Sections and cones

TL;DR

This paper proves the existence of a continuous section of norm exactly 1 for surjective maps between C*-algebras and explores properties of cone C*-algebras, including liftings and quasidiagonality.

Contribution

It establishes a continuous section of norm exactly 1 and investigates cone C*-algebras' properties, including liftings, quasidiagonality, and hyperlinear traces.

Findings

Existence of a continuous section of norm exactly 1.

Any *-homomorphism from a cone over a separable C*-algebra to a quotient C*-algebra lifts to a contractive asymptotic homomorphism.

All hyperlinear traces on cones are MF.

Abstract

By Bartle-Graves theorem every surjective map between C*-algebras has a continuous section, and Loring proved that that there exists a continuous section of norm arbitrary close to 1. Here we prove that there exists a continuous section of norm exactly 1. This result is used in the second part of the paper which is devoted to properties of cone C*-algebras. It is proved that any $\ast$-homomorphism from the cone over a separable C*-algebra to a quotient C*-algebra always lifts to a contractive asymptotic homomorphism. As an application we give a short proof and strengthen the result of Forough-Gardella-Thomsen that states that any cpc (order zero) map has an asymptotically cpc (order zero, respectively) lift. As another application we give unified proofs of Voiculescu's result that cones are quasidiagonal and Brown-Carrion-White's result that all amenable traces on cones are quasidiagonal. We also prove that all hyperlinear traces on cones are MF.