# nLab KK-theory

cohomology

## Topics in Functional Analysis

#### Noncommutative geometry

noncommutative geometry

(geometry $\leftarrow$ Isbell duality $\to$ algebra)

# Contents

## Idea

KK-theory is a “bivariant” joint generalization of operator K-theory and K-homology: for $A, B$ two C*-algebras, the KK-group $KK(A,B)$ is a natural homotopy equivalence class of $(A,B)$-Hilbert bimodules equipped with an additional left weak Fredholm module structure. These KK-groups $KK(A,B)$ behave in the first argument as K-homology of $A$ and in the second as K-cohomology/operator K-theory of $B$.

Abstractly, KK-theory is an additive category of C*-algebras which is the split-exact and homotopy-invariant localization of C*Alg at the compact operators. Hence, abstractly KK-theory is a fundamental notion in noncommutative topology, but its standard presentation by Fredolm-Hilbert bimodules as above is rooted in functional analysis. A slight variant of this localization process is called E-theory.

Due to this joint root in functional analysis and (noncommutative) cohomology/homotopy theory (“noncommutative stable homotopy theory”), KK-theory is a natural home of index theory, for elliptic operators on smooth manifolds as well as for their generalization to equivariant situations, to foliations and generally to Lie groupoid-theory (via their groupoid convolution C*-algebras) and noncommutative geometry.

As a special case of this, quantization in its incarnation as geometric quantization by push-forward has been argued to naturally proceed by index theory in KK-theory (Landsman 03, Bos 07). Also the coupling of D-branes and their Chan-Paton bundles in twisted K-theory with RR-charge in string theory is naturally captured by the coupling between K-homology and K-cohomology in KK-theory (e.g. Szabo 08).

## Definition

We state first the original and standard definition of $KK$-groups in terms of equivalence classes of Fredholm-Hilbert C*-bimodules in

Then we state the abstract category-theoretic characterization by localization in

An equivalent and explicity homotopy theoretic characterization akin to that of the standard homotopy category Ho(Top) is in

### In terms of Fredholm-Hilbert $C^\ast$-bimodules

###### Definition

In all of the following, ”$C^\ast$-algebra” means separable C*-algebra. We write C*Alg for for the category whose objects are separable $C^\ast$-algebras and whose morphisms are $\ast$-homomorphisms between these.

###### Example

We write

• $\mathcal{B} \coloneqq \mathcal{B}(\mathcal{H})$ for the $C^\ast$-algebra of bounded operators on a complex, infinite-dimensional separable Hilbert space;

• $\mathcal{K} \coloneqq \mathcal{K}(\mathcal{H}) \hookrightarrow \mathcal{B}(\mathcal{H})$ for the compact operators.

###### Definition

For $B \in$ C*Alg, a Hilbert C*-module over $B$ is

1. a complex vector space $H$;

2. equipped with a C*-representation of $B$ from the right;

3. equipped with a sesquilinear map (linear in the second argument)

$\langle -,-\rangle \colon H \times H \to B$

(the $B$-valued inner product)

such that

1. $\langle -,-\rangle$ behaves indeed like a positive definitine inner product over $B$:

1. $\langle x,y\rangle^\ast = \langle y,x\rangle$

2. $\langle x,x\rangle \geq 0$ (in the sense of positive elements in $B$)

3. $\langle x,x\rangle = 0$ precisely if $x = 0$;

4. $\langle x,y \cdot b\rangle = \langle x,y \rangle \cdot b$

2. $H$ is complete with respect to the norm:

${\Vert x \Vert_H} \coloneqq {\Vert \langle x,x\rangle\Vert_B}$.

###### Definition

For $A,B \in C^\ast Alg$ an $(A,B)$-Hilbert C*-bimodule is an $B$-Hilbert C*-module, def. 2 $(H, \langle \rangle)$ equipped with a C-star representation of $A$ from the left such that all $a \in A$ are “adjointable” in the $B$-valued inner product, meaning that

$\langle a^\ast \cdot x,y\rangle = \langle x, a y\rangle \,.$
###### Definition

For $A, B \in$ C*Alg, Kasparov $(A,B)$-bimodule is a $\mathbb{Z}_2$-graded $(A,B)$-Hilbert bimodules $\mathcal{H}, \langle -,-\rangle$, def. 3, equipped with an adjointable odd-graded bounded operator $F \in \mathcal{B}_A(\mathcal{H})$ such that

1. $(F^2 - 1)\pi(a) \in \mathcal{K}_A(\mathcal{H})$

2. $[F, \pi(a)] \in \mathcal{K}_A(\mathcal{H})$

3. $(F - F^\ast) \pi(a)\in \mathcal{K}_A(\mathcal{H})$

for all $a \in A$,

hence such that $F$ squares to the identity, commutes with multiplication operators and is self-adjoint up to compact operators.

For instance (Blackadar 99, p. 144).

###### Example

For $B = \mathbb{C}$ a Kasparov $(A,B)$-bimodule is equivalently an $A$-Fredholm module for an essentially self-adjoint Fredholm operator

###### Definition

A homotopy between two Kasparov $(A,B)$-bimodules is an $(A, C([0,1],B))$-bimodule which interpolates between the two.

(…)

###### Definition

Writes $KK(A,B)$ for the set of equivalence classes of Kasparov $(A,B)$-bimodules under homotopy, def. 5.

###### Proposition

$KK(A,B)$ is naturally an abelian group under direct sum of bimodules and operators.

###### Proposition

There is a composition operation

$KK(A,B) \times KK(B,C) \to KK(A,C)$

such that (…). This is called the Kasparov product.

A streamlined version of the definition of the Kasparov product is in (Skandalis 84).

###### Remark

From the point of view of E-theory the Kasparov product is equivalently just the composition of homotopy classes of completely poistive asymptotic C*-homomorphisms. See at E-theory for more on this.

###### Remark

On the other hand, at least between $C^\ast$-algebras which are algebras of functions on smooth manifolds $A_i = C(X_i)$ , KK-classes are presented by correspondences $X_1 \leftarrow Z \to X_2$ and the Kasparov product is given just by the fiber product-composition operation on correspondences (Connes-Skandalis 84, theorem 3.2, Block-Weinberger 99, section 3).

### Universal category-theoretic characterization

###### Proposition

The Kasparov product, def. 2, is associtative. Thus under the Kasparov product

$KK(-,-) \;\colon\; C^\ast Alg \times C^\ast Alg \to C^\ast Alg$

is the hom-functor of an additive category.

The category $KK$ is a kind of localization of the category of C-star-algebras:

###### Theorem

The canonical functor

$Q \colon C^\ast Alg \to KK$

exhibits $KK$ as the universal category receiving a functor from C*-algebras such that

1. $KK$ is an additive category;

2. $Q$ is homotopy-invariant;

3. $Q$ inverts the tensor product with the C*-algebra of compact operators

(for all $C^\ast$-homomorphisms of the form $id \otimes e \langle e,- \rangle \;\colon A\; \to A \otimes \mathcal{K}$ the morphism $Q(id \otimes e \langle e)$ is an isomorphism).

4. $Q$ preserves split short exact sequences.

This is due to (Higson 87, theorem 4.5). The generalization to the equivariant case is due to (Thomsen 98).

###### Remark

The localization conditions here are analogous to those that define the localization of stable (∞,1)-categories to noncommutative motives (see there for more).

###### Corollary

The minimal tensor product of C-star-algebras

$\otimes \colon C^\ast Alg \times C^\ast Alg \to C^\ast Alg$

extends uniquely to a tensor product $\otimes_{KK}$ on $KK$ such that there is a commuting diagram of functors

$\array{ C^\ast Alg \times C^\ast Alg &\stackrel{Q \times Q}{\to}& KK \\ \downarrow^{\mathrlap{\otimes}} && \downarrow^{\mathrlap{\otimes_{KK}}} \\ C^\ast Alg &\stackrel{Q}{\to}& KK } \,.$

For more discussion of more explicit presentations of this localization process for obtaining KK-theory see at homotopical structure on C*-algebras and also at model structure on operator algebras.

### In terms of homotopy-classes of $\ast$-homomorphisms

Theorem (Cuntz)

If $A,B$ are C-star-algebras with $A$ separable and $B$ $\sigma$-unital, then

$KK(A,B) \simeq [q A, B \otimes \mathcal{K}] \,,$

where

• $q A$ is the kernel of the codiagonal $A \star A \to A$,

• $\mathcal{K}$ is the $C^\ast$-algebra of compact operators.

• $[-,-]$ is the set of homotopy equivalence classes of $\ast$-homomorphisms.

(reviewed in (Joachim-Johnson07)).

### In terms of correspondences/spans of groupoids

At least to some extent, KK-classes between C*-algebras of continuous functions on manifolds/spaces, and maybe more generally between groupoid convolution algebras can be represented by certain equivalence classes of spans/correspondences

$X \leftarrow (Z,E) \to Y$

of such spaces.

See the corresponding references below.

Such a description by abelianizations of correspondences is reminiscent of similar constructions of motivic cohomology. See below. For more on this see also the pointers at at motivic quantization.

(…)

• category of equivariant correspondences equipped with cocycle: $\hat F_{\mathcal{G}}^\ast$ (theorem 2.26);

• specifically for K-theory cocycles: $\widehat {KK}_{\mathcal{G}}^\ast$ (section 4, page 27)

• pull-push from correspondences to KK in proof of theorem 4.2, bottom of p. 27

(…)

### As an analog of motives in noncommutative topology

To some extent KK-theory/E-theory look like an analogue in noncommutative topology of what in algebraic geometry is the category of motives. (Connes-Consani-Marcolli 05). (Meyer 06).

Specifically the characterization in terms of spans/correspondences above is reminiscent to the definition of pure motives, see the rferences below: In terms of correspondences. A relation between bivariant algebraic K-theory and motivic cohomology is discussed in (Garkusha-Panin 11).

A universal functor from KK-theory to noncommutative motives

$KK \longrightarrow NCC_{dg}$

was given in (Mahanta 13). This sends a C*-algebra to the dg-category of perfect complexes over (the unitalization of) its underlying associative algebra.

### Equivariant KK-theory

Pretty much all of KK-theory has a generalization to equivariant cohomology where all algebras and modules are equipped with actions of a given topological group or more generally topological groupoid $\mathcal{G}$, and all operators are suitably invariant/equivariant under this action. See at equivariant KK-theory for more.

The Baum-Connes conjecture and the Green-Julg theorem assert that under some conditions $\mathcal{G}$-equivariant KK-theory is equivalent to the plain KK-theory of the groupoid convolution algebras of the corresponding action groupoids. See at Green-Julg theorem for details.

## Examples

### Basic examples

###### Example

For $f \colon A \to B$ a homomorphism of $\mathbb{Z}_2$ graded C*-algebras, take $B$ as a right Hilbert module over itself and equip it with the left action of $A$ induced by $f$. This makes it a Hilbert bimodule. Together with the 0-Fredholm operator, this represents an element

$(B, f, 0) \in KK(A,B) \,.$

For instance (Blackadar 99, example 17.1.2 a)).

###### Example

For

$(H_i, F_i) \in KK(A,B)$

a Fredholm $(A_i,B)$-Hilbert bimodule for $i \in \{1,2\}$, the direct sum is

$(H_1 \oplus H_2, F_1 \oplus F_2) \in KK(A_1\oplus A_2, B) \,.$

For instance (Blackadar 99, example 17.1.2 c)).

### The archetypical examples

###### Example

Let $(X,g)$ be a closed smooth Riemannian manifold, and let $V_0, V_1$ be two smooth vector bundles over $X$ with Hermitian strucure (associated to a chosen unitary group-principal bundle).

Then given an elliptic pseudodifferential operator

$P \colon \Gamma(V_0) \to \Gamma(V_1)$

on smooth sections it extends to an essentially unitary Fredholm operator on square integrable sections $L^2(V_i)$.

Consider then the $\mathbb{Z}_2$-graded Hilbert space

$H \coloneqq L^2(V_0) \oplus L^2(V_1)$

equipped with the evident action of $C(X)$ (by “multiplication operators?”). Then with $P$ a parametrix for $Q$, the operator

$F \coloneqq \left[ \array{ 0 & Q \\ P & 0 } \right]$

is a Fredholm operator on $H$, so that

$\left( L^2(V_1) \oplus L^2(V_2), \left[ \array{ 0 & Q \\ P & 0 } \right] \right) \in KK(C(X),\mathbb{C}) \,.$
###### Example

Let $(X,g)$ be an almost complex manifold and let $D \colon \overline{\partial} + \overline{\partial}^\ast$ be the Dolbeault-Dirac operator. This extends to an operator on

$H \coloneqq L^2(\Omega^{0,\bullet})$

and

$F \coloneqq \frac{D}{\sqrt{1 + D^2}}$

(defined by functional calculus) is then a Fredholm operator on that. Then

$\left( L^2(\Omega^{0,\bullet}), \frac{\overline{\partial} + \overline{\partial}^\ast}{\sqrt{1+ (\overline{\partial} + \overline{\partial}^\ast)^2}} \right) \in KK(C(X), \mathbb{C}) \,.$

## Properties

### Relation to operator K-cohomology, K-homology, twisted K-theory

KK-theory is a joint generalization of operator K-theory, hence also of topological K-theory, as well as of K-homology and of twisted K-theory.

For $A \in$ C*Alg we have that

• $KK(\mathbb{C}, A) \simeq K_0(A)$

is the operator K-theory group of $A$ in degree 0 and

• $KK(C(\mathbb{R}^1),A) \simeq K_1(A)$

is the operator K-theory group of $A$ in degree 1. (e.g. (Introduction, p. 20). If here $A = C(X)$ is the C*-algebra of functions on a suitable topological space $X$, then this is the topological K-theory of that space

• $KK(\mathbb{C}, C(X)) \simeq K^0(X)$

• $KK(C(\mathbb{R}), C(X)) \simeq K^1(X)$.

More generally, if $A = C_r(\mathcal{G}_\bullet)$ is the reduced groupoid convolution algebra of a Lie groupoid, then

• $KK(\mathbb{C}, C_r(\mathcal{G}_\bullet)) \simeq K^0(\mathcal{G})$

is the K-theory of the corresponding differentiable stack. If moreover $c \colon \mathcal{G} \to \mathbf{B}^2 U(1)$ is a circle 2-group-principal 2-bundle ($U(1)$-bundle gerbe) over $\mathcal{X}$ and if $A = C(\mathcal{X}_\bullet, c)$ is the twisted groupoid convolution algebra of the corresponding centrally extended Lie groupoid, then

• $KK(\mathbb{C}, C_r(\mathcal{X}_\bullet,x)) = K^0(\mathcal{X}, c)$

is the corresponding twisted K-theory (Tu, Xu, Laurent-Gengoux 03).

On the other hand, with $A$ in the first argument and the complex numbers in the second we have that

• $K(A,\mathbb{C}) \simeq K^0(A)$

ar equivalence classes of $A$-Fredholm modules and hence the K-homology of $A$.

(…)

### Relation to extensions

There is an isomorphism

$KK(A,B) \simeq Ext^1(A,B)$

to a suitable group of suitable extensions of $A$ by $B$. (Kasparov 80, reviewed in Inassaridze).

### Triangulated sructure and $KU$-module structure

###### Proposition

$KK$ is naturally a stable triangulated category.

###### Proposition

There is a functor

$\mathbb{K}(-) \;\colon\; C^\ast Alg \to Ho(Spectra)$

to the stable homotopy category such that

1. $\pi_n(\mathbb{K})(A) \simeq K_n(A)$, for all $A \in C^\ast Alg$, (hence the spectrum is a cohomology spectrum for the operator K-theory of $A$);

2. $\mathbb{K}(\mathbb{C})$ is naturally a ring spectrum;

3. $\mathbb{K}(A)$ is naturally a symmetric $\mathbb{K}(\mathbb{C})$-module spectrum

4. $\mathbb{K}$ lifts to a lax monoidal functor

$\mathbb{K} \;\colon\; C^\ast Alg \to Ho(\mathbb{K}(\mathbb{C}) Mod)$

to the homotopy category of module spectra, and this in turn extends to a lax monoidal functor on the KK-category

$\mathbb{K} \;\colon\; KK \to Ho(\mathbb{K}(\mathbb{C}) Mod) \,.$
5. $\mathbb{K}$ restricts to a fully faithful functor on the thick subcategory of the triangulated category $KK$ generated by the tensor unit (the “bootstrap category”).

This is the main result of (DEKM 11, section 3).

###### Remark

Since $\mathbb{K}$ is a lax monoidal functor in particular it preserves dual objects and dual morphisms, hence Poincaré duality algebras and their Umkehr maps.

### Künneth theorem

The thick subcategory of the triangulated category $KK$ generated from the tensor unit is called the bootstrap category $Boot \hookrightarrow KK$. For $A \in Boot \hookrightarrow KK$ one has that $KK(A,B)$ satisfies a Künneth theorem. See at bootstrap category for more.

### Excision and relation to E-theory

###### Definition

Given a short exact sequence of C*-algebras one says that $KK$ satisfies excision or that it is excisive for this sequence if it preserves its exactness in the middle.

###### Example

By theorem 1, $KK$ is excisive over split exact sequences.

###### Proposition

$KK$ is excisive for nuclear C*-algebras in the first argument.

This is discussed (Kasparov 80, section 7), (Cuntz-Skandalis 86).

More generally:

###### Proposition

$KK$ is excisive for K-nuclear C*-algebras in the first argument.

###### Remark

It is not expected that excision is satisfied fully generally by $KK$. Instead, the universal improvement of $KK$-theory under excision can be constructed. This is called E-theory. See there for more.

### Poincaré duality and Thom isomorphism

###### Definition

A C*-algebra is a Poincaré duality algebra if it is a dualizable object in the symmetric monoidal category $KK$ with dual its opposite algebra.

###### Proposition

Let $X$ be a smooth manifold which is compact. Then the C*-algebra $C(X) \otimes C_0(T^\ast X)$ (the tensor product of the algebra of functions of compact support on $X$ and on its cotangent bundle) is isomorphic, in $KK$, to $\mathbb{C}$:

$d \colon C(X) \otimes C_0(T^\ast X) \stackrel{\simeq}{\to} \mathbb{C} \,.$
###### Corollary

For $X$ a compact smooth manifold, there is a natural isomorphism (Thom isomorphism)

$K_0( C_0(T^\ast X)) \simeq KK(\mathbb{C}, C_0(T^\ast X)) \stackrel{KK(C,(-)\otimes C(X))}{\to} KK(C(X), C(X) \otimes C_0(T^\ast X) ) \underoverset{\simeq}{KK(C(X), d)}{\to} KK(C(X), \mathbb{C} ) \,.$

For more discussion see at Poincaré duality algebra.

### Push-forward in KK-theory

Umkehr map in KK-theory (Brodzki-Mathai-Rosenberg-Szabo 07, section 3.3)

If $A$, $B$ are Poincaré duality algebras, def. 8, then for $f \colon A \to B$ a morphism, the corresponding Umkehr map is (postcomposition) with the dual morphism of its opposite algebra version:

$f! \coloneqq (f^op)^\ast \,.$

For more and a discussion of twisted Umkehr maps see at Poincaré duality algebra and at Freed-Witten-Kapustin anomaly cancellation.

## Further Theorems

geometric contextuniversal additive bivariant (preserves split exact sequences)universal localizing bivariant (preserves all exact sequences in the middle)universal additive invariantuniversal localizing invariant
noncommutative algebraic geometrynoncommutative motives $Mot_{add}$noncommutative motives $Mot_{loc}$algebraic K-theorynon-connective algebraic K-theory
noncommutative topologyKK-theoryE-theoryoperator K-theory

## References

### General

KK-theory was introduced by Gennady Kasparov in

• Gennady Kasparov, The operator $K$-functor and extensions of $C^{\ast}$-algebras, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 3, 571–636, 719, MR81m:58075, Zbl, abstract, english doi, free Russian original: pdf

prompted by the advances in Brown-Douglas-Fillmore theory, especially in the last 1977 article.

Some streamlining of the definitions appeared in

• Georges Skandalis, Some remarks on Kasparov theory, J. Funct. Anal. 59 (1984) 337-347.

A textbook account is in

Introductions and surveys include

• Gennady Kasparov, Operator K-theory and its applications: elliptic operators, group representations, higher signatures $C^\ast$-extensions, Proceedings ICM 1983 Warszawa, PWN-Elsevier (1984) 987-1000.

• Nigel Higson, A primer on KK-theory. Proc. Sympos. Pure Math. 51, Part 1, 239–283. (1990) (pdf)

• Georges Skandalis, Kasparov’s bivariant K-theory and applications Exposition. Math. 9, 193–250 (1991) (pdf slides)

• Introduction to KK-theory and E-theory, Lecture notes (Lisbon 2009) (pdf slides)

• Heath Emerson, R. Meyer (notes taken by S. Hong), KK-theory and Baum-Connes conjecture, Lectures at Summer school on operator algebras and noncommutative geometry (June 2010) (pdf)

• R. Meyer, How analysis and topology interact in bivariant K-theory, 2006 (pdf)

### Excision

Excision for KK-theory is further studied in

• Georges Skandalis, Une notion de nuclearité en K-theorie, K-Theory 1 (1988) 549-574.

### In Category theory and Homotopy theory

KK-theory is naturally understood in terms of universal properties in category theory and in homotopy theory.

That $KK(A,B)$ is naturally thought of as a collection of “generalized homomorphisms” of $C^\ast$-algebras was amplified in

• Joachim Cuntz, Generalized Homomorphisms Between $C^\ast$-algebras and KK-theory, Springer Lecture Notes in Mathematics, 1031 (1983), 31-45.

• Joachim Cuntz, K-theory and C-algebras_, Springer Lecture Notes in Mathematics, 1046 (1984), 55-79.

That under the Kasparov product these are indeed the hom-objects in a category was first observed in

• Nigel Higson, A characterization of KK-theory, Pacific J. Math. Volume 126, Number 2 (1987), 253-276. (EUCLID)

where moreover this category is realized as the universal additive and split exact “localization” of $C^\ast Alg$ at the $C^\ast$-algebra of compact operators.

The generalization of this statement to equivariant KK-theory is in

Characterization of KK-theory as the satellites of a functor is in

A triangulated category structure for KK-theory is discussed in

A model category realization of KK-theory is discussed in

A category of fibrant objects-structure on C*Alg which unifies the above homotopical pictures is discussed in

More on this is at homotopical structure on C*-algebras.

Further discussion in the context of stable homotopy theory and E-theory is in

• Martin Grensing, Noncommutative stable homotopy theory (arXiv:1302.4751)

• Snigdhayan Mahanta, Higher nonunital Quillen $K'$-theory, KK-dualities and applications to topological $\mathbb{T}$-duality, Journal of Geometry and Physics, Volume 61, Issue 5 2011, p. 875-889. (pdf)

Refinement of operator K-theory to cohomology spectra is discussed in

• Ulrich Bunke, Michael Joachim, Stephan Stolz, Classifying spaces and spectra representing the K-theory of a graded $C^\ast$-algebra, High-dimensional manifold topology, World Sci. Publ., River Edge, NJ, 2003, pp. 80–102

This construction is functorial (only) for essential $\ast$-homomorphisms of C*-algebras.

A refinement of the KK-category to a spectrum-enriched category ($\sim$ stable (∞,1)-category) is claimed in

• Michael Joachim, Stephan Stolz, An enrichment of $KK$-theory over the category of symmetric spectra Münster J. of Math. 2 (2009), 143–182 (pdf)

and the generalization of this to equivariant K-theory over geometrically discrete groupoids is discussed in

• Paul Mitchener, $KK$-theory spectra for $C^\ast$-categories and discrete groupoid $C^\ast$-algebras (arXiv:0711.2152)

but this construction is stated to be mistaken on p. 3 of

This article in turn considers a variant of the construction in (Bunke-Joachim-Stolz 03) which gives operator K-theory spectra that are functorial for general $\ast$-homomorphisms.

Observations relating to a genuine stable (∞,1)-category structure maybe at least of E-theory are in

### In the context of the Novikov conjecture

• Jonathan Rosenberg, Group C-algebras and Topological Invariants_ , Proc. Conf. in Neptun, Romania, 1980, Pitman (London, 1985)

### In the context of the Atiyah-Singer index theorem

The classical Atiyah-Singer index theorem is reviewed in operator K-theory (with some hints towards KK-theory) in

Generalization to the relative case in KK-theory, hence for indices of fiberwise elliptic operators on Hilbert C*-module-fiber bundles is in

• Jody Trout, Asymptotic Morphisms and Elliptic Operators over $C^\ast$-algebras, K-theory, 18 (1999) 277-315 (arXiv:math/9906098)

### For convolution algebras and In geometric quantization

Discussion of KK-theory with an eye towards C-star representations of groupoid convolution algebras in the context of geometric quantization by push-forward is in

• Rogier Bos, Groupoids in geometric quantization PhD Thesis (2007) (pdf)

with a summary/exposition in

• Klaas Landsman, Functoriality of quantization: a KK-theoretic approach, talk at ECOAS, Dartmouth College, October 2010 (web)

The KK-theory of twisted convolution algebras and its relation to twisted K-theory of differentiable stacks is discussed in

Discussion of groupoid 1-cocycles and their effect on the groupoid algebra KK-theory is discussed in

### In terms of correspondences/spans

#### For plain KK-theory

KK-classes between algebras of functions on smooth manifolds are described in terms of equivalence classes of correspondence manifolds carrying a vector bundle in section 3 of

This generalizes the Baum-Douglas geometric cycles from K-homology to KK-theory.

A further generalization of this, where one algebra $C(Y)$ is generalized to $C(Y) \otimes A$ for $A$ a unital separable $C^\ast$-algebra, is in section 3 of

In section 5 of

this is reviewed and then a characterization in terms of co-spans of C*-algebras is given. This version is effectively a restatement of the characterization by Cuntz as reproduced in (Blackadar 99, corollary 17.8.4).

For similar structures see also at motive in the section Relation to bivariant K-theory.

#### For equivariant KK-theory

Generalization of such correspondence-presentation to equivariant KK-theory (and hence, by the Green-Julg theorem essentially to KK-theory of groupoid algebras of action groupoids of compact topological groups) – was introduced in

based on

based on technical aspects of the construction of pushforward along and comoposition of equivariant correspondences in

Further developments of this are in

### Relation to motives and algebraic KK-theory

The general analogy between KK-cocycles and motives is noted explicitly in

and also very briefly in (Meyer 06).

A relation between motivic cohomology and bivariant algebraic K-theory is discussed in

• Snigdahayan Mahanta, Higher nonunital Quillen $K'$-theory, KK-dualities and applications to topological $\mathbb{T}$-duality, Journal of Geometry and Physics, Volume 61, Issue 5 2011, p. 875-889. (pdf)

(in the context of noncommutative motives).

In

• Snigdhayan Mahanta, Higher nonunital Quillen $K'$-theory, KK-dualities, and applications to topological T-duality, J. Geom. Phys., 61 (5), 875-889, 2011 (pdf, talk notes)

it is shown that there is a universal functor $KK \longrightarrow NCC_{dg}$ from KK-theory to the category of noncommutative motives, which is the category of dg-categories and dg-profunctors up to homotopy between them. This is given by sending a C*-algebra to the dg-category of perfect complexes of (the unitalization of) its underlying associative algebra.

### In D-brane theory

KK-theory also describes RR-field charges and sources in D-brane theory.

A review is in

based on

• D-Branes, RR-Fields and Duality on Noncommutative Manifolds, Commun. Math. Phys. 277:643-706,2008 (arXiv:hep-th/0607020)

Noncommutative correspondences, duality and D-branes in bivariant K-theory, Adv. Theor. Math. Phys.13:497-552,2009 (arXiv:0708.2648)

D-branes, KK-theory and duality on noncommutative spaces, J. Phys. Conf. Ser. 103:012004,2008 (arXiv:0709.2128)

### Smooth refinement and spectral triples

Discussion of KK-theory for spectral triples is discussed in

Revised on February 8, 2014 13:46:55 by Urs Schreiber (82.113.106.88)