Approximately Multiplicative Decompositions of Nuclear Maps

TL;DR

This paper studies the approximate multiplicative decompositions of nuclear maps, focusing on conditions like quasidiagonality that enable decompositions with well-behaved maps, extending concepts to a W*-analog.

Contribution

It introduces conditions under which nuclear maps can be decomposed into order zero maps with multiplicative preservation, extending the theory to a W*-context.

Findings

Decomposition of nuclear maps with controlled multiplicative behavior

Conditions like quasidiagonality facilitate such decompositions

Extension of nuclear dimension concepts to W*-algebras

Abstract

We expand upon work from many hands on the decomposition of nuclear maps. Such maps can be characterized by their ability to be approximately written as the composition of maps to and from matrices. Under certain conditions (such as quasidiagonality), we can find a decomposition whose maps behave nicely, by preserving multiplication up to an arbitrary degree of accuracy and being constructed from order zero maps (as in the definition of nuclear dimension). We investigate these conditions and relate them to a W*-analog.