nLab KK-theory



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

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).


We state first the original and standard definition of KKKK-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 *C^\ast-bimodules


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


We write

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

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


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

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

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

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

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

    (the BB-valued inner product)

such that

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

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

    2. x,x0\langle x,x\rangle \geq 0 (in the sense of positive elements in BB)

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

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

  2. HH is complete with respect to the norm:

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


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

a *x,y=x,ay. \langle a^\ast \cdot x,y\rangle = \langle x, a y\rangle \,.

For A,BA, B \in C*Alg, Kasparov (A,B)(A,B)-bimodule is a 2\mathbb{Z}_2-graded (A,B)(A,B)-Hilbert bimodules ,,\mathcal{H}, \langle -,-\rangle, def. , equipped with an adjointable odd-graded bounded operator F B()F \in \mathcal{B}_B(\mathcal{H}) such that

  1. (F 21)π(a)𝒦 B()(F^2 - 1)\pi(a) \in \mathcal{K}_B(\mathcal{H})

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

  3. (FF *)π(a)𝒦 B()(F - F^\ast) \pi(a)\in \mathcal{K}_B(\mathcal{H})

for all aAa \in A,

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

For instance (Blackadar 99, p. 144).


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


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



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


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


There is a composition operation

KK(A,B)×KK(B,C)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).


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.


On the other hand, at least between C *C^\ast-algebras which are algebras of functions on smooth manifolds A i=C(X i)A_i = C(X_i) , KK-classes are presented by correspondences X 1ZX 2X_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


The Kasparov product, def. , is associative. Thus under the Kasparov product

KK(,):C *Alg×C *AlgC *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 KKKK is a kind of localization of the category of C-star-algebras:


The canonical functor

Q:C *AlgKK Q \colon C^\ast Alg \to KK

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

  1. KKKK is an additive category;

  2. QQ is homotopy-invariant;

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

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

  4. QQ 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).


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.


The minimal tensor product of C-star-algebras

:C *Alg×C *AlgC *Alg \otimes \colon C^\ast Alg \times C^\ast Alg \to C^\ast Alg

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

C *Alg×C *Alg Q×Q KK KK C *Alg Q KK. \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,BA,B are C-star-algebras with AA separable and BB σ\sigma-unital, then

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


(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(Z,E)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. For more on this see also the pointers at at motivic quantization.


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

  • specifically for K-theory cocycles: KK^ 𝒢 *\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

KKNCC dg 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 𝒢\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 𝒢\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.


Basic examples


For f:ABf \colon A \to B a homomorphism of 2\mathbb{Z}_2 graded C*-algebras, take BB as a right Hilbert module over itself and equip it with the left action of AA induced by ff. This makes it a Hilbert bimodule. Together with the 0-Fredholm operator, this represents an element

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

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



(H i,F i)KK(A,B) (H_i, F_i) \in KK(A,B)

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

(H 1H 2,F 1F 2)KK(A 1A 2,B). (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


Let (X,g)(X,g) be a closed smooth Riemannian manifold, and let V 0,V 1V_0, V_1 be two smooth vector bundles over XX with Hermitian strucure (associated to a chosen unitary group-principal bundle).

Then given an elliptic pseudodifferential operator

P:Γ(V 0)Γ(V 1) 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)L^2(V_i).

Consider then the 2\mathbb{Z}_2-graded Hilbert space

HL 2(V 0)L 2(V 1) H \coloneqq L^2(V_0) \oplus L^2(V_1)

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

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

is a Fredholm operator on HH, so that

(L 2(V 1)L 2(V 2),[0 Q P 0])KK(C(X),). \left( L^2(V_1) \oplus L^2(V_2), \left[ \array{ 0 & Q \\ P & 0 } \right] \right) \in KK(C(X),\mathbb{C}) \,.

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

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


FD1+D 2 F \coloneqq \frac{D}{\sqrt{1 + D^2}}

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

(L 2(Ω 0,),¯+¯ *1+(¯+¯ *) 2)KK(C(X),). \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}) \,.


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 AA \in C*Alg we have that

  • KK(,A)K 0(A)KK(\mathbb{C}, A) \simeq K_0(A)

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

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

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

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

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

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

  • KK(,C r(𝒢 ))K 0(𝒢)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:𝒢B 2U(1)c \colon \mathcal{G} \to \mathbf{B}^2 U(1) is a circle 2-group-principal 2-bundle (U(1)U(1)-bundle gerbe) over 𝒳\mathcal{X} and if A=C(𝒳 ,c)A = C(\mathcal{X}_\bullet, c) is the twisted groupoid convolution algebra of the corresponding centrally extended Lie groupoid, then

  • KK(,C r(𝒳 ,x))=K 0(𝒳,c)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 AA in the first argument and the complex numbers in the second we have that

  • K(A,)K 0(A)K(A,\mathbb{C}) \simeq K^0(A)

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


Relation to extensions

There is an isomorphism

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

to a suitable group of suitable extensions of AA by BB. (Kasparov 80, reviewed in Inassaridze).

Triangulated structure and KUKU-module structure


KKKK is naturally a stable triangulated category.

(Meyer 07, Uuye 10, theorem 2.29).


There is a functor

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

to the stable homotopy category such that

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

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

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

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

    𝕂:C *AlgHo(𝕂()Mod) \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

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

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


Since 𝕂\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 KKKK generated from the tensor unit is called the bootstrap category BootKKBoot \hookrightarrow KK. For ABootKKA \in Boot \hookrightarrow KK one has that KK(A,B)KK(A,B) satisfies a Künneth theorem. See at bootstrap category for more.

Excision and relation to E-theory


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


By theorem , KKKK is excisive over split exact sequences.


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

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

More generally:


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

(Skandalis 88)


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

Poincaré duality and Thom isomorphism

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


Let XX be a smooth manifold which is compact. Then the C*-algebra C(X)C 0(T *X)C(X) \otimes C_0(T^\ast X) (the tensor product of the algebra of functions of compact support on XX and on its cotangent bundle) is isomorphic, in KKKK, to \mathbb{C}:

d:C(X)C 0(T *X). d \colon C(X) \otimes C_0(T^\ast X) \stackrel{\simeq}{\to} \mathbb{C} \,.

(Kasparov 80)


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

K 0(C 0(T *X))KK(,C 0(T *X))KK(C,()C(X))KK(C(X),C(X)C 0(T *X))KK(C(X),d)KK(C(X),). 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 AA, BB are Poincaré duality algebras, def. , then for f:ABf \colon A \to B a morphism, the corresponding Umkehr map is (postcomposition) with the dual morphism of its opposite algebra version:

f!(f op) *. 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

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 addMot_{add}noncommutative motives Mot locMot_{loc}algebraic K-theorynon-connective algebraic K-theory
noncommutative topologyKK-theoryE-theoryoperator K-theory



KK-theory was introduced by Gennady Kasparov in

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 *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:


Excision for KK-theory is further studied in

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

  • Joachim Cuntz, Generalized Homomorphisms Between C *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 *AlgC^\ast Alg at the C *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 KK'-theory, KK-dualities and applications to topological 𝕋\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 *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

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

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

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

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 *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

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

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)C(Y) is generalized to C(Y)AC(Y) \otimes A for AA a unital separable C *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.

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

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

For a collection of literature see also paragraph 1.5 in

(in the context of noncommutative motives).


  • Snigdhayan Mahanta, Higher nonunital Quillen KK'-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 KKNCC dgKK \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

based on

Smooth refinement and spectral triples

Discussion of KK-theory for spectral triples is discussed in

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