nLab asymptotic C-star-homomorphism

Idea

In noncommutative topology the standard notion of homomorphism of C*-algebras is too restrictive for some applications, related to the fact that some noncommutative $C^\ast$-algebras correspond to “locally badly behaved” noncommutative topological spaces. The notion of asymptotic $C^\ast$-homomorphism is more flexible than that of plain $C^\ast$-homomorphisms and designed to correct this problem. Homotopy classes of asymptotic $C^\ast$-homomorphisms are the hom-sets in a category called E-theory. See there for more details.

Definition

Definition

For $A,B$ two C*-algebras, an asymptotic homomorphism between them is a $[0,\infty)$-parameterized collection of continuous functions $\{\phi_t \colon A \to B\}_{t \in [0, \infty)}$, such that

• for each $a \in A$, the function $t \mapsto \phi_t(a)$ is a continuous function;

• in the limit $t \to \infty$, $\phi_t$ becomes a star-algebra homomorphism.

As for ordinary $C^\ast$-algebra homomorphisms one puts:

Definition

For $f_t, g_t \colon A \to B$ to asymptotic $C^\ast$-homomorphisms, def. , a (right) homotopy between them is an asyptotic homomorphism $\eta_t \colon A \to C([0,1],B)$ which restricts to $f$ at 0 and to $g$ at $1$, hence such that it fits into a commuting diagram of the form

$$\array{ && B \\ & {}^{\mathllap{f_t}}\nearrow & \uparrow^{\mathrlap{ev_0}} \\ A &\stackrel{\eta_t}{\to}& C([0,1], B) \\ & {}_{\mathllap{g_t}}\searrow & \downarrow^{\mathrlap{ev_1}} \\ && B } \,.$$

Homotopy of asymptotic $C^\ast$-homomorphisms is clearly an equivalence relation. Write $[A,B]$ for the set of homotopy-equivalence classes of asymptotic homomorphisms $A \to B$.

Proposition

For $A,B \in$ C*Alg, the set $[A, C_0((0,1), B)]$ is naturally an abelian group under the composition operation which sends the homotopy classes presented by $f,g \colon A \times (0,1) \to B$ to the homotopy class of

$$f + g \;\colon\; A \times (0,1) \stackrel{\cdot 2}{\to} A \times (0,2) \stackrel{x \mapsto \left\{ \array{f(x) & x \lt 1 \\ g(x-1) & x \gt 1 } \right. }{\to} B \,.$$

Remark

The $t$-wise composition of two asymptotic $C^\ast$-homomorphisms is not in general itself an asymptotic $C^\ast$-homomorphims. However, every asympotic homomorphism is homotopic to one which is an equicontinuous function, and $t$-wise composition of equicontinuous asymptotic $C^\ast$-homomorphisms is again an asymptotic homomorphism.

Examples

Example

Two asymptotic $C^\ast$-homomorphisms which differe just by a reparameterization of $[0,\infty)$ while having the same limit can be related by a homotopy, def. .

Example

(self-adjointification homotopy)

For $f_t \colon A \to B$ an asymptotic $C^\ast$-homomorphism, there is a homotopy to the asymptotic morphism

$$\tilde f_t(a) \coloneqq \tfrac{1}{2}\left(f(a) + f(a^\ast)^\ast\right) \,.$$

References

The notion was introduced in

• Alain Connes, Nigel Higson, Déformations, morphismes asymptotiques et $K$-théorie bivariante, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), no. 2, 101–106, MR91m:46114, pdf

A review is for instance around p. 23 of

• Introduction to KK-theory and E-theory, Lecture notes (Lisbon 2009) (pdf slides)