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

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

• $𝒦≔𝒦(\mathscr{H})\hookrightarrow ℬ(\mathscr{H})$ 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)

   $$⟨-,-⟩:H\times H\to B$$\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 $(A,B)$-Hilbert C*-bimodule is an $B$-Hilbert C*-module, def. 2 $(H,⟨⟩)$ 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

$$⟨a^{*}\cdot x,y⟩=⟨x,ay⟩.$$\langle a^\ast \cdot x,y\rangle = \langle x, a y\rangle \,.

Definition

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

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

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

Example

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

Definition

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

(…)

Definition

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

Proposition

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

Proposition

There is a composition operation

$$\mathrm{KK}(A,B)\times \mathrm{KK}(B,C)\to \mathrm{KK}(A,C)$$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 $C^{*}$-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

$$\mathrm{KK}(-,-):C^{*}\mathrm{Alg}\times C^{*}\mathrm{Alg}\to C^{*}\mathrm{Alg}$$KK(-,-) \;\colon\; C^\ast Alg \times C^\ast Alg \to C^\ast Alg

is the hom-functor of an additive category.

(Higson 87, theorem 4.1)

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 $\mathrm{id}\otimes e⟨e,-⟩:A\to A\otimes 𝒦$ the morphism $Q(\mathrm{id}\otimes e⟨e)$ 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 (infinity,1)-categories to noncommutative motives (see there for more).

Corollary

The minimal tensor product of C-star-algebras

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

$$\begin{array}{ccc}{C}^{*}\mathrm{Alg}\times C^{*}\mathrm{Alg}& \stackrel{Q\times Q}{\to }& \mathrm{KK}\\ {\downarrow }^{\otimes }& & {\downarrow }^{{\otimes }_{\mathrm{KK}}}\\ C^{*}\mathrm{Alg}& \stackrel{Q}{\to }& \mathrm{KK}\end{array}.$$\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 $*$-homomorphisms

Theorem (Cuntz)

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

$$\mathrm{KK}(A,B)\simeq [qA,B\otimes 𝒦],$$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.

• $[-,-]$ 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

$$X\leftarrow (Z,E)\to Y$$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. See below.

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 Chow motives. A relation between bivariant algebraic K-theory and motivic cohomology is discussed in (Garkusha-Panin 11).

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

• $\mathrm{KK}(ℂ,A)\simeq K_0(A)$

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

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

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

• $\mathrm{KK}(ℂ,C(X))\simeq K^0(X)$

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

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

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

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

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

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

• $K(A,ℂ)\simeq K^0(A)$

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

(…)

Relation to extensions

There is an isomorphism

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

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.

(Meyer 07, Uuye 10, theorem 2.29).

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.

(Skandalis 88)

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 $C(X)\otimes C_0(T^{*}X)$ (the tensor product of the algebra of functions of compact supposer on $X$ and on its cotangent bundle) is isomorphic, in $\mathrm{KK}$, to $ℂ$:

$$d:C(X)\otimes C_0(T^{*}X)\stackrel{\simeq }{\to }ℂ.$$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)

$$K_0(C_0(T^{*}X))\simeq \mathrm{KK}(ℂ,C_0(T^{*}X))\stackrel{\mathrm{KK}(C,(-)\otimes C(X))}{\to }\mathrm{KK}(C(X),C(X)\otimes C_0(T^{*}X))\underset{\simeq }{\overset{\mathrm{KK}(C(X),d)}{\to }}\mathrm{KK}(C(X),ℂ).$$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} ) \,.

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 Roseberg 80).

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

Related concepts

• index theory

• bivariant cohomology theory

• 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 ${\mathrm{Mot}}_{\mathrm{add}}$	noncommutative motives ${\mathrm{Mot}}_{\mathrm{loc}}$	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 $C^{*}$-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 $C^{*}$-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

• 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 $\mathrm{KK}(A,B)$ 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

• 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 $C^{*}$-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)

• Snigdahayan Mahanta, Higher nonunital Quillen $K\prime$-theory, KK-dualities and applications to topological $𝕋$-duality, Journal of Geometry and Physics, Volume 61, Issue 5 2011, p. 875-889. (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

• 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 $C^{*}$-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

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, no. 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 $C^{*}$-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).

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

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

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)

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

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)

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 $C^{*}$-algebras (arXiv:0906.5400)

• Snigdahayan Mahanta, Higher nonunital Quillen $K\prime$-theory, KK-dualities and applications to topological $𝕋$-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).

• pdf

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 (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. 277:643-706,2008 (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 triple

Discussion of KK-theory for spectral triples is discussed in

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