Projectional Skeletons of Fourier Algebras

TL;DR

This paper proves that preduals of group von Neumann algebras of locally compact groups have a projectional skeleton where each dual map is a conditional expectation.

Contribution

It establishes that all such preduals possess a projectional skeleton with the property that each dual map is a conditional expectation.

Findings

Preduals of group von Neumann algebras have a projectional skeleton.

Each dual map in the skeleton is a conditional expectation.

Affirms the property for preduals of locally compact groups' von Neumann algebras.

Abstract

The preduals of $W^*$-algebras are 1-Plichko spaces. A natural question arises: does every predual possess a projectional skeleton (PS) $\{P_s:s\in J\}$ such that each $P_s^*$ is a conditional expectation? In this note, we answer this question affirmatively for the preduals of the group von Neumann algebras of locally compact groups.