nLab KK-theory

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

• In terms of Fredholm Hilbert C-star-bimodules

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

• Universal category-theoretic characterization.

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

• In terms of homotopy classes of star-homomorphisms.

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

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

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

Definition

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

1. a complex vector space $\mathcal{H}$;

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

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

   $$\langle -,-\rangle \colon \mathcal{H} \times \mathcal{H} \to B$$

   (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. $(\mathcal{H}, \langle \rangle)$ 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. , equipped with an adjointable odd-graded bounded operator $F \in \mathcal{B}_B(\mathcal{H})$ such that

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

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

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

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. .

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 positive 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. , is associative. 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.

(Higson 87, theorem 4.1)

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 and see the references below.

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

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

(Higson 87, theorem 4.8)

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 structure and $KU$-module structure

Proposition

$KK$ is naturally a stable triangulated category.

(Meyer 07, Uuye 10, theorem 2.29).

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 , $KK$ 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.

(Skandalis 88)

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.

(Brodzki-Mathai-Rosenberg-Szabo 07, def. 2.1)

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 $\mathbb{C}$:

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

(Kasparov 80)

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

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

(Brodzki-Mathai-Rosenberg-Szabo 07, p. 14)

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

Further Theorems

• The Baum-Connes conjecture is naturally formulated within KK-theory.

• The Novikov conjecture has been verified in many cases using KK-theory. (see for instance Rosenberg 80).

• The Atiyah-Singer index theorem is naturally formutaled in KK-theory/E-theory. (See (Higson-Roe)).

Related concepts

• KK-bootstrap category

• equivariant KK-theory

• index theory

• bivariant cohomology theory

• bivariant algebraic K-theory, K-motive

• operator K-theory

• E-theory

• homotopical structure on C*-algebras, model structure on operator algebras

• assembly map, Baum-Connes conjecture

geometric context	universal additive bivariant (preserves split exact sequences)	universal localizing bivariant (preserves all exact sequences in the middle)	universal additive invariant	universal localizing invariant
noncommutative algebraic geometry	noncommutative motives $Mot_{add}$	noncommutative motives $Mot_{loc}$	algebraic K-theory	non-connective algebraic K-theory
noncommutative topology	KK-theory	E-theory	operator 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

• Bruce 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 $C^\ast$-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)

On an approach using quasi-homomorphisms:

• Joachim Cuntz, James Gabe. Generalized homomorphisms and KK with extra structure (2024). (arXiv:2404.06840).

Excision

Excision for KK-theory is further studied in

• Joachim Cuntz, Georges Skandalis, Mapping cones and exact sequences in KK-theory, J. Operator Theory 15 (1986) 163-180.

• 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. doi:10.1007/BFb0072109

• 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

• Klaus Thomsen, The universal property of equivariant KK-theory (K-theory preprint)

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

• Hvedri Inassaridze, Universal property of Kasparov bivariant K-theory (pdf, ps)

A triangulated category structure for KK-theory is discussed in

• Ralf Meyer, Categorical aspects of bivariant K-theory, (arXiv:math/0702145)

• Ralf Meyer, Ryszard Nest, Homological algebra in bivariant K-theory and other triangulated categories (arXiv:math/0702146)

• Ralf Meyer, KK-theory as a triangulated category, Lecture notes (2009) (pdf)

A model category realization of KK-theory is discussed in

• Michael Joachim, Mark Johnson, Realizing Kasparov’s KK-theory groups as the homotopy classes of maps of a Quillen model category (arXiv:0705.1971)

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

• Otgonbayar Uuye, Homotopy theory for $C^\ast$-algebras (arXiv:1011.2926)

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.

As the homotopy category of a stable $\infty$-category

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

• Ivo Dell'Ambrogio, Heath Emerson, Tamaz Kandelaki, Ralf Meyer, A functorial equivariant K-theory spectrum and an equivariant Lefschetz formula (arXiv:1104.3441)

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

• Snigdhayan Mahanta, Noncommutative stable homotopy and semigroup $C^*$-algebras, ESI preprint 2394 (arXiv:1211.6576)

A complete realization of (equivariant, even) KK-theory as the homotopy category of a stable $(\infty,1)$-category:

• Ulrich Bunke, Alexander Engel, Markus Land, A stable $\infty$-category for equivariant KK-theory $[$arXiv:2102.13372$]$

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

• Nigel Higson, John Roe, Lectures on operator K-theory and the Atiyah-Singer Index Theorem (2004) (pdf)

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

• Klaas Landsman, Quantization as a functor (arXiv:math-ph/0107023)

• Klaas Landsman, Functorial quantization and the Guillemin-Sternberg conjecture , Proc. Bialowieza 2002 (arXiv:math-ph/0307059)

• 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)

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

• Jean-Louis Tu, Ping Xu, Camille Laurent-Gengoux, Twisted K-theory of differentiable stacks (arXiv:math/0306138)

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

• Bram Mesland, Groupoid cocycles and K-theory (arXiv:1005.3677)

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

• Alain Connes, Georges Skandalis, The longitudinal index theorem for foliations, Publ. Res. Inst. Math. Sci. 20 6, 1139-1183 (1984) (pdf)

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:

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

In section 5 of

• Jacek Brodzki, Varghese Mathai, Jonathan Rosenberg, Richard Szabo, Noncommutative correspondences, duality and D-branes in bivariant K-theory, Adv. Theor. Math. Phys.13:497-552,2009 (arXiv:0708.2648)

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.

A ‘generators and relations’ description of KK-theory in terms of spans is given in

• Bernhard Burgstaller, The generators and relations picture of KK-theory, (2016) arXiv:1602.03034

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

• Heath Emerson, Ralf Meyer, Bivariant K-theory via correspondences, Adv. Math. 225 (2010), 2883-2919 (arXiv:0812.4949)

based on

• Heath Emerson, Ralf Meyer Equivariant embedding theorems and topological index maps, Adv. Math. 225 (2010), 2840-2882 (arXiv:0908.1465)

• Heath Emerson, Ralf Meyer, Dualities in equivariant Kasparov theory (arXiv:0711.0025)

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

• Paul Baum, Jonathan Block, Equivariant bicycles on singular spaces. C.R. Acad. Sci. Paris, t. 311 Serie I, 1990 (pdf)

• Heath Emerson, Ralf Meyer, Equivariant embedding theorems and topological index maps, Adv. Math. 225 (2010), 2840-2882 (arXiv:0908.1465)

Further developments of this are in

• Heath Emerson, Duality, correspondences and the Lefschetz map in equivariant KK-theory: a survey (arXiv:0904.4744)

• Heath Emerson, Robert Yuncken, Equivariant correspondences and the Borel-Bott-Weil theorem (arXiv:0905.1153)

Relation to motives and algebraic KK-theory

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

• Alain Connes, Caterina Consani, Matilde Marcolli, Noncommutative geometry and motives: the thermodynamics of endomotives (arXiv:math/0512138)

• Alain Connes, Matilde Marcolli, Noncommutative Geometry, Quantum Fields and Motives

and also very briefly in (Meyer 06).

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

• Guillermo Cortiñas, Andreas Thom, Bivariant algebraic K-theory, J. Reine Angew. Math. 510 (2007), 71–124. (arXiv:math/0603531)

• Snigdahayan Mahanta, Noncommutative correspondence categories, simplicial sets and pro $C^\ast$-algebras (arXiv:0906.5400)

• 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)

• Grigory Garkusha, Ivan Panin, K-motives of algebraic varieties (arXiv:1108.0375)

• Grigory Garkusha, Algebraic Kasparov K-theory. II (arXiv:1206.0178)

For a collection of literature see also paragraph 1.5 in

• Andrew Blumberg, David Gepner, Gonçalo Tabuada, A universal characterization of higher algebraic K-theory (arXiv:1001.2282)

(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.

See also at motivic quantization and motives in physics.

In D-brane theory

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

A review is in

• Richard Szabo, D-branes and bivariant K-theory, Noncommutative Geometry and Physics 3 1 (2013): 131. (arXiv:0809.3029)

based on

• Jacek Brodzki, Varghese Mathai, Jonathan Rosenberg, Richard Szabo,

  D-Branes, RR-Fields and Duality on Noncommutative Manifolds, Commun. Math. Phys. 277 (2008) 643-706 [arXiv:hep-th/0607020, doi:10.1007/s00220-007-0396-y]

  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

• Bram Mesland, Spectral triples and KK-theory: A survey (arXiv:1304.3802)