nLab asymptotic C-star-homomorphism



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



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

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


For f t,g t:ABf_t, g_t \colon A \to B to asymptotic C *C^\ast-homomorphisms, def. , a (right) homotopy between them is an asyptotic homomorphism η t:AC([0,1],B)\eta_t \colon A \to C([0,1],B) which restricts to ff at 0 and to gg at 11, hence such that it fits into a commuting diagram of the form

B f t ev 0 A η t C([0,1],B) g t ev 1 B. \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 *C^\ast-homomorphisms is clearly an equivalence relation. Write [A,B][A,B] for the set of homotopy-equivalence classes of asymptotic homomorphisms ABA \to B.


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

f+g:A×(0,1)2A×(0,2)x{f(x) x<1 g(x1) x>1B. 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 \,.

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



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


(self-adjointification homotopy)

For f t:ABf_t \colon A \to B an asymptotic C *C^\ast-homomorphism, there is a homotopy to the asymptotic morphism

f˜ t(a)12(f(a)+f(a *) *). \tilde f_t(a) \coloneqq \tfrac{1}{2}\left(f(a) + f(a^\ast)^\ast\right) \,.


The notion was introduced in

A review is for instance around p. 23 of

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

Last revised on April 10, 2013 at 20:33:13. See the history of this page for a list of all contributions to it.