1. Introduction
The systematic study of various tensor norms on the tensor product of Banach spaces was begun with the work of Schatten [1] , which was later studied by Grothendieck in the context of locally convex topological space. One of the most natural and useful tensor norm is the Banach space projective tensor norm. For a pair of arbitrary Banach spaces
and
and
an element in the algebraic tensor product
, the Banach space projective tensor norm is defined to be

will denote the completion of
with respect to this norm. For operator spaces
and
, the operator space projective tensor product of
and
is denoted by
and is defined to be the completion of
with respect to the norm:
![]()
the infimum taken over
and all the ways to write
, where
,
,
and
and
.
Kumar and Sinclair defined an embedding
from
into
, and using the non-com- mutative version of Grothendieck’s theorem to the setting of bounded bilinear forms on
-algebras, it was
shown that this embedding satisfies
([2] , Theorem 5.1). Recently, analogue of Grothen-
dieck’s theorem for jointly completely bounded (jcb) bilinear forms was obtained by Haagerup and Musat [3] . Using this form for jcb, the canonical embedding for the operator space projective tensor product have been
studied by Jain and Kumar [4] , and they showed that the embedding
from
into
satisfies
.
In Section 2, an alternate approach for the bi-continuity of the canonical embedding of
into
has been presented with an improved constant. Our proof essentially uses the fact that the dual of the Banach space projective tensor norm is the Banach space injective tensor norm. We also consider the corresponding operator space version of this embedding and discuss its isomorphism. As a consequence, one can obtain the equivalence between the Haagerup tensor norm and the Banach space projective tensor norm (resp. operator space projective tensor norm).
In the next section, it is shown that if the number of all closed ideals in one of the
-algebras is finite then every closed ideal of
is a finite sum of product ideals. One can obtain all the closed ideals of
as
,
and
, and the closed
ideals of
as
, where
or
,
for each
, for an infinite dimensional separable Hilbert space
and locally compact Hausdorff topological space
. Similarly, the closed ideal structure of
, where
is any
-algebra and
is the multiplier algebra of
,
being a nonunital, non-element- ary, separable, simple AF
-algebra, can be obtained. We may point that such result fails for
, the minimal tensor product of
-algebras
and
.
Section 4 is devoted to the inner automorphisms of
and
for
-algebras as well as for operator algebras. Recall that the Haagerup norm on the algebraic tensor product of two operator spaces
and
is defined, for
, by
![]()
where infimum is taken over all the ways to write
![]()
where
. The Haagerup tensor product
is defined to be the comple-
tion of
in the norm
[5] .
2. Isomorphism of Embeddings
For Banach spaces
and
and
,
, define a linear map
as
, for
and
. Using ([6] , Proposition 1.2), it is easy to see that
is well defined. Also, clearly this map is linear and contractive with respect to
, and in fact
, and hence can be extended to
with
. A bilinear form
in
is called nuclear if
, and the nuclear norm of
is defined to be
. The
Banach space of nuclear bilinear forms is denoted by
. For
-algebras
and
, consider the canonical map
from
into
, the dual of the Banach space injective tensor product of
and
, defined by
![]()
where
is the natural isometry of
into
,
is as above with
and
,
is the natural inclusion of
into
, the space of integral bilinear forms.
Lemma 2.1 For
-algebras
and
, the canonical map
the space of integral operators from
to
satisfies
for all
. In par-
ticular,
is bi-continuous.
Proof: The inequality of the right hand side follows directly from the definition of
. Let
and
. By the Hahn Banach Theorem, there exists
with
such that
.
Since
, so
, for some
, for all
and
with
. By ([7] , Proposition 2.1(2)), there is a net
of finite rank operators from
to
such that
and
for any
.
Now, for each
, corresponding to
we can associate
. For
, there is
such that
for all
. Thus
. Since
is a finite rank operator, so let
. Choose an Auerbach basis
for
with associated coordinate functionals
in
. Thus, for any
,
,
for
.
By using
, it follows that
,
for
. Therefore, for any
and
,
![]()
where
is the canonical isometric map from
to
. Thus
and so
. Moreover, for
, we have
![]()
Since
is arbitrary, so
. ![]()
Next, we consider the map
defined by
.
Proposition 2.2 For
-algebras
and
, the natural map
is bi-conti-
nuous and
, for all
.
Proof: By the above lemma, we have a map
with
for all
. Also, ([6] , Proposition 3.21) shows that the natural inclusion map
is isometric. We will show that
. For
,
,
and
,
,
since
for
. Thus
. Therefore, by linearity and continuity,
, and hence the map
satisfies
for all
. ![]()
Haagerup proved that every bounded bilinear form on
can be extended uniquely to a separately normal norm preserving bounded bilinear form on
([7] , Corollary 2.4), so we have a continuous iso-
metric map
. Set
![]()
where
is the natural embedding of
into
. Kumar and Sinclair proved that this
embedding is a bi-continuous map with lower bound
([2] , Theorem 5.1). We re-establish its bi-continuity
with an alternate proof and an improved lower bound
.
Theorem 2.3 For
-algebras
and
, the natural embedding
satisfies
for all
.
Proof: We know that the natural embedding
is isometric. Thus, by
the Hahn Banach theorem,
is a quotient map. We will show that
![]()
where
is as in Proposition 2.2. Since
and
are linear and continuous, it suffices to show that
and
agree on
. Note that, for
,
,
and
,
![]()
where
is the bilinear form corresponding to
.
Since
and
so, by Goldstine’s Lemma, there are nets
and
such that
converges to
in
and
converges to
in
. The separate
-continuity of the
bilinear form
and the equality
shows that
. Thus,
. Hence, by Proposition 2.2, we deduce that
. ![]()
Remark 2.4 (i) Note that, for a
-algebra
having Completely positive approximation property, the canonical embedding of
into
is isometric by ([8] , Theorem 3.6) and ([9] , Theorem 3.6). However, for the largest Banach space tensor norm, the embedding
is isometic if one of the
-algebra has the metric approximation property, which follows directly by using ([6] , Theorem 4.14) in the above theorem.
(ii) For a locally compact Hausdorff topological group
, let
and
be the group
- algebra and the reduced group
-algebra of
, respectively. Then, for any
-algebra
and a discrete
amenable group
, the natural embedding of
into
is isometric by ([8] , Theorem 4.2); and for any amenable group
, the natural embedding of
into ![]()
is isometric by ([8] , Proposition 4.1).
(iii) The natural embedding
is isomorphism if
has the approximation property,
has the Radon Nikodym property and every bilinear form on
is nuclear. This follows directly by observing that if
has the Radon Nikodym property then ([6] , Theorem 5.32) gives us
![]()
where
and
denote the Pietsch integral and nuclear operators from
to
, respectively [6] . Clearly, bijectivity follows if we show that
is an onto map. For this, let
so it is nuclear. Since
has the approximation property, so there exists an element
such that
![]()
where
is an isometric isomorphism from
to
([6] , Corollary 4.8). Consider the canonical map
. Of course
on
, and hence by linearity and con- tinuity
.
We now discuss the operator space version of the above embedding, which is already discussed in [4] . Note that in this case the embedding is positive, and becomes an isomorphism under the conditions weaker than that required in case of the Banach space projective tensor product. For operator spaces
and
, an operator from
into
is called completely nuclear if it lies in the image of the map
[10] . The space of completely nuclear operators will be denoted by
. This space has the natural operator
space structure determined by the identification
.
For
-algebras
and
, consider the map
from
into the dual of operator space injective tensor product
given by
![]()
where
is the natural completely isometric map,
and
[10] . Making use of the fact that the dual of the operator space projective tensor norm is the operator space injective ([10] , Proposition 8.1.2) and an application of Grothendieck’s theorem for jcb ([11] , Proposition 1) and the techniques of Lemma 2.1, we obtain the following:
Lemma 2.5 For
-algebras
and
, the canonical map
satisfies
for all
. In particular,
is bi-continuous.
Proposition 2.6 For
-algebras
and
, the natural map
, defined by
, is bi-continuous satisfying
for all
.
Proof. By ([10] , Theorem 15.3.1) we have
is locally reflexive operator space. Therefore, ([10] , Theorem
14.3.1) implies that
can be identified with
, where
denotes the space of
completely integral operators from
to
. Now, the result follows by using the techniques of Proposition 2.2 and ([10] , Proposition 15.4.4).
By ([4] , Proposition 2.5), we have a continuous completely isometric map
. Let
![]()
where
is the natural embedding of
into
. Then clearly
.
For a matrix ordered space
and its dual space
, we define
-operation on
by
,
and
. Note that, for
-algebras
and
,
is a Banach
-algebra ([12] , Proposition 3).
Theorem 2.7 For
-algebras
and
, the natural embedding
is
-preserving positive bounded
map which satisfies
for all
.
Proof: Given
,
,
,
and
,
![]()
On the other hand,
. So in order to prove that
is
- preserving, we have to show that
.
Note that, for
and
,
,
and hence the result follows from the separate
-continuity of the bilinear forms corresponding to
and
.
Now given an algebraic element
, where
is defined as in [13] . For the positivity of
, we have to show that
for
. By ([13] , Theorem 1.9), it suffices to show that if
then
![]()
where
for all
,
and
for all
,
.
Since
is
-dense in
, so given
we obtain a net
which is
-convergent to
. Now note that
. Hence the result follows.
The bi-continuity of the map
follows as in Theorem 2.3. ![]()
Remark 2.8 By ([14] , Theorem 2.2), the natural embedding
is completely isometric if one of the
-algebras has the
MAP.
We now discuss the isomorphism of this embedding. For
, the map
, for
and
, has a unique continuous extension to a map
, with
. The next proposition does not have counterpart in the Banach space context.
Proposition 2.9 For
-algebras
and
, the family
is total on
.
Proof: Suppose that
such that
for all
. Let
with
. Since
, so
for some
, for all
and
, with
. If
is taken in the universal representation of
then
satisfies the ![]()
by ([10] , Theorem 15.1) and ([5] ,
1.4.10). So there exists a net
of finite rank
-continuous mapping from
to
such that
, and
for all
. Thus for
and
, there exists
such that
for all
. Since
, we have
such that
. Since
is a finite rank operator so, as in Lemma 2.1,
for
and
. Thus, for
![]()
where
,
,
, and
, a norm convergent representation
in
[10] ,
. Given
for all
. Therefore,
for any
. Thus
, giving that
for all
, and hence
. ![]()
In particular, the map
defined above is 1-1. Thus
.
Now, as in Remark 2.4(iii), we have the following:
Corollary 2.10 Let
and
be
-algebras such that every completely bounded operator from
to
is completely nuclear and the map
defined in the Proposition 2.6 is onto. Then the natural embedding
is an isomorphism map.
Remark 2.11 The embedding in the case of the Haagerup tensor product turns out to be completely isometric, which can be seen as below. For operator spaces
,
, using the fact that
and ([5] ,
1.6.7), the
map
is completely isometric. Set
![]()
where
. Then, clearly
. By the self-duality of the Haagerup norm, the map
is completely isometric. As in Theorem 2.3,
,
where
is the completely isometric map from
to
, which further gives
for any
. Thus
is completely isometric.
3. Closed Ideals in ![]()
It was shown in ([4] , Theorem 3.8) that if
or
is a simple
-algebra then every closed ideal of
is the product ideal, i.e. of the form
or
for closed ideal
of
and
of
. In the following, we generalize this result to the
-algebra which has only a finite number of closed ideals. More precisely, it is shown that if one of the
-algebras
and
has only finitely many closed ideals, then
every closed ideal in
is precisely of the form
, for some
and closed ideals
in
,
in
,
. Thus obtaining the complete lattice of closed ideals of
,
,
, where
is an infinite dimensional separable Hilbert space,
is a locally compact Hausdorff space,
is any
-algebra and
is the multiplier algebra of
,
being a nonunital, non-elementary, separable, simple AF
-algebra ([15] , Theorem 2). We would like to remark that in [4] the lattice of closed ideals of
has already been explored.
Proposition 3.1 Let
and
be
-algebras and
a closed ideal in
. If
, the closure of
in
, then
, where
is the natural map from
into
.
Proof: Since
so there exists a sequence
such that
as
tends to infinity. Consider the identity map
and
. Of course,
on
, and hence by continuity
. Thus
and so
by ([12] , Theorem 6). ![]()
The following lemma can be proved as a routine modification to the arguments of ([16] , Lemma 1.1).
Lemma 3.2 For closed ideals
of
and
of
,
.
In order to prove our main result. We first investigate the inverse image of product ideals of
for
-algebras
and
, which is largely based on the ideas of ([10] , Proposition 7.1.7)
Proposition 3.3 For
-algebras
, and
and the complete quotient maps
,
. Let
and
be closed ideals in
and
, respectively. Then
![]()
Proof: By ([4] , Proposition 3.2) and the Bipolar theorem, it suffices to show that
![]()
Let
then
and
. Since
, so
for some
, for all
and
. Define a bilinear map
as
![]()
where
and
. Clearly,
is well defined. Note that, for
,
and
, we have
. For any
, there are
and
with
,
such that
. We can find
such that
,
. By defini- tion, we may write
and![]()
where
,
both have norm
1. Thus
, and so
. This shows that
is jcb bilinear form. Thus it will determine a
. We have
for all
and
. This implies that
on
, and so by continuity
. Now let
. We may assume that
. Then
and
. So
with
,
and
,
[10] . Since
and
are complete quotient maps and
, so it follows that
. Hence
![]()
Since the annihilator is reverse ordering, so converse is trivial. ![]()
Now we are ready to prove the main result.
Theorem 3.4 If
and
are
-algebras such that number of closed ideals in
is finite. Then every closed ideal in
is a finite sum of product ideals.
Proof. Proof is by induction on
, the number of closed ideals in
counting both
and
. If
then the result follows directly by ([4] , Theorem 3.8). Suppose that the result is true for all
- algebras with
. Let
be a
-algebra with
.
Since there are only finitely many closed ideals in
so there exists a minimal non-zero closed ideal, say
, which is simple by definition. Let
be a closed ideal in
then
is a closed ideal in
.
So it is equal to
for some closed ideal
in
by ([4] , Theorem 3.8). Consider the closed ideal
,
the closure of
in
, where
is an injective map ([11] , Theorem 1). Then
for some closed ideal
in
by ([17] , Proposition 5.2). We first show that
. Since the map
is injective so
. Thus
,
which by using ([18] , Corollary 4.6), ([19] , Proposition 4), and Lemma 3.2, gives that
and so
. To see the equality, let
. Take any
then
so it belongs to
by Proposi- tion 3.1. Thus
. Hence
.
As in ([17] , Theorem 5.3),
for
. Thus
by Lemma 3.2. Since
cannot contain
, so
. Thus
, which is a closed
ideal in
, is a finite sum of product ideals by induction hypothesis. Let
then clearly
contains
. Corresponding to the complete quotient map
, we have a quotient map
with kernel
and
is a closed ideal of
([19] , Lemma 2).
Also
and so by the induction hypothesis
![]()
where
and
are closed ideals in
and
, for
, respectively. Thus, by ([19] , Lemma 2)
and Theorem 3.3,
. So
is a finite sum of product ideal and hence closed by ([4] , Proposition 3.2).
We now claim that
.
Let
. Since the closed ideal
has a bounded approximate identity
so there exist
such that
and
belongs to the least closed ideal of ![]()
containing
([20] ,
11, Corollary 11). This implies that
so
. Hence
. Therefore
is a finite sum of product ideals.
4. Inner Automorphisms of ![]()
For unital
-algebras
and
, isometric automorphism of
is either of the form
or
, where
,
,
and
are isometric isomorphisms ([11] , Theorem 4). In the following, we characterize the isometric inner
-automorphisms of
completely.
Proposition 4.1 For unital
-algebras
and
, the map
is inner automorphism of
(resp.
) if and only if
is inner automorphism of
and
is inner automorphism of
.
Proof: Suppose that
is implemented by
. We will show that
is implemented by
, where
is
-homomorphism from
into
[11] . It is easy to see that
.
So, for
,
. As
is
-
dense in
, so
is implemented by
. Hence the result follows from ([21] , Theorem 1). Converse is trivial. ![]()
We now characterize the isometric inner automorphism of
for
-algebras
and
other than
.
Theorem 4.2 For unital
-algebras
and
other than
for some
, the isometric inner
- automorphism of
is of the form
, where
and
are inner
-automorphisms of
and
, respectively.
Proof: Suppose that
is the isometric inner
-automorphism of
. So
![]()
where
and
are
-automorphisms of
and
, respectively or
![]()
where
and
are
-isomorphisms,
is a flip map [11] . In view of Proposition 4.1, it suffices to show that the second case will never arise for
-algebras
and
other than
. Let
be a proper closed ideal in
and
, which is a closed ideal in
by ([12] , Theorem 5). Since
is inner so it preserves
. Thus, for any
,
. Therefore
[19] , which further gives that
. Hence
and so
is simple. Similarly, one can show that
is simple. By hypothesis there exists
which implements
so
that
for all
and
. Choose
such that
,
, and
. Thus, for
and
, we have
, hence
.
Now choose
such that
. Therefore,
for all
. Take any
,
being an isomorphism, there exists a unique
such that
. Thus
for any
. Now define a finite dimensional subspace
of
by
![]()
The above inequality implies that
, where
is the closed ball center at
and radius
. If
is proper then Riesz Lemma implies that for
there exists
such that
and
. Since
for any
, so we can choose
such that
, and, because
, a contradiction arises. Therefore,
. Thus, by the classical Wedderburn-Artin
Theorem,
for some
. Similarly,
for some
. ![]()
However, by ([11] , Theorem 5), for unital
-algebras
and
with at least one being non-commutative, isometric inner automorphism of
is of the form
, where
and
are inner automorphisms of
and of
, respec- tively.
Corollary 4.3 For an infinite dimensional separable Hilbert space
, every inner automorphism of
is of the form
, where
and
are inner automorphisms of
.
We now give an equivalent form of Proposition 4.1 in case of operator algebras. For operator algebras
and
, we do not know if
(or
) is inner then
and
are inner or not. However, if one of the automorphism is an identity map then we have an affirmative answer for the Haagerup tensor product. In order to prove this, we need the following results.
Proposition 4.4 For operator spaces
and
, the family
is total on
.
Proof: For
, assume that
for all
. We can assume that
. Therefore,
for
a norm convergent representation in
, where
and
are strongly independent with
and
. Then we have
for all
. From the
strongly independence of
, choose linear functionals
such that
![]()
where
are the standard basis for
by the equivalent form of ([17] , Lemma 2.2). Thus
and so
. Because
was arbitrary, we conclude that
for each
, hence
.
Corollary 4.5 For operator algebras
and
, if
and
are completely contractive automorphisms of
and
, respectively. Then
is a completely contractive automorphism of
.
Proof: By the functoriality of the Haagerup tensor product, the map
is comple-
tely contractive. One can see that
is an algebra homomorphism. Let
be a norm con-
vergent representation in
. Since
and
are bijective maps, so there exist unique
and
, for each
, such that
. By [22] , there is a new norm on
and
with respect
to that
and
become a new operator algebras, say
and
, and the natural maps
from
to
,
from
to
and their inverses are completely bounded, and the maps
and
are completely isometric. Therefore,
is completely isometric, so for all positive integers ![]()
![]()
This shows that the partial sums of
form a Cauchy sequence in
, and so we may define an element
. Then, clearly
. Thus the map
is onto. To prove the injectivity of the map
, let
for
. Then, for
a norm con- vergent representation in
, we have
. Thus, for any
,
. But
is one-to-one, so
. Now Proposition 4.4 yields that
. Again by applying the same technique we obtain
. ![]()
By the above corollary, for operator algebras
and
and automorphisms
of
and
of
, it is clear that if
and
are inner then
is.
In the following, by a
-reduced operator algebra we mean an operator algebra having isometric involution with respect to which it is
-reduced, and for any
-reduced operator algebra
having approximate identity, we denote by
the set of all pure states of
.
Corollary 4.6 For
-reduced operator algebra
having approximate identity and any operator algebra
, the family
is total on
.
Proof: Using ([23] , Proposition 2.5.5), we have
, where
is the set of continuous positive forms on
of norm less than equal to 1. Therefore, if
for all
then ![]()
for all
. Thus
for any
and
. Since the algebra
is
-reduced, so
it admits a faithful
-representation, say
, on some Hilbert space, say
. For a fix
in the closed unit ball of
, define
as
for
. One can easily verify that
. As
is
faithful so
is one-to-one. Therefore,
for any
and hence the result follows from Proposition 4.4. ![]()
Corollary 4.7 For any operator algebra
and
-reduced operator algebra
having approximate identity, the family
is total on
.
The following can be proved on the similar lines as those in ([21] , Lemma 2) by using ([23] , Proposition 2.5.4), so we skip the proof.
Lemma 4.8 For unital Banach
-algebra
and any Banach algebra
and a pure state
of
, we have
for
and
(Similarly, for any Banach algebra
and unital Banach
-algebra
,
for
and
,
).
Theorem 4.9 Let
and
be unital operator algebras. Suppose that
is
-reduced and
has a completely contractive outer automorphism. Then
has a completely contractive outer automorphism.
Proof: Let
be a completely contractive outer automorphism. Define a map
from
into
as
. By Corollary 4.5,
is a completely contractive automorphism of
. Assume that
is a inner automorphism implemented by
. Then
. As
so we can find the pure state
on
such that
by Corollary 4.7. Let
. Note that for any
we have
. This implies that
, the relative commutant of
in
, which is
by ([18] , Corollary 4.7). For
,
by the module property of the slice map. Since
is invertible, so
is invertible by Lemma 4.8. Therefore,
and hence
is inner, a contradiction. Thus
is an outer automorphism. ![]()
NOTES
*2010 Mathematics Subject Classification. Primary 46L06, Secondary 46L07, 47L25.