# nLab KK-theory

cohomology

## Topics in Functional Analysis

#### Noncommutative geometry

noncommutative geometry

(geometry $←$ 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 $\mathrm{KK}\left(A,B\right)$ is a natural homotopy equivalence class of $\left(A,B\right)$-Hilbert bimodules equipped with an additional left weak Fredholm module structure. These KK-groups $\mathrm{KK}\left(A,B\right)$ 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 $\mathrm{KK}$-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 ${C}^{*}$-bimodules

###### Definition

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

###### Example

We write

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

• $𝒦≔𝒦\left(ℋ\right)↪ℬ\left(ℋ\right)$ 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)

$⟨-,-⟩:H×H\to B$\langle -,-\rangle \colon H \times H \to B

(the $B$-valued inner product)

such that

1. $⟨-,-⟩$ behaves indeed like a positive definitine inner product over $B$:

1. $⟨x,y{⟩}^{*}=⟨y,x⟩$

2. $⟨x,x⟩\ge 0$ (in the sense of positive elements in $B$)

3. $⟨x,x⟩=0$ precisely if $x=0$;

4. $⟨x,y\cdot b⟩=⟨x,y⟩\cdot b$

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

$\parallel x{\parallel }_{H}≔\parallel ⟨x,x⟩{\parallel }_{B}$.

###### Definition

For $A,B\in {C}^{*}\mathrm{Alg}$ an $\left(A,B\right)$-Hilbert C*-bimodule is an $B$-Hilbert C*-module, def. 2 $\left(H,⟨⟩\right)$ 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

$⟨{a}^{*}\cdot x,y⟩=⟨x,ay⟩\phantom{\rule{thinmathspace}{0ex}}.$\langle a^\ast \cdot x,y\rangle = \langle x, a y\rangle \,.
###### Definition

For $A,B\in$ C*Alg, Kasparov $\left(A,B\right)$-bimodule is n ${ℤ}_{2}$-graded $\left(A,B\right)$-Hilbert bimodules $ℋ,⟨-,-⟩$, def. 3, equipped with an adjointable odd-graded bounded operator $F\in {ℬ}_{A}\left(ℋ\right)$ such that

1. $\pi \left(a\right)\left({F}^{2}-1\right)\in {𝒦}_{A}\left(ℋ\right)$

2. $\left[\pi \left(a\right),F\right]\in {𝒦}_{A}\left(ℋ\right)$ for all $a\in A$.

###### Example

For $B=ℂ$ a Kasparov $\left(A,B\right)$-bimodule is equivalently an $A$-Fredholm module.

###### Definition

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

(…)

###### Definition

Writes $\mathrm{KK}\left(A,B\right)$ for the set of equivalence classes of Kasparov $\left(A,B\right)$-bimodules under homotopy, def. 5.

###### Proposition

$\mathrm{KK}\left(A,B\right)$ is naturally an abelian group under direct sum of bimodules and operators.

###### Proposition

There is a composition operation

$\mathrm{KK}\left(A,B\right)×\mathrm{KK}\left(B,C\right)\to \mathrm{KK}\left(A,C\right)$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.

### Universal category-theoretic characterization

###### Proposition

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

$\mathrm{KK}\left(-,-\right)\phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}{C}^{*}\mathrm{Alg}×{C}^{*}\mathrm{Alg}\to {C}^{*}\mathrm{Alg}$KK(-,-) \;\colon\; C^\ast Alg \times C^\ast Alg \to C^\ast Alg

is the hom-functor of an additive category.

The category $\mathrm{KK}$ is a kind of localization of the category of C-star-algebras:

###### Theorem

The canonical functor

$Q:{C}^{*}\mathrm{Alg}\to \mathrm{KK}$Q \colon C^\ast Alg \to KK

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

1. $\mathrm{KK}$ is an additive category;

2. $Q$ is homotopy-invariant;

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

(for all ${C}^{*}$-homomorphisms of the form $\mathrm{id}\otimes e⟨e,-⟩\phantom{\rule{thickmathspace}{0ex}}:A\phantom{\rule{thickmathspace}{0ex}}\to A\otimes 𝒦$ the morphism $Q\left(\mathrm{id}\otimes e⟨e\right)$ 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).

###### Corollary

The minimal tensor product of C-star-algebras

$\otimes :{C}^{*}\mathrm{Alg}×{C}^{*}\mathrm{Alg}\to {C}^{*}\mathrm{Alg}$\otimes \colon C^\ast Alg \times C^\ast Alg \to C^\ast Alg

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

$\begin{array}{ccc}{C}^{*}\mathrm{Alg}×{C}^{*}\mathrm{Alg}& \stackrel{Q×Q}{\to }& \mathrm{KK}\\ {↓}^{\otimes }& & {↓}^{{\otimes }_{\mathrm{KK}}}\\ {C}^{*}\mathrm{Alg}& \stackrel{Q}{\to }& \mathrm{KK}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $*$-homomorphisms

Theorem (Cuntz)

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

$\mathrm{KK}\left(A,B\right)\simeq \left[qA,B\otimes 𝒦\right]\phantom{\rule{thinmathspace}{0ex}},$KK(A,B) \simeq [q A, B \otimes \mathcal{K}] \,,

where

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

• $𝒦$ is the ${C}^{*}$-algebra of compact operators.

• $\left[-,-\right]$ is the set of homotopy equivalence classes of $*$-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

$X←\left(Z,E\right)\to Y$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. For more on this see at motive in the section motive – Relation to KK-theory.

## 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

• $\mathrm{KK}\left(ℂ,A\right)\simeq {K}_{0}\left(A\right)$

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

• $\mathrm{KK}\left(C\left({ℝ}^{1}\right),A\right)\simeq {K}_{1}\left(A\right)$

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

• $\mathrm{KK}\left(ℂ,C\left(X\right)\right)\simeq {K}^{0}\left(X\right)$

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

More generally, if $A={C}_{r}\left({𝒢}_{•}\right)$ is the reduced groupoid convolution algebra of a Lie groupoid, then

• $\mathrm{KK}\left(ℂ,{C}_{r}\left({𝒢}_{•}\right)\right)\simeq {K}^{0}\left(𝒢\right)$

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

• $\mathrm{KK}\left(ℂ,{C}_{r}\left({𝒳}_{•},x\right)\right)={K}^{0}\left(𝒳,c\right)$

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

• $K\left(A,ℂ\right)\simeq {K}^{0}\left(A\right)$

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

(…)

### Relation to extensions

There is an isomorphism

$\mathrm{KK}\left(A,B\right)\simeq {\mathrm{Ext}}^{1}\left(A,B\right)$

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

### Triangulated (stable) structure

###### Proposition

$\mathrm{KK}$ is naturally a stable triangulated category.

### Excision and relation to E-theory

###### Definition

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

###### Example

By theorem 1, $\mathrm{KK}$ is excisive over split exact sequences.

###### Proposition

$\mathrm{KK}$ is excisive for nuclear C*-algebras in the first argument.

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

More generally:

###### Proposition

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

###### Remark

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

### Poincaré duality and Thom isomorphism

###### Proposition

Let $X$ be a smooth manifold which is compact. Then the C*-algebra $C\left(X\right)\otimes {C}_{0}\left({T}^{*}X\right)$ (the tensor product of the algebra of functions of compact supposer on $X$ and on its cotangent bundle) is isomorphic, in $\mathrm{KK}$, to $ℂ$:

$d:C\left(X\right)\otimes {C}_{0}\left({T}^{*}X\right)\stackrel{\simeq }{\to }ℂ\phantom{\rule{thinmathspace}{0ex}}.$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)

${K}_{0}\left({C}_{0}\left({T}^{*}X\right)\right)\simeq \mathrm{KK}\left(ℂ,{C}_{0}\left({T}^{*}X\right)\right)\stackrel{\mathrm{KK}\left(C,\left(-\right)\otimes C\left(X\right)\right)}{\to }\mathrm{KK}\left(C\left(X\right),C\left(X\right)\otimes {C}_{0}\left({T}^{*}X\right)\right)\underset{\simeq }{\overset{\mathrm{KK}\left(C\left(X\right),d\right)}{\to }}\mathrm{KK}\left(C\left(X\right),ℂ\right)\phantom{\rule{thinmathspace}{0ex}}.$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} ) \,.

## References

### General

KK-theory was introduced by Gennady Kasparov in

• Gennady Kasparov, The operator $K$-functor and extensions of ${C}^{*}$-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

• B. Blackadar, K-theory for Operator Algebras, 2nd ed. Cambridge University Press, Cambridge (1999)

Introductions and surveys include

• Gennady Kasparov, Operator K-theory and its applications: elliptic operators, group representations, higher signatures ${C}^{*}$-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)

### 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 $\mathrm{KK}\left(A,B\right)$ is naturally thought of as a collection of “generalized homomorphisms” of ${C}^{*}$-algebras was amplified in

• Joachim Cuntz, Generalized Homomorphisms Between C‘-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 ${C}^{*}\mathrm{Alg}$ at the ${C}^{*}$-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)

• Snigdahayan Mahanta, Higher nonunital Qullen $K\prime$-theory, KK-dualities and applications to topological $𝕋$-duality pdf

### 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 ${C}^{*}$-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)

See also the related references at Guillemin-Sternberg geometric quantization conjecture.

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

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

A generalization of this, where one algebra $C\left(Y\right)$ is generalized to $C\left(Y\right)\otimes A$ for $A$ a unital separable ${C}^{*}$-algebra, is in

• Jonathan Block, S. Weinberger, Arithmetic manifolds of positive scalar curvature., J. Diff. Geom. 52, no. 2, 375–406 (1999).

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).

Similar correspondence-presentation of equivariant KK-theory – hence at least of something close to KK-classes between groupoid algebras of action groupoids – was introduced in

Further developments of this are in

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

### In D-brane theory

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

A review is in

based on

### Smooth refinement and spectral triple

Discussion of KK-theory for spectral triples is discussed in

### Algebraic KK-theory

Analogues of Kasparov K-theory in the spirit of algebraic K-theory are developed in

• Grigory Garkusha, Algebraic Kasparov K-theory. I, pdf, II, pdf; Universal bivariant algebraic K-theories, J. Homotopy Relat. Struct. 8(1) (2013), 67-116 pdf

Revised on May 18, 2013 03:32:44 by Urs Schreiber (82.113.99.177)