On complements of Kazhdan projections in semisimple groups

TL;DR

This paper proves that for specific groups acting isometrically on certain Banach spaces, the subspace of invariant vectors has a 1-complemented complement, advancing understanding of group representations in Banach spaces.

Contribution

It establishes that the complement of invariant vectors in these representations is always 1-complemented, a novel result in the context of semisimple groups and Banach spaces.

Findings

The complement of invariant vectors is 1-complemented in certain Banach space representations.

Applicable to isometric representations of some groups.

Provides new insights into the structure of group actions on Banach spaces.

Abstract

We prove that for an isometric representation of some groups on certain Banach spaces, the complement of the subspace of invariant vectors is 1-complemented.