatment of B., 287, 300.

Breuning, Johann Lorenz von: I, 98.

Breuning, Johann Philipp von: I, 98.

Breuning, Johann Lorenz (Lenz) von: I, 99, 119, 198;

reports to Romberg about B., 199;
B’s lines in his album, 201, 202.

Breuning, Madame von: I, 99;

selects B. as teacher for her children, 100;
influence over B., 100, 188, 119, 303;
death of, 100;
dedication of the pianoforte arrangement of the Violin Concerto,
II, 134.

Breuning, Marie von: III, 213.

Breuning, Stephen von: I, 99;

intimacy with B., 119, 191, 198;
returns to Vienna, 288, 301;
B. advises his employment by the Teutonic Order, 303;
his relations with B. in Vienna, 310 et seq.;
B’s injustice toward him, 311.
—II, becomes clerk in Austrian war department, 14;
 quarrels with B., 27 et seq.;
reconciliation, 32;
receives miniature from B., 33;
poem for the second performance of “Fidelio,” 61;
letter concerning the opera, 57;
B’s concern for his health, 155;
death of his wife, 155;
dedication of the Violin Concerto, 162;
warns B. against his brother Karl, 322.
—III, 24, 197;
intimacy with B resumed, 213;
persuades B. to resign guardianship, 264;
objects to unqualified bequest to Nephew Karl, 279;
finds B’s bank stock and the love-letter, 376.

Bridgetower, George Augustus Polgreen, Violinist: I, 186;

his career, II, 8 et seq.;
his notes on the “Kreutzer Sonata,” 10.

Bridi, Joseph Anton: II, 391.

British Museum: Sketches in I, 205, 206, 209, 210, 261.

Broadwood, Thomas:
Presents pianoforte to B., II, 390 et seq.; III, 201, 237.

Browne, Count: I, 199;

B. calls him his “first Mæcenas,” 222, 244; II, 20.
—Countess, I, 200, 209, 227, 244.

Brühl, Count: III, 153.

Brunswick, Count Franz: I, 322;

the Rasoumowsky quartets, II, 104;
letters from B., 105, 124, 202, 219, 245, 266;
ruined by theatrical management, 154;
 dedication of Fantasia Op. 77, 195; III, 24, 170;
offers summer sojourn in Hungary to B., 179.

Brunswick, Count Géza: I, 340, 341.

Brunswick, Countess Marie: I, 340.

Brunswick, Countess Therese: I, xvi, 279;

her relations with B., 317, 322, 335 et seq.;
sends her portrait to B., 335;
B’s message to her brother, “Kiss your sister Therese,” II, 105,
161, 173;
dedication of the Sonata Op. 78, 195;
portrait of, 202;
letter to, 203, 239.

Bryant, William Cullen: Quoted I, 252.

Buda-Pesth: National Museum of, gets B’s Broadwood pianoforte,

II, 392.

“Buona Figliuola, La”: Opera by Piccini, I, 25, 32.

Burbure, Léon: Supplies information concerning the Belgian

Beethovens, I, 42.

Bureau d’Arts et Industrie: Established, II, 35.

Burney, Dr. Charles: “Present State, etc.,” quoted, I, 174.

Cache, Singer at first performance of “Fidelio”, II, 51.

Cäcilien-Verein of Frankfort: Subscription to the Mass in D, III,

104, 106, 110, 180.
“Calamità di Cuori, La”: Opera by Galuppi, I, 26.

Caldara: Opera “Gioas, Re di Giuda,” I, 184.

Campbell, Thomas: “The Battle of the Baltic,” II, 203.

Capponi, Marchese: I, 341.

Carlyle, Thomas: II, 360.

Carpani:
Italian text for Haydn’s “Creation,” II, 116;
introduces Rossini to B., 360.

Carriere, Moriz: Dubious of the genuineness of B’s letters to

Bettina von Arnim, II, 185.

Cassel: B. invited to become chapelmaster at, II, 122, 124, 135 et

seq., 141.

Cassentini, Dancer: I, 285.

Castelli:
On failure of the Concerto in E-flat, II, 215;
torchbearer and poet at B’s funeral, III, 312.

Castlereagh, Viscount: II, 291.

Catalani: II, 310.

Catalogue, Classified, of B’s works: II, 38.

Catherine II, Empress of Russia: II, 81.

Champein: I, 86.
Channing: B. asks for full report of speech on his death-bed, III,
283.

Chantavoine, Jean: I, 211, 228, 337.

Chappell, Music publisher in London: II, 413.

Charles XIV (Bernadotte), King of Sweden: III, 130.

Cherubini, Luigi:
On B’s playing, I, 220, 324;
engaged to compose operas for Vienna, II, 3, 47;
B’s respect for, 48;
opinion of “Fidelio,” 63, 64, 202;
“Les deux Journées,” II, 3, 36; III, 139;
“Lodoiska,” II, 3;
“Elise,” 3;
“Medea,” 3;
“Faniska,” 110;
asked by B. to urge subscription to Mass on King of France, III,
100, 126;
on B. and Mozart, 205;
Schlesinger on, 206;
on B’s quartets, 216;
his “Requiem” sung at B’s funeral, 312.

Cherubini, Madame: On B’s social conduct, I, 121.

Chorley, Henry F.: Receives and publishes B’s letter to Bettina von
Arnim, II, 182, 184, 316.

Churchill, John, Earl of Marlborough: I, 6.

Church Music: B’s views on, III, 203.

Cibbini, Antonia: B’s offer of marriage to, III, 205, 207.

Cimarosa, Domenico:
“L’Italiana in Londra,” I, 32;
“Il Matrimonio segreto,” 164.

Clam-Gallas, Count Christian: I, 194.

Clari, Countess Josephine di: I, 194.

Clemens August, Elector of Cologne: I, 1;

his extravagance, 7;
succeeds to the Electorship, 7;
career of, 7;
life in Rome, 8;
Grand Master of the Teutonic Order, 7, 98;
opens strong-box of the Order, 8;
falls ill while dancing and dies, 7, 8;
entry into Bonn, 9;
his music-chapel, 9;
appoints Van den Eeden Court Organist, 10;
increases salary of B’s grandfather, 10;
music in his reign, 14;
his theatre, 30;
appoints B’s grandfather Court Musician, 43.

Clement, Franz, Violinist: II, 2;

conductor, 42;
B’s Violin Concerto, 76;
succeeds Häring as conductor in Vienna, 112;
produces “Mount of Olives,” 156, 209, and the Ninth Symphony,
III, 157 et seq.

Clementi, Muzio: I, 33;

encounter between him and B., II, 23, 38, 75;
contract with B. for compositions, 102;
tardy payment of debt, 131, 158;
 B. on his pianoforte studies, 375;
B. sends them to Gerhard von Breuning, III, 214.

Collard, F. W., Partner of Clementi: II, 102, 103.

Collin, von:
“Coriolan,” II, 101, 102;
“Bradamante,” II, 119;
“Macbeth,” II, 119, 151;
“Jerusalem Delivered,” II, 119, 151;
Letter to, II, 149;
asked to write a drama for Pesth, 88, 201.

Cologne:
Electors of in the 18th century, I, 1 et seq.;
Archbishop Engelbert, 3;
civil income of Electorate, 7.

“Colonie, Die” (“L’Isola d’Amore”): Opera by Sacchini, I, 108.

Complete Editions of B’s Works planned: II, 18, 38, 192; III, 36,
54, 190, 205, 237;
Archduke Rudolph’s Collection, II, 200.

Congress of Vienna: II, 288, 289.

Consecutive Fifths: B’s dictum on II, 89.

“Contadina in Corte, La”: Opera by Sacchini, I, 26.

Conti: I, 282; II, 2.

“Convivo, Il”: Opera by Cimarosa, I, 107.

“Corsar aus Liebe”: Opera by Weigl, I, 268; II, 2.

Courts of Europe: Invited to subscribe to the Mass in D, III, 93 et
seq.

Court Composers: Their duties in the 18th century, I, 13.

Court Theatres of Vienna: B. asks appointment as composer for,

II, 98.

Cramer, F., Violinist: I, 186; II, 12.

Cramer, John Baptist: I, 186;

sketch of, 218, 219;
makes B’s acquaintance, 218;
his admiration for B., 219;
on B’s playing, 210; II, 318;
B’s opinion of him as pianist, 381.

“Creation, The,” Haydn’s oratorio: I, 243, 266, 282, 284, 285; II,
89, 116, 120;
receives the first metronomic marks, 223.

Cressner, George, English Ambassador at Bonn: I, 65.

Cromwell: I, viii; II, 360.

Czapka: Magistrate to whom B. appeals, III, 265.

Czartoryski, Prince: I, 271.

Czernin, Count: I, 172.

Czerny, Carl: I, 85;

anecdote about B. and Gelinek, 152;
on B’s extempore playing, 196;
use of high registers of pianoforte, 223, 236;
pupil of B., 314;
 duet playing with Ries, 314;
testimonial from B., 315;
memory of, 315;
rebuked by B. for changing his music, 316;
letters, 316, 322;
on the reception of the “Eroica,” II, 35;
on the Rasoumowsky Quartets, 75;
on B’s playing and teaching, 90;
on B’s character, 91;
on the theme of the Credo in the Mass in C, 107;
on the first performance of the Choral Fantasia, 130, 215, 314;
rebuked by B. for changing his music, 337;
letter, 338;
B’s advice as to instruction of Nephew Karl, 374;
inaccuracies as biographer, 376;
visits B., III, 203;
torchbearer at B’s funeral, 312.

Czerny, Wenzel: I, 236.

Czerwensky, Oboist: I, 239.

Dalayrac: Operas “Nina,” I, 107, 108;

“Les deux petits Savoyards,” 109.

Danhauser: Makes bust of B., II, 221;

death-mask, III, 310.

Dardanelli, Singer: III, 77, 169.

Decker: Makes crayon drawing of B., III, 176.

Degen, Aëronaut: III, 62.

Deiters, Dr. Hermann, German translator of Thayer’s work: I,

Dedication; 88;
writes conclusion of the biography, xv, 75, 103;
discusses date of a letter to Wegeler, 177;
on the C-sharp minor Sonata, 292;
B’s letters to Bettina von Arnim, II, 197;
B’s conduct towards Simrock, III, 53.

De la Borde: Opera “Die Müllerin,” I, 109.

Deler (Teller, Deller?): “Eigensinn und Launen der Liebe,” opera, I,

31.

Dembscher: III, 193;

canon, “Muss es sein?” 224, 244.

Demmer: Singer at first performance of “Fidelio,” II, 50, 61; III, 83.

Demmer, Joseph: Petitions for the post of B’s grandfather, I, 22;

appointed, 23.

Denmark, King of: Subscribes for the Mass in D, III, 102, 105.

Desaides: Opera “Julie,” I, 29, 107;

“Die Reue vor der That,” 32;
his operas in Bonn, 86;
“Les trois Fermiers,” 107.

Descriptive Music: B. and, II, 120.

“Déserteur, Le”: Opera by Monsigny, I, 31, 46.

Dessauer, Joseph: Buys autograph score of “Eroica,” II, 24.

“Deux Journées, Les”: Opera by Cherubini, II, 3; III, 139.

Devenne: “Battle of Gemappe,” II, 252.

Deym, Countess Isabelle: I, 342; II, 105.

Deym, Countess Josephine: I, 279, 322, 342; II, 203.

Diabelli, Anton: II, 314;

III, negotiations with B. 107;
variations on his waltz, 127 et seq.;
commissions Sonata for four hands, 183.

Dickens, Mrs., English singer: II, 310.

“Die beiden Caliphen”: Opera by Meyerbeer, II, 297.

“Die Müllerin”: Opera by De la Borde, I, 109.

Dietrichstein, Count: Tries to have B. appointed Imperial Court

Composer, III, 115;
sends B. texts for missal hymns, 116.

Dittersdorf: Operas “Doktor und Apotheker,” I, 108, 109;

“Hieronymus Knicker,” 109;
“Das rothe Käppchen,” 109, 139, 176, 183.

Dobbeler, Abbé Clemens: Carries Trio Op. 3 to England, I, 134,

145.

Dobbler’s Dramatic Company: I, 28.

“Dr. Murner”: Opera by Schuster, I, 108.

“Doktor und Apotheker”: Opera by Dittersdorf, I, 108, 109.

Doležalek, Johann Emanuel: I, 239;

sketch of, 368;
on the first performance of the Choral Fantasia, II, 130; III, 294.
Donaldson, Edinburgh publisher: III, 42.

“Don Giovanni”: Opera by Mozart, I, 91, 107, 163, 193; II, 204;
III, 42.

“Donne sempre Donne, Le”: Opera by Lucchesi, I, 26.

Dont, Jacob: II, 399.

Dont, Joseph Valentine: II, 399.

Donzelli, Singer: III, 169.

“Dorfbarbier, Der”: Opera by Hiller, I, 36.

“Dorfdeputirten, Die”: Opera by Schubauer, I, 109.

Dousmoulin. (See Touchemoulin.)

“Dragomira”: Drama by Grillparzer, III, 118, 120, 122.

Dragonetti, Domenico:
Makes B’s acquaintance, I, 218;
skill on double-bass, 218; II, 124;
Trio in Fifth Symphony, 126, 256;
recitatives in Ninth Symphony, III, 207.

Drama, German: Cultivated in the time of Max Friedrich, I, 28 et

seq.

Drechsler, Chapelmaster: III, 131;

pallbearer at B’s funeral, 312.

Dresden: B’s intended visit to, I, 192.

Drewer, Ferdinand, Violinist: I, 23, 24.

Drieberg, Baron F. J.: “Les Ruines de Babilone,” II, 202.

Drosdick, Baroness: II, 86.

Duncker, Friedrich: “Leonore Prohaska,” II, 298.

Duni: Opera, “Die Jäger und das Waldmädchen,” I, 29.

Duport: Director of the Kärnthnerthor Theatre, and the Ninth

Symphony, III, 157.

Duport, Pierre, Violoncellist: I, 195, 205.

Dürck, F.: His lithograph of Stieler’s portrait, III, 42.

Duschek, Madame: I, 194, 226.

Düsseldorf: Electoral archives at, I, 5.

Dutillier: Operas, “Nanerina e Pandolfo,” I, 165;

“Trionfo d’Amore,” 165.

Eberl, Anton: I, 172; II, 2.

Ecclesiastical States of Germany: Former, I, 1, 15.

Edwards, F. G.: His sketch of Bridgetower’s career, II, 11.

Egyptian Text: Preserved by B., II, 168.

“Ehrenpforte, Die”: Drama by Treitschke, II, 317.

“Eifersucht auf der Probe”: Opera by Anfossi, I, 32.

“Eifersüchtige Liebhaber, Der” (“L’Amant jaloux”): Opera by
Grétry, I, 31, 107.

“Eigensinn und Launen der Liebe”: Opera by Deler (?), I, 31.

“Einsprüche, Die”: Opera by Neefe, I, 36.

Electoral Chapels: Appointments in, I, 9.

Electors of Cologne: I, 1 et seq.

Ella, John: II, 12; III, 32.

Embel, F. X.: III, 142.

“Ende gut, Alles gut”: Opera by d’Antoine, I, 109.

Engelbert, Archbishop of Cologne: I, 3.

England: B’s plan to visit, II, 142;

his admiration for the English people and government, III, 36, 76,
181, 303;
court of, not invited to subscribe to Mass in D, 104, 112.
(See “Prince Regent.”)

English plays produced at Max Friedrich’s court: I, 29, 30, 31.

“Entführung aus dem Serail”: Opera by Mozart, I, 32, 107, 109.

Eppinger, Heinrich, Amateur violinist: I, 235, 274, 306; II, 2.

Eppinger, Dr. Joseph: II, 335.

Erard, Sébastien: Presents pianoforte to B., II, 21.

Erdödy, Count: I, 172;
continued friendship for B., II, 82, 215, 271.

Erdödy, Countess Marie: Said by Schindler to have been one of

B’s loves, I, 324;
sketch, II, 82, 124;
dedication of Trios Op. 70, 132;
proposes plan to keep B. in Vienna, 136, 141;
letter of apology from B., 144, 162, 315, 319;
B’s letter of condolence of death of her child, 339;
dedication of Op. 102, 357; III, 21;
dedication, 23.

Erk and Böhme: “Deutscher Liederhort,” I, 278.

“Erlkönig”: Song by Schubert, I, 230; III, 236.

Ernst, Violinist: Purchaser of the Heiligenstadt Will, I, 351;

and B’s last quartets, III, 139.

“Ernst und Lucinda” (“Eraste et Lucinde”): Opera by Grétry, I, 31.

Ertmann, Baroness Dorothea: Pupil of B., I, 322; II, 2, 83, 215;

B. consoles her grief by playing the pianoforte, 356;
dedication of Sonata Op. 101, 356, 365.

Esterhazy, Count Franz: I, 170.

Esterhazy, Prince Franz Anton: I, 172;

Princess, I, 172.

Esterhazy, Count Johann Nepomuk: I, 170.

Esterhazy, Prince Nicholas: I, 169; II, 98;

commissions B. to write a mass, 100;
letters from B., 107;
 criticism of the Mass in C, 108, 116.

Esterhazy, Count Niklas: II, 98, 225.

Esterhazy, Prince Paul Anton: I, 166, 171, 189;

invited to subscribe to the Mass in D, III, 103.

Esterhazy, Princess: Dedication of the Marches Op. 45, I, 351; II,

40, 108.

“Esther”: Opera by S. F. A. Auber, I, 14.

“Euryanthe”: Opera by Weber, III, 139, 140.

“Évènements imprévus, Les”: Opera by Grétry, I, 32.

Ewer and Co.: III, 13.

Eybler, Joseph: I, 165;

B’s respect for him, 242;
pallbearer at B’s funeral, III, 312.

Facius, the Brothers: Amateurs in Bonn, I, 38.

Falsification of B’s age: I, 55, 70, 71.

“Falstaff, ossia le Tre Burli”: Opera by Salieri, I, 227.

“Faniska”: Opera by Cherubini, II, 110.

“Fassbinder, Der”: Opera by Oudinet, I, 29.

“Fausse Magie, La”: Opera by Grétry, I, 107.

“Faust,” Goethe’s: II, 119; III, 75, 220.

“Félix, ou l’Enfant trouvé”: Opera by Monsigny, I, 32, 109.

Felsburg, Count Stainer von: II, 338; III, 156.

“Fermiers, Les trois”: Opera by Desaides, I, 107.

Fidelissimo Papageno: Nickname for Schindler, III, 102.

“Fiesco”: Drama by Schiller, III, 117.

“Filosofo di Campagna”: Opera by Galuppi, I, 25.

Finanz-Patent, Austrian: Its effect on B’s annuity, II, 211 et seq.

“Finta Giardiniera, La”: Opera by Paisiello, I, 108.

Fischer, Cäcilie: I, xviii, 57, 58.

Fischer, Gottfried: I, xvii, 43, 47, 50, 51, 61, 66.

Fischer Manuscript: I, xvii, 43, 47, 50, 51, 61, 66.

Fischer: Opera, “Swetard’s Zaubergürtel,” II, 49.

Fodor, Singer: III, 121.

Fontaine, Mortier de: II, 73.

Forkel, J. N.: Biography of Bach, I, 303;

“History of Music in Examples,” II, 34.

Forray, Baron Andreas von: II, 220.

Förster, Emanuel Aloys: I, 172;

influences B’s chamber music, 273;
 his son’s lessons from B., II, 31, 125, 315, 380.

Forti, Singer: II, 286.

Fouche, Mary de: I, 186.

Fouqué, Baron de la Motte: II, 330.

“Four Elements, The”: Oratorio planned by Kuffner, III, 219.

Fox, Mrs. Jabez: Acquires Thayer’s posthumous papers, I, xiv;

her copy of Mähler’s portrait of B., II, 16.

“Fra due Litiganti”: Opera by Sarti, I, 86, 109.

France, King of: Subscribes for Mass in D and strikes medal, III,
99, 105, 230.

Frank, Dr.: Treats B., I, 300.

Frank, Joseph: I, 243.

Frank, Madame. (See Gerardi.)

Frankfort: Cäcilien-Verein in, III, 104, 106, 111, 180.

Franz (Francis), Emperor of Austria: I, 214; III, 296.

Franzensbrunn: II, 223.

“Frascatana, La”: Opera by Paisiello, I, 107.

Frederick II, King of Prussia: I, 195;

reputed father of B., III, 214.

Frederick III, of Prussia, German Emperor: marries Princess

Victoria of England; a Wedding Song, III, 13.

Frederick William III, King of Prussia: I, 194, 195, 205.

“Freischütz, Der”: Opera by Weber, III, 121, 135.

Freudenberg, Karl Gottfried: Visits B., III, 202.

Freund, Philip: Variation, I, 300.

“Freundschaft auf der Probe” (“L’Amitié á l’Épreuve”): Opera by

Grétry, I, 131.

Friedelberg: I, 199; “Ein grosses, deutsches Volk sind wir,” 200.

Friedlowsky, Clarinettist; I, 329.

Fries, Count Moritz: I, 172;

dedication of Violin Sonatas, 290;
Quintet Op, 29, 294;
collects funds for Bach’s daughter, 308.

Frimmel: “Beethoven Jahrbuch,” I, 255;

“Beethoven’s Wohnungen,” 269;
on the Bagatelles, 362, 337;
on Beethoven’s portraits, II, 15.

Fritzieri (Fridzeri, Frizer): Opera “Die seidenen Schuhe,” I, 32,

86.

Fry, William Henry, American critic: II, 358.

Fuchs, Aloys: I, 194, 276;

anecdote of B. and Haydn, 285;
owner of Heiligenstadt Will, 351;
solo singer in Troppau, II, 208, 368.
Fuchs: “Battle of Jena” arranged for two flutes, II, 252.

Fugger, Countess: Favorite of Elector Joseph Clemens, I, 3.

Fugues: B’s opinion on, II, 289.

Fürstenberg, Cardinal: I, 3;
his government of the Electorate, 5;
political vicissitudes, 5, 14.

Fuss, Johann: Opera “Romulus and Remus,” II, 304.

Fux, Joseph: “Gradus ad Parnassum,” I, 158, 159.

Galitzin, Prince George: III, 230.

Galitzin, Prince Nicolas Boris: III, 73;

dedication, 81;
the last Quartets, 87, 183;
asked to appeal to Czar for subscription, 102;
controversy over payment for the Quartets, 226 et seq.

Gallenberg, Count Wenzel Robert: Marries Countess Guicciardi,

I, 320, 324;
associated with Barbaja, 320;
as a composer, II, 42; III, 130.

Gallenberg, Count: Son of Countess Guicciardi, I, 340.

Galuppi: Operas “Il Filosofo di Campagna,” I, 25;

“La Calamità di Cuori,” 26;
“Tre Amanti ridicoli,” 27.

Gänsbacher: On Vogler’s playing, II, 15;

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