Article
Kyungpook Mathematical Journal 2021; 61(1): 111137
Published online March 31, 2021
Copyright © Kyungpook Mathematical Journal.
The Universal Property of Inverse Semigroup Equivariant KKtheory
Bernhard Burgstaller
Departamento de Matematica, Universidade Federal de Santa Catarina, CEP 88.040900 FlorianÓpolisSC, Brasil
email : bernhardburgstaller@yahoo.de
Received: January 23, 2020; Revised: June 11, 2020; Accepted: October 5, 2020
Abstract
 Abstract
 1. Introduction
 2. Equivariant
KK theory  3. The Unitization of a
G algebra  4. The Splitexactness, Stability and Homotopy Invariance of
KK^{G}  5. Cocycles
 6. The Isomorphism
u_{#}  7. The Cocyle Set
${\mathbb{E}}^{G}(A,B)$  8. The Isomorphism
$\Phi $  9. The Map Ψ
 10. The Abelian Group Homomorphism
Ψ'  11. The Natural Transformation ξ
 12. The Universality Theorem
 13. Nonunital Inverse Semigroups
 Acknowledgements.
 References
Higson proved that every homotopy invariant, stable and split exact functor from the category of C^{*}algebras to an additive category factors through Kasparov's KKtheory. By adapting a group equivariant generalization of this result by Thomsen, we generalize Higson's result to the inverse semigroup and locally compact, not necessarily Hausdorff groupoid equivariant setting.
Keywords: universal property, stable split exact homotopy functor, KKtheory, inverse semigroup, locally compact groupoid, nonHausdorff.
1. Introduction
 Abstract
 1. Introduction
 2. Equivariant
KK theory  3. The Unitization of a
G algebra  4. The Splitexactness, Stability and Homotopy Invariance of
KK^{G}  5. Cocycles
 6. The Isomorphism
u_{#}  7. The Cocyle Set
${\mathbb{E}}^{G}(A,B)$  8. The Isomorphism
$\Phi $  9. The Map Ψ
 10. The Abelian Group Homomorphism
Ψ'  11. The Natural Transformation ξ
 12. The Universality Theorem
 13. Nonunital Inverse Semigroups
 Acknowledgements.
 References
In [3], Cuntz noted that if
In [12], Thomsen generalized Higson's findings to the group equivariant setting by replacing everywhere in the above statement algebras by equivariant algebras, *homomorphisms by equivariant *homomorphisms and
In this note we extend Higson's universality result to the inverse semigroup equivariant setting.
More specifically, the following will be shown. Let
Let
Let
Proposition 1.1.
Let
Theorem 1.2.
Let
Theorem 1.3.
Let
Corollary 1.4.
The above results are also valid for countable nonunital inverse semigroups
A very brief overview of this note is as follows. (We give further short summaries at the beginning of each section).
In Section 2 we briefly recall the definitions of inverse semigroup equivariant
Sections 411 present an adaption of the content of Thomsen's paper [12]; this goes mostly without saying. Section 12 is essentially taken from Higson's paper [4].
We finally would like to clarify the implications of our results to groupoid equivariant
Actually, all our results of this paper work for all topological, not necessarily Hausdorff groupoids equivariant
Indeed, let
A
But the introduced actions, or similar constructions are just:

(1) Cocycles: The Definition 5.1 has to be replaced by the analogous Definition 1.5 below.

(2) Unitization: One replaces Definition 3.3 by the corresponding definition of [8].

(3) Direct sum, internal, external tensor product: It is clear that these constructions are also continuous for groupoids.

(4) For an element
$[T,\mathcal{E}]\in K{K}^{\mathcal{G}}(A,B)$ one has the condition that the bundle$g\mapsto g({T}_{s(g)}){T}_{r(g)}\in \mathcal{K}({\mathcal{E}}_{r(g)})$ is in${r}^{*}\mathcal{K}(\mathcal{E})$ . Here one has also additionally to check continuity. 
(5) One has also to consider
C_{0}(X) structure of the base spaceX of the groupoid, which recall is homomorphims${C}_{0}(X)\to \mathcal{Z}\mathcal{M}(A)$ (center of the multiplier algebra ofA ). Any possible necessary computations one makes analogously as we do it for the restrictedG action$E\to \mathcal{Z}\mathcal{M}(A)$ to the idempotent elementsE of the inverse semigroup by replacinge ∈ E by$f\in {C}_{0}(X)$ .
Definition 1.5.
Let
in
In this way it is (almost) clear that the results of this paper hold also in the locally compact, not necessarily Hausdorff groupoid equivariant setting.
Corollary 1.6.
Let
2. Equivariant KK theory
 Abstract
 1. Introduction
 2. Equivariant
KK theory  3. The Unitization of a
G algebra  4. The Splitexactness, Stability and Homotopy Invariance of
KK^{G}  5. Cocycles
 6. The Isomorphism
u_{#}  7. The Cocyle Set
${\mathbb{E}}^{G}(A,B)$  8. The Isomorphism
$\Phi $  9. The Map Ψ
 10. The Abelian Group Homomorphism
Ψ'  11. The Natural Transformation ξ
 12. The Universality Theorem
 13. Nonunital Inverse Semigroups
 Acknowledgements.
 References
Our reference for inverse semigroup equivariant
Definition 2.1.
Let
We denote the set of idempotent elements of
Definition 2.2.
A
Throughout we shall identify the
Definition 2.3.
A

(a)
$\u27e8{U}_{g}(\xi ),{U}_{g}(\eta )\u27e9={\beta}_{g}(\u27e8\xi ,\eta \u27e9)$ 
(b)
U_{g}(ξ b) = U_{g}(ξ) β_{g}(b) 
(c)
${U}_{g{g}^{1}}(\xi )b=\xi {\beta}_{g{g}^{1}}(b)$
for all
Lemma 2.4.
In the last definition, automatically
For a positive approximate unit
so that
for all
Definition 2.5.
Given a
It is useful to notice that every
Lemma 2.6.
Let
Definition 2.7.
A *homomorphism
Definition 2.8.
A
Definition 2.9.
Let
for all
We equip the multiplier algebra of a
Definition 2.10.
Given a
We also write
Definition 2.11.
Write
Since every
Definition 2.12.
Let
Note that then
Lemma 2.13.
Let
for all
Lemma 2.14.
Let
Let us point out that we have all the necessary techniques for inverse semigroup equivariant
From now on all
3. The Unitization of a G algebra
 Abstract
 1. Introduction
 2. Equivariant
KK theory  3. The Unitization of a
G algebra  4. The Splitexactness, Stability and Homotopy Invariance of
KK^{G}  5. Cocycles
 6. The Isomorphism
u_{#}  7. The Cocyle Set
${\mathbb{E}}^{G}(A,B)$  8. The Isomorphism
$\Phi $  9. The Map Ψ
 10. The Abelian Group Homomorphism
Ψ'  11. The Natural Transformation ξ
 12. The Universality Theorem
 13. Nonunital Inverse Semigroups
 Acknowledgements.
 References
We shall later need a unitization of a
Definition 3.1.
Let
Lemma 3.2.
Let
where
Then the linear direct sum
for all
We leave it to the reader to show that
Since
for
defines a *isomorphism (
If
(
Note that the canonical projection
Finally, a straightforward check shows that
Definition 3.3.
For a
Because
4. The Splitexactness, Stability and Homotopy Invariance of KK^{G}
 Abstract
 1. Introduction
 2. Equivariant
KK theory  3. The Unitization of a
G algebra  4. The Splitexactness, Stability and Homotopy Invariance of
KK^{G}  5. Cocycles
 6. The Isomorphism
u_{#}  7. The Cocyle Set
${\mathbb{E}}^{G}(A,B)$  8. The Isomorphism
$\Phi $  9. The Map Ψ
 10. The Abelian Group Homomorphism
Ψ'  11. The Natural Transformation ξ
 12. The Universality Theorem
 13. Nonunital Inverse Semigroups
 Acknowledgements.
 References
The aim of this section is the proof of Proposition 1.1.
Lemma 4.1.
Let
an exact sequence of
where
Definition 4.2.
We recall that
Lemma 4.3.
The functor
where the
where
by the
for all
Lemma 4.4.
The functor
its image under
Similarly, there is an element
It was checked in [7] that
Let
which is in the image of
is split exact.
By Lemmas 4.3 and 4.4 and the evidence of homotopy invariance we obtain the main result of this section:
Corollary 4.5.
Proposition 1.1 is true.
5. Cocycles
 Abstract
 1. Introduction
 2. Equivariant
KK theory  3. The Unitization of a
G algebra  4. The Splitexactness, Stability and Homotopy Invariance of
KK^{G}  5. Cocycles
 6. The Isomorphism
u_{#}  7. The Cocyle Set
${\mathbb{E}}^{G}(A,B)$  8. The Isomorphism
$\Phi $  9. The Map Ψ
 10. The Abelian Group Homomorphism
Ψ'  11. The Natural Transformation ξ
 12. The Universality Theorem
 13. Nonunital Inverse Semigroups
 Acknowledgements.
 References
When we shall later introduce a Cuntz picture of Kasparov theory, the corresponding transformation produces a
Definition 5.1.
Let
hold in
Lemma 5.2.
Let

(a) Then we have
$${\alpha}_{g{g}^{1}}={u}_{g}^{*}{u}_{g}={u}_{g}{u}_{g}^{*}={u}_{g{g}^{1}}\in \mathcal{M}(A).$$ In particular, everyu_{g} is a partial isometry and the source and range projection ofu_{g} both agree with${\alpha}_{g{g}^{1}}$ and are in the center of$\mathcal{M}(A)$ . 
(b) In particular, every
u_{e} = α_{e} is a selfadjoint projection in the center of$\mathcal{M}(A)$ for alle ∈ E . 
(c) We may replace the second identity of (5.1) by the identity
$$\overline{{\alpha}_{g}}({u}_{{g}^{1}})={u}_{g}^{*}$$ without changing the definition of a cocycle.
which checks Lemma 5.2(c). The second identity of (5.1) is on the other hand easily obtained from this new identity. The identity
Definition 5.3.
Given an αcocycle
Definition 5.4.
For an αcocycle
Notice that
6. The Isomorphism u_{#}
 Abstract
 1. Introduction
 2. Equivariant
KK theory  3. The Unitization of a
G algebra  4. The Splitexactness, Stability and Homotopy Invariance of
KK^{G}  5. Cocycles
 6. The Isomorphism
u_{#}  7. The Cocyle Set
${\mathbb{E}}^{G}(A,B)$  8. The Isomorphism
$\Phi $  9. The Map Ψ
 10. The Abelian Group Homomorphism
Ψ'  11. The Natural Transformation ξ
 12. The Universality Theorem
 13. Nonunital Inverse Semigroups
 Acknowledgements.
 References
In this section we shall see that the objects
Definition 6.1.
Consider the two corner embeddings
Definition 6.2.
Let
That is, under a stable functor 'the actions α and
Lemma 6.3.
Consider the stable functor
respectively, where the occurring Hilbert
where
of
Lemma 6.4.
Let
7. The Cocyle Set
${\mathbb{E}}^{G}(A,B)$
${\mathbb{E}}^{G}(A,B)$
 Abstract
 1. Introduction
 2. Equivariant
KK theory  3. The Unitization of a
G algebra  4. The Splitexactness, Stability and Homotopy Invariance of
KK^{G}  5. Cocycles
 6. The Isomorphism
u_{#}  7. The Cocyle Set
${\mathbb{E}}^{G}(A,B)$  8. The Isomorphism
$\Phi $  9. The Map Ψ
 10. The Abelian Group Homomorphism
Ψ'  11. The Natural Transformation ξ
 12. The Universality Theorem
 13. Nonunital Inverse Semigroups
 Acknowledgements.
 References
Until Section 10 assume that
The
Definition 7.1.
Let
where

(a)
u_{+} andu_{} denote βcocycles$G\to \mathcal{M}(B)$ , 
(b)
${\phi}_{\pm}$ denoteG equivariant *homomorphisms$A\to (\mathcal{M}(B),{u}_{\pm}\overline{\beta}{u}_{\pm}^{*})$ , respectively, 
(c)
${\phi}_{+}(a){\phi}_{}(a)\in B$ , 
(d)
${u}_{+}{}_{g}{u}_{}{}_{g}\in B$
for all
Definition 7.2.
Two
In the rest of the paper we shall identify isomorphic
Definition 7.3.
The set of isomorphism classes of
Definition 7.4.
An
Definition 7.5.
Two
where
Definition 7.6.
For
where
Lemma 7.7.
Up to homotopy of
in
Definition 7.8.
Let
8. The Isomorphism $\Phi $
 Abstract
 1. Introduction
 2. Equivariant
KK theory  3. The Unitization of a
G algebra  4. The Splitexactness, Stability and Homotopy Invariance of
KK^{G}  5. Cocycles
 6. The Isomorphism
u_{#}  7. The Cocyle Set
${\mathbb{E}}^{G}(A,B)$  8. The Isomorphism
$\Phi $  9. The Map Ψ
 10. The Abelian Group Homomorphism
Ψ'  11. The Natural Transformation ξ
 12. The Universality Theorem
 13. Nonunital Inverse Semigroups
 Acknowledgements.
 References
In this section we shall isomorphically substitute Kasparov theory by its Cuntz picture in form of
Definition 8.1.
There is a map
where the
Lemma 8.2.
The just defined
because of the third identity of (5.1) and because
We have
because
The
for
Proposition 8.3.
Every element of
for all
Let
is homotopic in
Since the
where
is another
where
Identifying
By considering the same homotopies as in the nonequivariant case, see [5, p. 125] (notice that
Define an automorphism
and define a
Remark 8.4.
We use the last proposition as a basis for a Cuntz picture of
Theorem 8.5.
The set
by
We are going to show that
To check that
for all
For an idempotent
This shows that
the first identity of (5.1). Similarly we get the second identity and the third one computes as
We note that, since
is in
is in
Since
which verifies item (b) of Definition 7.1.
Now notice that indeed
We are going to prove injectivity of
Because at the endpoints of the cycle
and an isomorphism
for
Our next goal is to make
Define isometries
and
Consider the unitary
defines a path of
in
Definition 8.6.
For an equivariant *homomorphism
Lemma 8.7.
Let
By unitality of
9. The Map Ψ
 Abstract
 1. Introduction
 2. Equivariant
KK theory  3. The Unitization of a
G algebra  4. The Splitexactness, Stability and Homotopy Invariance of
KK^{G}  5. Cocycles
 6. The Isomorphism
u_{#}  7. The Cocyle Set
${\mathbb{E}}^{G}(A,B)$  8. The Isomorphism
$\Phi $  9. The Map Ψ
 10. The Abelian Group Homomorphism
Ψ'  11. The Natural Transformation ξ
 12. The Universality Theorem
 13. Nonunital Inverse Semigroups
 Acknowledgements.
 References
In this section we shall see how elements of Kasparov theory
Definition 9.1.
Given an
with two
A
for
Definition 9.2.
We sloppily use
Lemma 9.3.
Definition 9.1 is valid.
This shows that
with the usual center properties, identities (5.1), and the identity of Lemma 5.2(c).
We show that
This proves that
Definition 9.4.
Let
where
Let
Define an abelian group homomorphism
by
Notice here that the occurrence of
Lemma 9.5.
The definition of
(
where
whence Lemma 6.4 applies to
By homotopy invariance of
Lemma 9.6.
Let
We may summarize Lemmas 9.5 and 9.6 as follows.
Corollary 9.7.
The map
by
Lemma 9.8.
For every unital *homomorphism
for
10. The Abelian Group Homomorphism Ψ'
 Abstract
 1. Introduction
 2. Equivariant
KK theory  3. The Unitization of a
G algebra  4. The Splitexactness, Stability and Homotopy Invariance of
KK^{G}  5. Cocycles
 6. The Isomorphism
u_{#}  7. The Cocyle Set
${\mathbb{E}}^{G}(A,B)$  8. The Isomorphism
$\Phi $  9. The Map Ψ
 10. The Abelian Group Homomorphism
Ψ'  11. The Natural Transformation ξ
 12. The Universality Theorem
 13. Nonunital Inverse Semigroups
 Acknowledgements.
 References
In this section we shall define a variation
From now on
Definition 10.1.
Fix a onedimensional projection
Definition 10.2.
Consider the canonical split exact sequence
where
Let
by
The occurrence of
by Lemma 9.8 and
Lemma 10.3.
For any *homomorphism
Lemma 10.4.
The map
is an abelian group homomorphism.
Lemma 10.5.
Let
in
Consider now Definition 9.4 with respect to
as the difference
11. The Natural Transformation ξ
 Abstract
 1. Introduction
 2. Equivariant
KK theory  3. The Unitization of a
G algebra  4. The Splitexactness, Stability and Homotopy Invariance of
KK^{G}  5. Cocycles
 6. The Isomorphism
u_{#}  7. The Cocyle Set
${\mathbb{E}}^{G}(A,B)$  8. The Isomorphism
$\Phi $  9. The Map Ψ
 10. The Abelian Group Homomorphism
Ψ'  11. The Natural Transformation ξ
 12. The Universality Theorem
 13. Nonunital Inverse Semigroups
 Acknowledgements.
 References
In this section we shall show Theorem 1.2.
Definition 11.1.
Let
defined by
for
That
Lemma 11.2.
Consider the maps
developed in Definitions 9.4 and 10.2, respectively, for the homotopy invariant, stable, splitexact functor
for all
where
Proposition 11.3.
Given
Now consider another natural transformation
We have a commuting diagram
for all homomorphisms
Thus
by Lemma 11.2.
Definition 11.1 and Proposition 11.3 sum then up to:
Corollary 11.4.
Theorem 1.2 is true.
12. The Universality Theorem
 Abstract
 1. Introduction
 2. Equivariant
KK theory  3. The Unitization of a
G algebra  4. The Splitexactness, Stability and Homotopy Invariance of
KK^{G}  5. Cocycles
 6. The Isomorphism
u_{#}  7. The Cocyle Set
${\mathbb{E}}^{G}(A,B)$  8. The Isomorphism
$\Phi $  9. The Map Ψ
 10. The Abelian Group Homomorphism
Ψ'  11. The Natural Transformation ξ
 12. The Universality Theorem
 13. Nonunital Inverse Semigroups
 Acknowledgements.
 References
In this section we shall deduce Theorem 1.3 as described in [4, Theorem 4.5].
Definition 12.1.
A functor
For convenience of the reader we recall another characterization of splitexact, homotopy invariant, stable functors into additive categories, see [4, p. 269].
Lemma 12.2.
A functor

(a)
F(f) is invertible for every corner embedding$f\in {C}^{*}(A,A\otimes \mathcal{K})$ , 
(b)
F(f)=F(g) for all homotopic$f,g\in {C}^{*}(A,B)$ , and 
(c) for every split exact sequence
$$0\stackrel{}{\to}A\stackrel{j}{\to}D\underset{s}{\overset{p}{\rightleftarrows}}B\stackrel{}{\to}0$$ the map$F(A)\oplus F(B)\to F(D)$ defined by$$F(j)\circ {p}_{1}+F(s)\circ {p}_{2}$$ is an isomorphism, wherep_{1},p_{2} denotes the projection maps.
Lemma 12.3.
Consider
for all
Actually, the last
Theorem 12.4.
Theorem 1.3 is true.
and the element
We obtain a natural transformation
Define the functor
for all
Since by definition
for
We compute the functoriality of
Then with Lemma 12.3, identity (11.1) and (12.3) we compute
Let
We are going to show uniqueness of
For
Hence (12.2) defines a natural transformation (12.1), which by Proposition 11.3 is uniquely determined. Hence
13. Nonunital Inverse Semigroups
 Abstract
 1. Introduction
 2. Equivariant
KK theory  3. The Unitization of a
G algebra  4. The Splitexactness, Stability and Homotopy Invariance of
KK^{G}  5. Cocycles
 6. The Isomorphism
u_{#}  7. The Cocyle Set
${\mathbb{E}}^{G}(A,B)$  8. The Isomorphism
$\Phi $  9. The Map Ψ
 10. The Abelian Group Homomorphism
Ψ'  11. The Natural Transformation ξ
 12. The Universality Theorem
 13. Nonunital Inverse Semigroups
 Acknowledgements.
 References
Corollary 13.1
Corollary 1.4 is true.
Similarly we may view a countable discrete groupoid
Acknowledgements.
 Abstract
 1. Introduction
 2. Equivariant
KK theory  3. The Unitization of a
G algebra  4. The Splitexactness, Stability and Homotopy Invariance of
KK^{G}  5. Cocycles
 6. The Isomorphism
u_{#}  7. The Cocyle Set
${\mathbb{E}}^{G}(A,B)$  8. The Isomorphism
$\Phi $  9. The Map Ψ
 10. The Abelian Group Homomorphism
Ψ'  11. The Natural Transformation ξ
 12. The Universality Theorem
 13. Nonunital Inverse Semigroups
 Acknowledgements.
 References
We thank the Universidade Federal de Santa Catarina in FlorianÓpolis for the support we received when developing the content of this paper in 2013. We have published a short version in arXiv under the same title in 2014. Since that version turned out to be too brief and sketchy for an oﬃcial publication, we decided to write this selfcontained paper. We thank Alain Valette for suggesting to write a selfcontained version.
References
 Abstract
 1. Introduction
 2. Equivariant
KK theory  3. The Unitization of a
G algebra  4. The Splitexactness, Stability and Homotopy Invariance of
KK^{G}  5. Cocycles
 6. The Isomorphism
u_{#}  7. The Cocyle Set
${\mathbb{E}}^{G}(A,B)$  8. The Isomorphism
$\Phi $  9. The Map Ψ
 10. The Abelian Group Homomorphism
Ψ'  11. The Natural Transformation ξ
 12. The Universality Theorem
 13. Nonunital Inverse Semigroups
 Acknowledgements.
 References
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