Pointwise approximation by elementary complete contractions

TL;DR

This paper characterizes when complete contractions on C*-algebras can be approximated pointwise by elementary contractions, linking this to contractivity conditions on tensor products and providing a lifting obstruction.

Contribution

It establishes a precise criterion for approximating complete contractions by elementary ones, involving tensor product contractivity and ideal preservation.

Findings

Characterization of approximation conditions for complete contractions.

Identification of a lifting obstruction in the approximation process.

Connection between ideal-preserving contractions and tensor product contractivity.

Abstract

A complete contraction on a C*-algebra A, which preserves all closed two sided ideals J, can be approximated pointwise by elementary complete contractions if and only if the induced map on the tensor product of B with A/J is contractive for every C*-algebra B, ideal J in A and C*-tensor norm on the tensor product. A lifting obstruction for such an approximation is also obtained.