# nLab homotopical structure on C*-algebras

Contents

### Context

#### Noncommutative geometry

noncommutative geometry

(geometry $\leftarrow$ Isbell duality $\to$ algebra)

# 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 $C^\ast$-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 $C^\ast$-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 $(C^\ast Alg, \otimes)$ with the maximal tensor product of C*-algebras.

Write $C^\ast Alg_{sep} \hookrightarrow C^\ast Alg$ for the full subcategory on the separable $C^\ast$-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

$U \;\colon\; C^\ast Alg \to Top \,.$
###### Definition

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

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

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

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

###### Proposition

For $A \in$ C*Alg and $X \in$ Top be compact, the there is a natural isomorphism

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

This defines a powering of $C^\ast Alg$ over $Top_{cpt}$, by

$A^X \coloneqq C(X) \otimes A \,.$
###### Proposition

For every $A \in C^\ast Alg$, the hom-functor

$C^\ast Alg(A,-) \;\colon\; C^\ast Alg \to Top$

preserves pullbacks.

### Plain homotopy theory of $C^\ast$-algebras

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

###### Definition

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

$\array{ && B \\ & {}^{\mathllap{f}}\nearrow& \uparrow^{\mathrlap{B^{i_0}}} \\ A &\stackrel{\eta}{\to}& B^I \\ & {}_{\mathllap{g}}\searrow & \downarrow^{\mathrlap{B^{i_1}}} \\ && B } \,.$
###### Remark

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

$[A,B] \simeq \pi_0 C^\ast Alg(A,B)$

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

###### Definition

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

###### Definition

Call a morphism $f \colon A \to B$ in C*Alg a Schochet fibration if for all $D \in C^\ast Alg$ the map

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

is a Serre fibration in Top.

###### Theorem

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

###### Remark

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

$\Omega A = C_0((0,1), A) \,.$

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

### Stabilization at the compact operators

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

###### Definition

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

Say $f$ is a stable Schochet fibration of $f \otimes id_{\mathcal{K}}$ is a Schochet fibration, def. .

###### Proposition

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

###### Remark

(…) kk-groups (…)

### KK-theory

###### Definition

Say a morphism $f \colon A \to B$ in $C^\ast Alg_{sep}$ is a KK-equivalence if for all $D \in C^\ast Alg_{sep}$ the morphism

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

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

###### Theorem

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

## References

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

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

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 $C^\ast$-algebras itselt but on l.m.c.-C*-algebras is discussed in

Last revised on July 22, 2018 at 15:45:43. See the history of this page for a list of all contributions to it.