homotopical structure on C*-algebras


Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts


Operator algebra

Noncommutative geometry



By Gelfand duality, commutative C*-algebras are equivalent, dually, to suitable topological spaces. In the spirit of noncommutative topology one may therefore regard general C *C^\ast-algebras as a formal dual to generalized topological spaces and ask if the standard homotopy theory of topological spaces – as notably expressed by the standard model structure on topological spaces – generalizes to noncommutative topology.

By the general discussion at simplicial localization, for a category to present a homotopy theory it is sufficient to equip it with the structure of a category with weak equivalences / homotopical category. Such structures we discuss here. The standard such “noncommutative homotopy” structures reproduce, after stabilization, KK-theory and E-theory of C *C^\ast-algebras as their triangulated homotopy categories.

For practical purposes it is, as usual, useful to enhance the data of the weak equivalences by further (co-)fibrationauxiliary data. While there cannot be a suitable model category structure on C*Alg itself (([Uuye 10]); there is however one on l.m.c.-C*-algebras (JoachimJohnson 07)), there are suitable structures of a category of fibrant objects (Uuye 10). These we discuss here. For actual model category-structures see at model structure on operator algebras.


Category-theoretic preliminaries

Write C*Alg for the category of C*-algebras and *-algebra-homomorphisms between them. We regard this as a monoidal category (C *Alg,)(C^\ast Alg, \otimes) with the maximal tensor product of C*-algebras.

Write C *Alg sepC *AlgC^\ast Alg_{sep} \hookrightarrow C^\ast Alg for the full subcategory on the separable C *C^\ast-algebras.

Write Top for the category of compactly generated weakly Hausdorff topological spaces.

The forgetful functor which forgets the associative algebra-structure and the *-algebra-structure lands in compactly generated weakly Hausdorff spaces and hence is of the form

U:C *AlgTop. U \;\colon\; C^\ast Alg \to Top \,.

For A,BA,B \in C*Alg, let

C *Alg(A,B)Top(U(A),U(B))Top C^\ast Alg(A,B) \hookrightarrow Top(U(A), U(B)) \in Top

be the set of C *C^\ast-algebra homomorphisms equipped with the subspace topology from the mapping space of the underlying topological spaces.

This makes C *AlgC^\ast Alg a Top-enriched category.


For AA \in C*Alg and XX \in Top be compact, the there is a natural isomorphism

Top(X,U(A))U(C(X)A). Top(X,U(A)) \simeq U(C(X) \otimes A) \,.

This defines a powering of C *AlgC^\ast Alg over Top cptTop_{cpt}, by

A XC(X)A. A^X \coloneqq C(X) \otimes A \,.

(Uuye, lemma 2.2)


For every AC *AlgA \in C^\ast Alg, the hom-functor

C *Alg(A,):C *AlgTop C^\ast Alg(A,-) \;\colon\; C^\ast Alg \to Top

preserves pullbacks.

(Uuye, corollary 2.6)

Plain homotopy theory of C *C^\ast-algebras

Write I[0,1]I \coloneqq [0,1] \in Top for the standard topological interval.


For f,g:ABf,g \colon A \to B two morphisms in C*Alg, a right homotopy η:fg\eta \colon f \Rightarrow g is a morphism η:AB I\eta \colon A \to B^I such that

B f B i 0 A η B I g B i 1 B. \array{ && B \\ & {}^{\mathllap{f}}\nearrow& \uparrow^{\mathrlap{B^{i_0}}} \\ A &\stackrel{\eta}{\to}& B^I \\ & {}_{\mathllap{g}}\searrow & \downarrow^{\mathrlap{B^{i_1}}} \\ && B } \,.

Right homotopy is an equivalence relation. We write [A,B][A,B] for the set of right homotopy-equivalence classes of morphisms ABA \to B. We have a natural isomorphism

[A,B]π 0C *Alg(A,B) [A,B] \simeq \pi_0 C^\ast Alg(A,B)

These are the hom-sets of a category, the homotopy category Ho(C *Alg)Ho(C^\ast Alg)


Call a morphism f:ABf \colon A \to B in C*Alg a homotopy equivalence of C *C^\ast-algebras if it is an isomorphism in Ho(C *Alg)Ho(C^\ast Alg).


Call a morphism f:ABf \colon A \to B in C*Alg a Schochet fibration if for all DC *AlgD \in C^\ast Alg the map

C *(D,f):C *(D,A)C *(D,B) C^\ast(D,f) \;\colon\; C^\ast(D,A) \to C^\ast(D,B)

is a Serre fibration in Top.

(Schochet 84, Uuye, def. 2.14, prop. 2.18)


The category C*Alg equipped with weak equivalences being the homotopy equivalences of def. 3 and with fibrations being the Schochet fibrations of def. 4 is a category of fibrant objects.

(Schochet 84, Uuye, theorem 2.19).


The canonical functorial path space object of a C *C^\ast-algebra AA in this structure is A IA^I. It follows that the corresponding loop space object is

ΩA=C 0((0,1),A). \Omega A = C_0((0,1), A) \,.

Dually, interpreted as a space in noncommutative topology, this corresponds to the suspension of the space that corresponds to AA.

Stabilization at the compact operators

Write 𝒦C *Alg\mathcal{K} \in C^\ast Alg for the C*-algebra of bounded operators on an infinite-dimensional separable Hilbert space.


Say a morphism f:ABf \colon A \to B in C*Alg is a stable homotopy equivalence if fid 𝒦f \otimes id_{\mathcal{K}} is a homotopy equivalence, def. 3.

Say ff is a stable Schochet fibration of fid 𝒦f \otimes id_{\mathcal{K}} is a Schochet fibration, def. 4.


The category C*Alg becomes a category of fibrant objects with weak equivalences the stable homotopy equivalences and fibrations the stable Schochet fibrations.

(Uuye, prop. 2.24)


(…) kk-groups (…)



Say a morphism f:ABf \colon A \to B in C *Alg sepC^\ast Alg_{sep} is a KK-equivalence if for all DC *Alg sepD \in C^\ast Alg_{sep} the morphism

KK(D,f):KK(D,A)K(D,B) KK(D,f) \;\colon\; KK(D,A) \to K(D,B)

is an isomorphism, where KKKK is the KK-theory-category.


The category C *Alg sepC^\ast Alg_{sep} carries the structure of a category of fibrant objects whose weak equivalences are the KK-equivalences of def. 6, and whose fibrations are the Schochet fibrations, def. 4.

(Uuye, theore, 2.29)


The various structures of a category of fibrant objects on C*Alg are discussed in

using notions from

  • Claude Schochet, Topological methods for C∗-algebras. III. Axiomatic homology, Pacific J. Math. 114 (1984), no. 2, 399–445. MR 757510 (86g:46102)

A model categorical approach is presented in

A model category structure presenting KK-theory not on C *C^\ast-algebras itselt but on l.m.c.-C*-algebras is discussed in

Revised on May 29, 2013 20:02:20 by Urs Schreiber (