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asymptotic C-star-homomorphism

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Context

Operator algebra

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

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Contents

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 *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.

Definition

Definition

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:

Definition

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.

Proposition

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 \,.
Remark

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.

Examples

Example

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. .

Example

(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) \,.

References

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.