homotopical structure on C*-algebras

**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**

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

**quantum mechanical system**, **quantum probability**

**interacting field quantization**

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

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*.

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

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

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.

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

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

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

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

preserves pullbacks.

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

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

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

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 $Ho(C^\ast Alg)$

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

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)

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).

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 $A$.

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

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. .

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

(…) kk-groups (…)

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.

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. .

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)