Mathematical Communications 4(1999), 257-268 257

Tensor products of C-algebras, operator spaces

and Hilbert C-modules
ckler
Franka Miriam Bru

Abstract. This article is a review of the basic results on tensor

products of C ? -algebras, operator spaces and Hilbert C ? -modules.

Key words: tensor products, C ? -algebras, operator spaces, Hilbert

?
C -modules

1. Preliminaries
Let us recall a few basic facts about tensor products, C -algebras, operator spaces
and Hilbert C -modules which shall be used in this short overview. Algebraic tensor
products shall be denoted by , while tensor products completed with respect to a
norm shall be denoted by . Bilinear maps on the Cartesian product of vector
spaces (algebras, . . . ) are canonically identified by linear maps on the corresponding
tensor products: if : X Y Z is bilinear, the corresponding (unique) linear
operator : XY Z is given on elementary tensors by (xy) = (x, y).
The standard tensor product of Hilbert spaces H and K (i.e. the tensor product
HK completed with respect to the norm induced by the inner product given on
elementary tensors by ( | 00) = ( | 0)H ( | 0)K ) shall be denoted by HK.

Let it be reminded that for the standard tensor product of Hilbert spaces H and K
there is a natural identification of tensor products of bounded linear operators on
H and K as bounded linear operators on HK, i.e. B(H)B(K) B(HK) via
(T S)() = T ()S() for T B(H), S B(K).
A C -algebra is a norm-closed selfadjoint subalgebra A of B(H) for some Hilbert
space H. Equivalently, it is a Banach -algebra whose norm satisfies the C -property

kak2 = ka ak

for all a A. C -algebras shall be denoted by A, B, C, . . . The dual space of a C -

algebra A (i.e. the space of all continuous linear functionals on A) shall be denoted
by A . The finite-dimensional C -algebras Mn (C) shall be denoted by Mn .
A -morphism between -algebras (i.e. algebras with an involution) is a lin-
ear operator that is multiplicative and preserves the involution. For a -morphism
The lecture presented at the Mathematical Colloquium in Osijek organized by Croatian

Mathematical Society - Division Osijek, April 16, 1999.

Department of Mathematics, University of Zagreb, Bijeni
cka 30, HR-10 000 Zagreb, e-mail:
bruckler@math.hr
258 ckler
F. M. Bru

between C -algebras the continuity (even contractivity) is automatic, so the -

morphisms between C -algebras shall be referred to as C -morphisms. A represen-
tation of a C -algebra is a C -morphism : AB(H) for some Hilbert space H.
Every C -algebra can be faithfully (i.e. isometrically) represented as a (norm-closed
selfadjoint) subalgebra of B(H) for some Hilbert space H. Every representation
of A on B(H) and vector H define a continuous linear functional f on A by
f(a) = ((a) | ). Such a functional is positive, i.e. takes positive values on pos-
itive elements in A (those are elements of the form xx for x A). What is more
important is that for every positive functional on A (positive linear functionals are
automatically continuous) there exist a Hilbert space Hf , a vector f Hf and a
representation f of A on Hf such that f(a) = (f (a)f | f ) for all a A (the con-
struction is known as the Gelfand-Naimark-Segal construction). For more details
on the general theory of C -algebras, see e.g. [8].
An operator space is any subspace X of some C -algebra (i.e. of B(H) for some
Hilbert space H). It is usually (but not always) required that an operator space is
norm-closed. The point is that one considers not only the vector-space structure,
but also the matricial structure. Namely, for any n N one may identify the space
Mn (B(H)) of n n matrices over B(H) with the space B(Hn ), where Hn denotes
the orthogonal sum of n copies of H. This way, for any subspace X of B(H) one may
consider Mn (X) as a subspace of Mn (B(H)) = B(Hn ) and thus one has a norm kkn
on Mn (X) for every n. Note that for any vector space V one can identify the space
Mn (V ) with the tensor product V Mn . When considering maps between operator
spaces, one is naturally led to consider their n-th amplifications i.e. for a linear
map T : XY its n-th amplification is the linear operator Tn : Mn (X)Mn(Y )
defined by
Tn ([aij ]) = [T (aij )]

for [aij ] Mn (X). Since Mn (X) and Mn (Y ) are normed for every n, one may
consider the operator norm kTn k. If supnN kTn k < , then the operator T is
called completely bounded and kT kcb = supnN kTn k is the completely bounded norm
(cb-norm) of T . The set of all completely bounded maps from X to Y (with the cb-
norm) is denoted CB(X, Y ). A completely bounded map T is a complete isometry
if all Tn are isometric and it is a completely isometric isomorphism if T is invertible
and kT kcb = kT 1 kcb = 1. Operator spaces are identified up to completely isometric
isomorphisms. For more information on completely bounded operators see [7] and
[15].
The most important result on operator spaces is Ruans theorem ([17]) which
gives an abstract characterization of operator spaces:

Theorem 1. (Ruan) A vector space X with a sequence of norms k kn on Mn (X)

(n N) is an operator space if and only if the following two conditions are satisfied
(for all n, m N):
(i) kAkn kkkAkn kk for all , Mn, A Mn(X) (the matrix multiplication
being the natural one);
(ii) kA Bkn+m = max{kAkn , kBkm } for all A Mn (X), B Mm (X) (where
 
A 0
A B denotes the matrix Mn+m (X)).
0 B
 Tensor products of C ? -algebras 259

A (right) pre-Hilbert C -module is a vector space E which is a (right) module

over a C -algebra A with an inner product < . | . >: E EA with the following
properties:

(H1) < x | y + z >=< x | y > + < x | z >

(H2) < x | ya >=< x | y > a
(H3) < x | y > =< y | x >
(H4) < x | x > 0 (in A)
(H5) < x | x >= 0 x = 0

for all x, y, z E, a A. These properties obviously generalize the ones required

for the inner product of a Hilbert space.
If E is complete in the norm defined by
p
kxk = k < x | x > k

then E is called a Hilbert A-module. Every Hilbert C -module is an operator space

(see e.g. [3]). The only classes of operators on Hilbert C -modules which shall be
considered in this paper are BA (E) and BA (E). BA (E) is the Banach algebra of
all bounded operators on E that are A-linear, i.e. T such that T (xa) = T (x)a for
all x E, a A. BA(E) is the C -algebra (subalgebra of BA (E)) of all operators
T : EE for which there exists a map T : EE (called the adjoint of T ) such
that < T x | y >=< x | T y > for all x, y E. An overview of the basic theory of
Hilbert C -modules can be found in [19].

2. Tensor products of C -algebras

The norm of a C -algebra is unique in the sense: on a given -algebra A there is at
most one norm which makes A into a C -algebra. Still, on a -algebra A there may
exist different norms satisfying the C -property. The completion with respect to any
of such norms results in a C -algebra which contains A as a dense subalgebra. This
is precisely what happens when the tensor product of C -algebras is considered: in
the general case there are many different norms on the algebraic tensor product
AB (which is a -algebra) with the C -property (these shall be referred to as
C -(tensor) norms and tensor products of C -algebras completed with respect to a
C -norm shall be referred to as C -tensor products).
One such norm is the spatial norm k k defined by the inclusion AB
B(H)B(K) B(HK), assuming that A and B are faithfully represented on
Hilbert spaces H and K respectively. This norm was introduced by T.Turumaru
in 1953. The definition does not depend on particular representations of A and B,
that is for all t AB
ktk = k()(t)kB(HK)

for any two faithful representations of A on H and of B on K.

The second natural norm on AB was introduced in 1965 by A. Guichardet. It
is the maximal C -norm k k defined as

ktk = sup{ktk : k k C -seminorm on AB}.

260 ckler
F. M. Bru

The maximal C -norm is obviously adequatly named. As for the spatial norm,
it is shown that it is the minimal C -norm on the tensor product of C -algebras
(and the norm is often referred to as the minimal C -norm). Being minimal resp.
maximal means
ktk ktk ktk
for every other C -norm k k , for all C -algebras A and B and all t AB. The
set of all C -norms on AB is a complete lattice (with respect to the ordering
k k k k if ktk ktk for all t AB) and all C -norms k k on tensor
products of C -algebras are cross-norms, i.e.
kabk = kakkbk
for all a A, b B.
The spatial C -norm has several good properties, the most important of them
being its injectivity, i.e. the restriction of the spatial norm on AB to the tensor
product A1 B1 of subalgebras A1 of A and B1 of B is the spatial norm on A1B1.
This may not be the case for other C -tensor norms: although the restriction of
any C -norm k k from AB to A1B1 is always a C -norm, it is not necessarily
the -norm on A1B1 .
Other good properties of the spatial norm include the following: the C -tensor
product of two C -algebras is a simple C -algebra if and only if both C -algebras
are simple and the norm is the spatial one; the tensor product of continuous linear
functionals on C -algebras A and B is always continuous on AB with respect to
the spatial C -norm (and consequently with respect to every C -norm):
| (fg)(t) | kfkkgkktk
for f A , g B , t AB; the tensor product of C -morphisms is always
continuous if the range is equipped with the spatial C -norm (and the domain is
equipped with any C -norm).
The maximal C -norm has its good properties, too. The most important is that
the representation defined by
X
n X
n
ai bi 7 (ai )(bi )
i=1 i=1

can be continuously extended to a representation of the C -algebra A B for any

pair of commuting representations and of A and B on the same Hilbert space.
A pair (, ) of representations is called commuting if (a)(b) = (b)(a) for every
a A and b B. An algebraic representation : ABB(H) which satisfies
k(ab)k kakkbk for all a A, b B is called a subtensor representation.
Since for every subtensor representation of AB there exists a pair of commuting
representations and of A and B such that (ab) = (a)(b) = (b)(a) for all
a A and b B and since every representation of A B is subtensor (for every
C -norm k k ), one gets
ktk = sup{k(t)k : subtensor representation of AB}
for t AB. This is the original Guichardets definition of the maximal C -norm
for the tensor product of C -algebras.
 Tensor products of C ? -algebras 261

As for other C -norms, they can all be obtained in the following way:
Let S(AB) be the set of all linear functionals f on AB such that f(t t) 0
for all t AB and sup{| f(ab) | : kak 1, kbk 1} = 1. To each f AB one can
associate an algebraic representation f : ABB(H) (and a vector f Hf ) such
that f(t) = (f (t)f | f ) for all t AB (the construction imitates the Gelfand-
Naimark-Segal construction and then proves the continuity of the resulting f (t)
for all t). If we define ktkf = kf (t)k, k kf is obviously a C -seminorm. For a
subset S of S(AB) let
ktkS = sup ktkf .
f S

Then the collection of all S S(AB) such that k kS is a C -norm is in 1-1-

correspondence to the set of all C -norms on AB, i.e. every C -norm is uniquely
determined by such set S (in particular, the maximal norm corresponds to the
whole S(AB) and the spatial one to S(AB) (A B )). The correspondence
was proven by Lance and Effros in 1977 ([10]).
There are C -algebras A for which the minimal and the maximal norm on AB
coincide for all C -algebras B (and consequently the C -norm on AB is unique).
Such C -algebras are called nuclear and their C -tensor products obviously combine
the good properties of the minimal and maximal C -product. Among the nuclear
C -algebras are all finite-dimensional C -algebras (in particular, the algebras Mn
for n N), all commutative ones, all GCR-algebras (in particular the C -algebra
of compact operators on a Hilbert space), inductive limits of nuclear C -algebras
(in particular AF-algebras), type I C -algebras,. . . But not all C -algebras are
nuclear. The first example of a non-nuclear algebra is due to Takesaki ([18]): the
C -algebra generated by the left regular representation on l2 (G) of a free group G
with two generators.

Example 1. Since Mn is a nuclear C -algebra, for any C -algebra A the C -norm

on AMn is unique (and thus equal to the spatial norm). Moreover, the algebraic
tensor product AMn is already complete in this norm1 . In particular, the algebra
Mn (A) is a C -algebra.

There are different characterisations of nuclear C -algebras proven by Choi,

Effros, Kirchberg, Lance and others in the 70es. They are all complicated in proof
and mostly also in formulation. One of the simpler ones is: A is nuclear if and
only if the von Neumann algebra2 A is injective (i.e. for all C -algebras B A
there is a continuous retraction3 BA ). S. Wassermann showed that for infinite-

dimensional separable H B(H) is not injective (On tensor products of certain
group C -algebras, J. Funct. Anal. 23 (1976) 239-254), so one gets another example
of a non-nuclear C -algebra, namely B(H) for infinite-dimensional separable H.
Most of the characterisations of nuclear C -algebras are of the following form:
the identity operator on A (or on A )
1 In fact, the proof that M is nuclear is the proof of the completeness of AM in the spatial
n n
norm.
2 A von Neumann algebra is a weakly closed selfadjoint subalgebra of B(H) for some Hilbert

space H. The weak topology on B(H) is defined by the family of seminorms {p, : , H},
where p, (a) = (a | ) for a B(H). The enveloping von Neumann algebra of a C -algebra A is
isometrically isomorphic to its bidual A .
3 A retraction is a left inverse of the inclusion morphism.
262 ckler
F. M. Bru

a) can be approximated (in a suitable topology) by a certain class of (finite rank)

operators, or
b) there is an approximate factorization for , where under approximate factoriza-
tions one understands that for any neigborhood of (in a suitable topology) there
exist an operator 0 and operators (into Mn for some n) and (with domain Mn )
such that 0 = .
More detailed overviews of the theory of tensor products of C -algebras can be
found in [13] and [19].

3. Tensor products of operator spaces

Let X and Y be two operator spaces and XY their algebraic tensor product.
We consider norms on the vector space XY which make it an operator space,
i.e. sequences of norms k kn on Mn(XY ) which satisfy the conditions of Ruans
theorem. We also pose an additional condition on these norms - that of being
cross-norms:
kABknm = kAkn kBkm
for all n, m N, A = [aij ] Mn (X), B = [bkl ] Mm (Y ), where AB denotes
the matrix [aij bkl ] Mnm (XY ). Such norms shall be called operator space
cross-norms.
Similarly as in the C -algebra case, it turns out there is a minimal (the spatial
norm) and a maximal (the projective norm) among operator space cross norms,
but only if one more property is required: that the dual norm of the norms in
question is also an operator space cross-norm. The dual norm of a norm k k is
the norm induced by the natural inclusion X 0 Y 0 into (X Y )0 where X 0 denotes
the standard dual of an operator space X, X 0 = CB(X, C). Besides the minimal
and maximal norm, there is another very important operator space cross-norm: the
Haagerup norm.
The Haagerup norm was the first one considered. Pn The motivation was the con-
sideration of operators of the form (a) = i=1 ui avi for a A where u1 , . . . , un,
v1 , . . . , vn are some fixed elements in A. These operators result from the action of
P
ui vi AAop on A (where Aop is the C -algebra A with the reversed product).
If A B(H) then for , H the Cauchy-Schwarz inequality implies
X  X 1/2 X 1/2
  2 2
|((a) | )| =  (avi | ui ) kavi k kui k .
P 2 P P 2
Further, kavi k kakkvi k, kvi k = (vi vi | ) k vi vi kkk and simi-
larly for ui. It follows that
X 1/2 X 1/2
   
|((a) | )| ||a||  ui ui   vi vi  |||| |||| ,
P 1/2 P 1/2
so kk || uiui || || vi vi || . P
One Pmay allow also infinite (countable) sequences of ui and vi , provided that uiui
and vi vi are norm convergent. The natural definition following from these con-
siderations is
( n )
X Xn X
n
1/2 1/2
ktkh = inf k ui ui k k vi vi k : n N, t = ui vi AB
i=1 i=1 i=1
 Tensor products of C ? -algebras 263

for t AB. The proof that kkh (the Haagerup norm) is a norm is not completely
trivial (the proof of the triangle inequality and the definiteness are non-trivial).
These properties are proven by Effros and Kishimoto in 1987 ([9]). The norm
was named after Uffe Haagerup, whose unpublished manuscript Decomposition of
completely bounded maps on operator algebras (1980) initialized the definition and
research of the properties of this norm.
The Haagerup norm is not a C -norm, but if the definition is repeated for
n N and t Mn (XY ) for operator spaces X and Y , it turns out that the
Haagerup norm is an operator space cross-norm (see e.g. [16]) with a number
of good properties. First of all, the definition can be somewhat simplified: for
U Mn (XY ) one has

kU kh = inf{kAkkBk : U = A  B, k N, A Mnk (X), B Mkn (Y )},

hP i
k
where A  B denotes the matrix r=1 air brj . In particular, the formula implies
that the first definition doesnt depend on the C -algebras on which X and Y are
represented (note that in the original formula adjoints of elements occur and that
in the definition of operator spaces one does not require that the operator space is
closed upon taking adjoints). Moreover, for U Mn(XY ) the infimum is achieved
for some particular k N, A Mnk(X) and B Mkn(Y ).
The Haagerup tensor product (i.e. XY equipped with the Haagerup norm) is
associative:
(Xh Y )h Z = Xh (Y h Z)
(completely isometrically isomorphic). Further, the norm is (completely) injective
i.e. for subspaces X0 X and Y0 Y the restriction of the Haagerup norm
from Xh Y to X0 Y0 is the Haagerup norm and consequently X0 h Y0 is com-
pletely isometrically embedded into Xh Y . The tensor product T S of completely
bounded operators T and S on X and Y is completely bounded on Xh Y , and
more: kT Skcb kT kcbkSkcb . The Haagerup norm is also selfdual, i.e. the dual
norm of the Haagerup norm on XY is the Haagerup norm on X 0 Y 0 .
What is perhaps most important is the correspondence between completely bo-
unded bilinear operators4 : XY B(H) and their linearisations : XY B(H)
when the tensor product XY is equipped with the Haagerup norm (a bilinear op-
erator : X Y B(H) is completely bounded if and only if the corresponding
linear operator : Xh Y B(H) is completely bounded and in that case their cb-
norms are equal). Moreover, any completely bounded map on the Haagerup tensor
product of two operator spaces is essentially operator multiplication:

Theorem 2. (Christensen-Sinclair [6]/Paulsen-Smith[16])

If : Xh YB(H) is completely contractive (i.e. kkcb 1) and X A, Y B,
then there exist representations : AB(K1 ) and : BB(K2 ), isometries vi :
HKi and a contraction t : K2K1 such that for all x X, y Y

(xy) = v1 (x)t(y)v2 .
4 A bilinear operator : X Y Z, where X, Y and Z are operator spaces, is completely

bounded if supnN kn k < , where P the n-th amplification of is defined as n : Mn (X)

Mn (Y )Mn (Z), n ([xij ], [yij ]) = [ n
k=1 (xik , ykj )]. The cb-norm of a completely bounded
bilinear map is defined by kkcb = supnN kn k.
264 ckler
F. M. Bru

Example 2. The multiplication on a C -algebra as an operator on AA (i.e. m :

AAA, m(ab) = ab) is not continuous with respect to any C -tensor norm
in general, but it is continuous (even completely contractive) with respect to the
Haagerup norm. This even characterizes operator algebras: if X is an operator
space with an algebra multiplication which is completely bounded with respect to the
Haagerup norm, then X is an operator algebra. This is proven in D. P. Blecher,
Z.-J. Ruan, A. M. Sinclair, A characterization of operator algebras, J. Funct.
Anal. 89(1990), 188201.

There are many more remarkable properties of the Haagerup norm, which re-
sulted in a wide range of applications and identities. For example,

Cm (X)h Rn(Y ) = Mmn (Xh Y )

for all operator spaces X and Y and m, n N, in particular

Cnh Rn = Mn .

On the other hand, Rnh Cn can be isometrically identified with the dual of Mn
equipped with the trace-class norm. Here, Cm (X) denotes the subspace of Mm (X)
consisting of all matrices with nonzero entries only in the first column. Similarly,
Rn (Y ) denotes the matrices in Mn (Y ) with nonzero entries only in the first row.
Cn denotes Cn (C) and Rn denotes Rn(C). Further, A. Chatterjee and A. M. Sin-
clair showed in 1992 that if X and Y are both subspaces of B(H), then Xh Y is
a subspace of the space of completely bounded operators on K(H) (the compact
operators), via the identification (xy)(K) = xKy. In fact, the only nice prop-
erty the Haagerup norm does not have seems to be the commutativity: Xh Y
is not completely isometrically isomorphic to Y h X. For more details about the
Haagerup norm, see [2], [4], [5], [11] and [16].
The spatial norm k k on the tensor product of operator spaces is defined in
the same way as in the C -algebra case, of course taking into account the matricial
structure: X Y B(H)B(K) B(HK). The corresponding formula is

kU k = sup{k(ST )n (U )k : H, K Hilbert spaces,

S : XB(H), T : Y B(K)complete contractions}

and it turns out that it is sufficient to consider only finite-dimensional Hilbert

spaces H and K. The spatial norm is the least between all operator space cross-
norms whose dual norm is also an operator space cross-norm. It has the property
that X Y can be completely isometrically embedded into CB(X 0 , Y ), resp. into
CB(Y 0 , X).
The projective norm k k for the tensor product of operator spaces is in such a
way constructed that (X Y )0 is completely isometrically isomorphic to the space
JCB(X Y, C) via the usual identification of bilinear maps with linear maps on
tensor products. JCB(X Y, C) is the set of all bilinear maps : X Y C such
that sup{k[(xij , ykl )]k : p, q N, A = [xij ] Mp (X), B = [ykl ] Mq (X), kAk
1, kBk 1} is finite, adequately equipped with an matricial structure which makes
JCB(X Y, C) a operator space. Therefore,

kU k = sup{k < U, V > k : m N, V Mm (JCB(X Y, C)), kV km 1}

 Tensor products of C ? -algebras 265

for U Mn (XY ) and where < U, V >= [Vij (Ukl )] Mmn . (X Y )0 is also
completely isometrically isomorphic to the spaces CB(X, Y 0 ) and CB(Y, X 0 ). The
projective norm is the largest operator space cross-norm. Its dual norm is the
spatial norm (but not the other way around).
Both the spatial and the projective norm are associative and commutative and
they also both give some useful identites, but they have far less good properties than
the Haagerup norm. For more details about the spatial and projective operator
space tensor norms (and results for operator space tensor norms in general) see [2]
and [4].

4. Tensor products of Hilbert C -modules

There are two tensor products which have been considered for Hilbert C -modules
: the interior and the exterior tensor product.
Let E be a Hilbert A-module and F a Hilbert B-module and let : ABB (F ) be
a C -morphism. Then F can be considered a left A-module, setting ax = (a)(x) for
x F and a A. Thus, the algebraic tensor product of a right and a left A-module
can be constructed: EAF is the quotient of the vector space tensor product EF
by the subspace N spanned by all the expresions of the form xay xay with
x E, y F and a A. Elementary tensors in EA F are denoted as xA y. One
can regard EA F as a right B-module, setting (xA y)b = xA yb. Defining

< xA y | x0Ay0 > :=< y | (< x | x0 >E )y0 >F

and extending by (B-)linearity one gets an inner product on EA F : all properties

except for definiteness are quite easy provable, and definiteness follows from the fact
(for the proof see [14]) that the subspace {t EF : < t | t >= 0} of EF coincides
with N . The resulting Hilbert B-module is called the interior tensor product E F
of E and F (using ). There is a natural C -morphism : BA (E)BB (E F )
defined by (T )(xy) = T (x)y (that is, = T IdF ). The morphism is injective
if is. It is interesting that when we consider Hilbert C -modules as operator
spaces, the interior tensor product coresponds to the (module) Haagerup tensor
product:
E F = EhA F
completely isometrically, where EhA F denotes EA F completed with respect to
the Haagerup norm.
The exterior tensor product of a Hilbert A-module E and a Hilbert B-module F
is defined directly imitating the definition of the tensor product of Hilbert spaces:
first the algebraic tensor product EF is organized into a right AB-module setting
(xy)(ab) := xayb and then it is defined

< xy | x0y0 >=< x | x0 > < y | y0 >

(and of course the definitions are extended by (AB-)linearity to the whole algebraic
tensor product). Here one encounters two problems - missing of the definiteness and
the fact that AB is not a C -algebra, but only dense in one (A B for this case), so
the module multiplication (xy)t is not defined for all t AB and consequently
the condition < u | vt >=< u | v > t makes no sense for t (A B) \ (AB). This
266 ckler
F. M. Bru

means that we only have a semi-inner-product module EF over a pre-C -algebra

AB. But, as it turns out, the problem is quite easily resolved: first the definiteness
is achieved taking the usual quotient by the subspace N = {t EF : < t | t >= 0}
(as a matter of fact, the semi-inner product is actually already definite - see [14]). On
1/2
the quotient EF/N one may define the norm ku + N k = k< u | u > k . Then for
t AB and u EF one has k(u + N )tk ku + N kktk, so the structure of the
right AB-module can be extended to the structure of the right A B-module on
EF/N and then it is easily shown that the condition < u+N | (v +N )t >=< u+N |
v + N > t holds for u, v EF and t AB. The other relations required for the
inner product also extend to the situation now considered. The resulting Hilbert
A B-module EextF is called the exterior tensor product of E and F . As to the
properties of the exterior tensor product with respect to (adjointable) operators
on its underlying modules, there is a natural C -morphism (isometric embedding)
BA (E)BB (F )BA B (EextF ) defined by (ST )(xy) = S(x)T (y). The
embedding will not in general be surjective (as is well known for the Hilbert space
case). When we consider Hilbert C -modules as operator spaces, the exterior tensor
product corresponds to the spatial tensor product:

EextF = E F

completely isometrically, so EextF can be considered as a subspace of CB(E 0 , F )

or CB(F 0, E). Both results on the correspondence of the operator space tensor
norms and Hilbert module tensor products are due to Blecher ([3]).
Both tensor products are more completely described in [14]. The exterior and
the interior tensor product of Hilbert modules coincide when one of the Hilbert
modules considered is a Hilbert space (and if both are Hilbert spaces, this tensor
product is equal to the standard Hilbert space tensor product).

5. Some examples
Let us give a few examples of interconnections between the theories of tensor prod-
ucts of C -algebras, operator spaces and Hilbert C -modules.

Example 3. The natural contractive morphism Ah BAB is injective ( D.

P. Blecher, Geometry of the tensor product of C -algebras, Math. Proc. Cam.
Phil. Soc. 104 (1988) 119-127). Under this map, a closed ideal I in Ah B induces
a closed ideal I (the closure of the image of I) in A B. Closed non-zero ideals
in A B must contain a non-zero elementary tensor and every elementary tensor
which lies in I must lie in I. Combining these results one gets that closed non-zero
ideals in Ah B must contain a non-zero elementary tensor. This fact has many
consequences for investigating the ideal structure of Ah B, e.g. it immediately
follows that if both A and B are simple, so is Ah B, a property belonging also to
the spatial tensor product of C -algebras.
There are also analogies in the ideal structure of Ah B with the ideal struc-
ture of A B, e.g. the closed ideal B(H) K(H) + K(H) B(H) is maximal in
B(H) B(H) when is the Haagerup or the maximal C-norm (for H infinite
dimensional separable Hilbert space).
These results are from [1].
 Tensor products of C ? -algebras 267

Example 4. ([3]) The metric characterization of Hilbert C -modules can be given

in terms of the Haagerup tensor product:

Theorem 3. Let E be a Banach space (resp. operator space) which is also a

(right) A-module. Suppose that A is faithfully and nondegenerately5 represented
on a Hilbert space H. Then E is a Hilbert A-module (with its Hilbert C -module
norm coinciding with the original norm) if and only if the following three conditions
hold:
(i) The Haagerup tensor product EhA Hc is a Hilbert space6 ;
(ii) The map : EB(H, EhA Hc ) given by (x)() = x is a (complete) isom-
etry;
(iii) For all x E we have (x)(x) A.
If these conditions hold, the (unique7 ) inner product on E is given by

< x | y >= (x) (y).

The proof of the theorem uses a factorization theorem for Hilbert A-modules and
some (rather simple) properties of the Haagerup norm.
Moreover, the inner product < | > on a Hilbert A-module E and the inner
product (. | .) on the Hilbert space EhA Hc are related by the formula (x |
y) = (< y | x > | )H , so EhA Hc coincides with the interior tensor product
E H, where is a (nondegenerate) representation of A on H (so H can be thought
of as a left A-module as well as a right Hilbert C-module). The more general
correspondence between the interior tensor product of Hilbert modules and their
Haagerup tensor product is already mentioned in the previous section.

Example 5. If H is a (separable) Hilbert space and A a C -algebra, one can

form their tensor product HA, which is a pre-Hilbert-A-module defining < a |
b >= ( | )a b. Its completion is a Hilbert A-module (it is in fact the interior
tensor product of the right A-module A, with the inner product < a | b >= a b, and
the Hilbert space H considered as a right Hilbert C-module). This Hilbert module
coincides with the Hilbert A-module of all countableP sequences in A with natural
operations and the inner product < (an ) | (bn) >= nN an bn. The module is usu-
ally denoted HA and called the standard Hilbert A-module. It obviously generalizes
the l2 Hilbert space. The standard Hilbert A-module HA has great importance in the
theory of Hilbert modules and their applications, e.g. in K- and KK-theory. One
of its important properties is e.g. that BA (HA ) is C -isomorphic to the multiplier
algebra8 M (A K(H)).

5 A representation of a C -algebra A on a Hilbert space H is called nondegenerate if the

closure of the linear span of all (a) (a A, H) equals H.

6 Hc denotes the Hilbert column space B(C, H) with its natural operator space structure. This

is one way to turn a Hilbert space into an operator space.

7 E.C. Lance proved (Unitary operators on Hilbert C -modules, Bull. London Math. Soc. 26

(1994), 363-366) that there is a 1 1 correspondence between norm and inner product on a Hilbert
C -module.
8 The multiplier algebra of a C -algebra is the C -algebra M (A) = {a A : ax, xa

A for all x A} (this is just one possible realization of the more abstract definition). When
considering A as a Hilbert A-module, it turns out that M (A) is C -isomorphic to BA (A).
268 ckler
F. M. Bru

References
[1] S. D. Allen, A. M. Sinclair, R. R. Smith The ideal structure of the
Haagerup tensor product of C -algebras, J. reine angew. Math. 442(1993), 111-
148.
[2] D. P. Blecher, Tensor products of operator spaces II, Can. J. Math. 44(1992),
7590.
[3] D. P. Blecher, A new approach to Hilbert C -modules, preprint, 1995.
[4] D. P. Blecher, V. I. Paulsen, Tensor products of operator spaces, J. Funct.
Anal. 99(1991) 262292.
[5] D. P. Blecher, R. R. Smith, The dual of the Haagerup tensor product, J.
London Math. Soc. 45(1992), 126144.
[6] E. Christensen, A. M. Sinclair, Representations of completely bounded
multilinear operators, J. Funct. Anal. 72(1987), 151181.
[7] E. Christensen, A. M. Sinclair, A survey of completely bounded operators,
Bull. London Math. Soc. 21(1989), 417448.
[8] J. Dixmier, Les C -algebres et leurs representations, Gauthier-Villars, Paris,
1964.
[9] E. G. Effros, A. Kishimoto, Module maps and Hochschild-Johnson coho-
mology, Indiana Univ. Math. J. 36(1987), 257276.
[10] E. G. Effros, E. C. Lance, Tensor products of operator algebras, Adv. Math.
25(1977), 134.
[11] E. G. Effros, Z.-J. Ruan, Self-duality for the Haagerup tensor product and
Hilbert space factorizations, J. Funct. Anal. 100(1991), 257284.
[12] A. Guichardet, Tensor products of C -algebras, Aarhus University Lecture
Notes Series No.12, 1969.
[13] E. C. Lance, Tensor products and nuclear C -algebras, Proceedings of Sym-
posia in Pure Mathematics, Vol.38 (1982), 379399.
[14] E. C. Lance, Hilbert C -modules A toolkit for operator algebraists, London
Mathematical Society Lecture Note Series 210, Cambridge University Press,
1995
[15] V. I. Paulsen, Completely bounded maps and dilations, Pitman Research
Notes in Mathematics No. 146, Longman, London, 1986.
[16] V. I. Paulsen, R. R. Smith,Multilinear maps and tensor norms on operator
systems, J. Funct. Anal. 73(1987), 258276.
[17] Z.-J. Ruan, Subspaces of C -algebras, J. Funct. Anal. 76(1988), 217230.
[18] M. Takesaki, On the cross-norm of the direct product of C -algebras, Tohoku
Math. Journ. 16(1964), 111122.
[19] N. E. Wegge-Olsen, C -algebras and K-theory for Pedestrians, Oxford Uni-
versity Press, 1993.