1. Introduction
The theory of completely positive maps plays an important part in operator algebras, operator spaces, and extensions of C*-algebras. Many fundamental concepts and theorems are defined and proved via completely positive maps respectively, such as nuclearity, invertible extension, Stinespring’s Theorem ( [1] [2]), Voiculescu’s Theorem ( [3]), etc.
On the other hand, as an effective tool to study the structure of C*-algebras and to classify C*-algebras, the theory extensions of C*-algebras originated from Busby’s work in 1960’s ( [4]). Subsequently, Brown, Douglas and Fillmore established their famous BDF theory ( [5] [6]) to study essentially normal operators on a separable infinite-dimensional Hilbert space and extensions of C*-algebra C(X) by compact operators, where C(X) is the C*-algebra of continuous functions on a compact metric space X. Since then, the theory of extensions of C*-algebras has developed rapidly, and becomes an important invariant for classifying C*-algebras together with K-theory and KK-theory (see [1] [7] [8] [9], etc.).
As we know, an extension of C*-algebras is determined by its Busby invariant with respect to unitary equivalence, so to an extent classifying extensions of C*-algebras is a sort of classifying homomorphisms between C*-algebras. It should be pointed out that the KK-groups were defined via homomorphisms in this way at the beginning ( [8]), and it was already used to classify homomorphisms (see [10] [11] [12], etc.). Completely positive maps can be seen as generalization of homomorphisms and what is particularly important is that Ext-groups were characterized by completely positive maps, so it is natural to consider classifying completely positive maps.
This note is engaging in classifying completely positive maps between certain C*-algebras. Specifically, several invariants for classifying completely positive maps are introduced. As a main result, one of them is isomorphic to the Ext-group of C*-algebra extensions. In addition, this invariant induces a functor from C*-algebras to abelian groups which is split-exact.
2. Preliminaries
In this section, we need to recall some notations and definitions for C*-algebras and extensions. One can also see [1] [7] [13] [14] [15] for more details.
Suppose that D is a C*-algebra. Recall that
is an inner isomorphism, if there are isometries
in D with
and
for
, such that
, namely,
for
, where
. Suppose that
and
are such elements. Then
and
, and hence
is a unitary in D.
Let A and B be C*-algebras. An extension of A by B is a short exact sequence
Denote this extension by e or
.
The extension
is called trivial, if the above sequence splits, i.e. if there is a homomorphism
such that
.
For an extension
, there is a unique homomorphism
such that
, where
is the multiplier algebra of B, and
is the inclusion map from B into
. The Busby invariant of
is a homomorphism
from A into the corona algebra
defined by
for
, where
is the quotient map, and
such that
.
Two extensions
and
are called (strongly) unitarily equivalent, denoted by
, if there exists a unitary
such that
for all
. Denote by
or
the set of (strong) unitary equivalence classes of extensions of A by B.
Let H be a separable infinite-dimensional Hilbert space and
the ideal of compact operators in
. If B is a stable C*-algebra (i.e.
, where
is the tensor product operation), then the sum of two extensions
and
is defined to be the homomorphism
, where
and the isomorphism
is induced by an inner isomorphism from
onto
, where
is the direct sum of C*-algebras.
The above sets of equivalence classes of extensions are commutative semigroups with respect to this addition when B is stable. One can similarly define these semigroups replacing B by
if B is not stable. Denote by
the quotient of
by the subsemigroup of trivial extensions.
3. Main Result
Suppose that D is a unital properly infinite C*-algebra, namely, there are two elements
such that
For every C*-algebra A, we denote by
the set of all completely positive maps from A into D.
Definition 3.1. Two elements
are called (unitarily) equivalent, denoted by
, if there is a unitary
such that
.
It is easy to check that
is an equivalence relation on
. Denote by
the equivalence class of
.
Definition 3.2.
is the equivalence classes in
under the equivalence relation
, i.e.
.
Now we can define a diagonal addition in
as follows:
where
is the inner isomorphism with
.
Proposition 3.3. Equipped with the above addition,
is an abelian semigroup.
Proof. The following is similar to the proof of ( [7], 3.2.3), and we give it here for the sake of completeness.
Suppose that
and
are in
such that
and
. Then there are unitary elements
such that
and
. Thus
Since
is a unitary in D, we have
It follows that the addition is well-defined.
Let
and
be two inner isomorphisms from
onto D with
and
. Then
and hence,
Therefore, the addition is independent of the choices of inner isomorphisms.
Suppose that
. Then
Let
Then
is an inner isomorphism from
onto D and hence
Suppose that
and let
be isometries with
and
. One can check the following computation:
and
Put
and
. Then
and
are two inner isomorphisms from
onto D. Note that
Since
is a unitary in D, it follows that
This completes the proof of associativity.
Therefore,
is an abelian semigroup.
Remark 3.4. Suppose that
. We write
or
Definition 3.5. Let
be the set of homomorphisms from A into D. An element is called degenerate in
if it is also in
.
Definition 3.6. Two elements
are called equivalent, denoted by
, if there are
such that
.
Then
is an equivalence relation. The equivalence class of
is denoted by
, or by
simply.
Definition 3.7.
is the equivalence classes in
under the equivalence relation
, i.e.
.
We define an addition + in
by
To see the addition is well-defined, suppose that
and
. Then there exist
such that
and hence
Since
Similarly,
It follows that the addition is well-defined.
Remark 3.8. 1) Suppose that
. Then
if and only if there exist
such that
is unitarily equivalent to
.
2) Suppose that
. Then
is the neutral element in
if and only if for each
there exist
such that
is unitarily equivalent to
.
Theorem 3.9.
is a unital abelian semigroup. An element
is the unit of
if and only if
.
Proof. Suppose that
. Then
It follows that
is a semigroup. It is clear that
is abelian.
Let
. For any
, take
and set
. Then
that is,
. Since
and
, we have
by Remark 3.8. Hence
is the unit of
.
Suppose that
such that
is the unit of
. For
,
is also the unit of
, and hence
. Thus there exist
such that
. Note that
is unitarily equivalent to
. Since
and
are both homomorphisms,
is a homomorphism. Furthermore,
is in
, and hence
is in
.
Remark 3.10. The only invertible element in
is the unit. In fact, suppose that
is an invertible element in
with the inverse
. Then
is the unit and
is a homomorphism by Theorem 3.9. Thus,
is also a homomorphism. Therefore
is in
. It follows that
is the unit.
Definition 3.11. Let B be a closed ideal of D and
the quotient map. We define a relation
on
as follows: for
, we write
if there exist
such that
,
, and
.
Suppose that
,
. Then there exist
such that
Since
, there exist
such that
. Thus there is a unitary
such that
Put
Then we have
and
where
is the inner isomorphism from
onto
induced by
.
It follows that
is transitive, and hence
is an equivalence relation on
. Denote the equivalence class of
by
, or by
simply.
Let
. It is natural that we define an addition in
as follows:
Remark 3.12. The addition defined in Definition 3.11 is well-defined: for
and
, there exist
,
,
,
such that
,
,
,
,
, and
. Then
and hence
It is easy to see that
is the unit of
. Thus
is a unital abelian semigroup. In particular, for
, we have
; and for
, we have
.
Definition 3.13. Let
be the set of invertible elements in
. Then
is an abelian group.
Theorem 3.14. Let
be in
. Then
in
if and only if there exist
and a unitary
such that
Proof. Suppose that
in
. Since
, there exist
such that
,
and
. Hence, by Theorem 3.9, we have
. Since
, there exist
such that
. Then there is a unitary
such that
Conversely, suppose that there exist
and a unitary
such that
Set
and
. Then
are both inner isomorphisms from
onto D. Therefore
Note that
. Thus
.
Remark 3.15. Suppose that
in
. By Theorem 3.14, we have
where
and
is induced by the quotient map
.
Set
, we have
and
.
Theorem 3.16. Let
. Then
if and only if there is
and
such that
Proof. Suppose that
with the inverse
. Let
. Since
, there exist
in
and a unitary
in
such that
Set
Then
is an inner isomorphism from
onto
and
is an inner isomorphism from
onto
. It follows that
Set
and
Then we have
Conversely, since
Then
Thus
Proposition 3.17. Suppose that
such that
. Then
is a homomorphism.
Proof. Suppose that
and
is the inverse of
. Set
By Theorem 3.14, there exist
and
such that
Hence
is a homomorphism, and thus
Set
and
. Then
that is,
Put
. Then
. Hence
This implies that
, and furthermore
. It follows that
is a homomorphism.
Lemma 3.18. ( [7], 3.2.9) Suppose that A is a separable C*-algebra and B is a stable C*-algebra. Let
. Then the following three statements are equivalent:
1)
is invertible in
.
2) There exists
such that
.
3) There exists
such that
It is well known that
and
are innerly isomorphic if B is a stable C*-algebra. Then we have the following result.
Theorem 3.19. Let A and B be C*-algebras with B stable. Then
Proof. Note that the condition that A is separable is not necessary in the proof of (1)
(2) in Lemma 3.18 ( [7], 3.2.9). Suppose that
such that
is invertible in
. Then there exists
such that
. We define a map
by
, where
.
1) Prove that
is well-defined.
Suppose that
such that
. Then there exist
such that
and
. If
, there exist
and
such that
Hence,
Since
is an inner isomorphism,
Then
, and hence
is well-defined.
2) Prove that
is a homomorphism.
Note that
Since
we have
It follows that
is a homomorphism.
3) Prove that
.
Suppose that
is an invertible element with the inverse
. Then we have
Therefore,
is invertible.
4) Prove that
is injective.
Suppose that
and
, where
and
.
If
in
, then there exist
such that
,
and
. Therefore there exist
and unitary elements
such that
Put
Then we have
and
Thus,
Set
One can check that
is a unitary in
. Then we have
It follows that
Therefore,
is injective.
5) Prove that
is surjective.
Suppose that
. Then by Theorem 3.16 there exist
and an inner isomorphism
with
and
, such that
. Since
and
, by Theorem 3.17,
and
are homomorphisms and
Thus
and
. This implies that
is surjective.
Similar to Lemma 3.18, we have the following result.
Corollary 3.20. Let A and B be C*-algebras with B stable and let
. Consider the following three statements:
1)
is invertible in
.
2) There exists
such that
.
3) There exist
and
such that
Then (1)
(3)
(2).
Proposition 3.21. Let A and C be C*-algebras and
. Then
1) The map
defined by
is a semigroup homomorphism.
2) The map
defined by
is a unital semigroup homomorphism. Furthermore, it is a group homomorphism from
into
.
Theorem 3.22. Let
be the category of C*-algebras and
the category of abelian semigroups. Define
by
and
for any
and
. Then
is a contravariant functor from
to
.
Proof. 1) For a C*-algebra A and
, we have
. Then
is the unit of
.
2) Let
and
. Set
. Then
Thus
is a contravariant functor.
Corollary 3.23. Let
be the category of abelian groups. Then
induces a contravariant functor
from
into
by
, and from
into
by
.
For a short exact sequence of C*-algebras
, the functor
from
to
is not exact, and it is even not split-exact. The following is a counterexample.
Example 3.24. Suppose that H is an infinite dimensional separable Hilbert space. Let
,
,
and
. Then
. Let
be the inclusion map and let
be the quotient map. Then the exact sequence
is split.
Take a nonzero element
. We define a map
by
and
. Then
and
. If
for some
, then there exist
and a unitary
such that
Put
Since
,
. Note that
are positive if e is positive in E. It follows that
Therefore
. Furthermore,
since there is a sequence in
which is convergent to I in the strong operator topology on
. Then
. Hence
and
. Therefore,
Since
is a homomorphism,
is also a homomorphism. However,
is not a homomorphism by the definition of
. Otherwise, if
is a homomorphism from
to
, then it follows that
since a completely positive map preserves self-adjoint elements. This is in contradiction to the fact that
.
Theorem 3.25. Suppose that
is a split short exact sequence, then
is also a split short exact sequence.
Proof. Since
, we have
.
Assume that
. For any
, let
and
. Then
. Note that
is invertible and
Hence,
. Similarly,
is also invertible.
Suppose that the inverses of
and
are
and
respectively. Let
. Now we show that
is the unit. Suppose that
and
such that
and
, where
and
are homomorphisms. Then
Since
is a homomorphism,
is the unit of
. Since
,
. Then
and hence
is the inverse of
. Therefore,
is the inverse of
. Since
,
. Thus,
Suppose that
. Then
, and there exist
and
such that
and
. Hence,
. Note that
and
. It follows that
and
is an injective homomorphism.
Suppose that
. Then we have
Therefore,
is surjective.
Define
Then
. Finally,
is a split short exact sequence.
Remark 3.26. For any C*-algebra B, we can define
,
, etc., to be
,
, respectively. Since for any stable C*-algebra its multiplier algebra is properly infinite, these invariants are well-defined.
Acknowledgements
This work was supported by the Shandong Provincial Natural Science Foundation (Grant No. ZR2020MA009).