nLab homotopical structure on C*-algebras

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

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

(Uuye, lemma 2.2)

Proposition

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

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

preserves pullbacks.

(Uuye, corollary 2.6)

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.

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

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.

(Schochet 84, Uuye, theorem 2.19).

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.

(Uuye, prop. 2.24)

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. .

(Uuye, theore, 2.29)

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

• Paul Arne Østvær, Homotopy theory of $C^*$-algebras, Frontiers in Mathematics, Springer Basel, 2010, (arxiv/0812.0154, pdf)

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