nLab
homotopical structure on C*-algebras

Context

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

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Operator algebra

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory: classical, pre-quantum, quantum, perturbative quantum

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Noncommutative geometry

Contents

Idea

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.

Definitions

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 \,.
Definition

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.

Proposition

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)

Proposition

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.

Definition

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 } \,.
Remark

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)

Definition

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).

Definition

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)

Theorem

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).

Remark

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.

Definition

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.

Proposition

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)

Remark

(…) kk-groups (…)

KK-theory

Definition

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.

Theorem

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)

References

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

Last revised on May 29, 2013 at 20:02:20. See the history of this page for a list of all contributions to it.