Homotopy lifting, asymptotic homomorphisms, and traces

TL;DR

This paper proves a homotopy lifting theorem for *-homomorphisms and asymptotic homomorphisms, and applies it to establish invariance and trace properties of C*-algebras, with implications for quasidiagonality and KK-theory.

Contribution

It introduces a homotopy lifting theorem for asymptotic homomorphisms and uses it to derive new invariance and trace properties of C*-algebras.

Findings

MF-property is homotopy invariant

Amenable traces on homotopy dominated algebras are quasidiagonal if on the dominating algebra

All hyperlinear traces on certain algebras are MF

Abstract

The following homotopy lifting theorem is proved: Let $\phi, \psi: B \to D/I$ be homotopic $\ast$-homomorphisms and suppose $\psi$ lifts to a (discrete) asymptotic homomorphism. Then $\phi$ lifts to a (discrete) asymptotic homomorphism. Moreover the whole homotopy lifts. We also prove a cp version of this theorem and a version where $\phi$ is replaced by an asymptotic homomorphism. We obtain a lifting characterization of several important properties of C*-algebras and use them together with the lifting theorem to get the following applications: 1) MF-property is homotopy invariant; 2) If either $A$ or $B$ is exact, $A$ is homotopy dominated by $B$ and all amenable traces on $B$ are quasidiagonal, then all amenable traces on $A$ are quasidiagonal; 3) If a C*-algebra $A$ is homotopy dominated by a nuclear C*-algebra $B$ and all (hyperlinear) traces on $B$ are MF, then all hyperlinear traces on $A$ are MF. 4) Some of the extension groups introduced by Manuilov and Thomsen coincide. 5) The C*-algebra $qA$ from Cuntz's picture of KK-theory is always quasidiagonal.