Herz-Schur Multipliers and Non-Uniformly Bounded Representations of Locally Compact Groups

TL;DR

This paper establishes the existence of strongly continuous representations for locally compact groups that realize Herz-Schur multipliers as coefficients, with controlled growth of the representation norm relative to the group's metric.

Contribution

It proves the existence of non-uniformly bounded representations realizing Herz-Schur multipliers with explicit norm bounds, extending previous results.

Findings

Existence of strongly continuous representations for Herz-Schur multipliers.

Representation norms grow at most exponentially with the group's metric distance.

Realization of Herz-Schur multipliers as coefficients of these representations.

Abstract

Let G be a second countable, locally compact group and let f be a continuous Herz-Schur multiplier on G. Our main result gives the existence of a (not necessarily uniformly bounded) strongly continuous representation on a Hilbert space, such that f is the coefficient of this representation with respect to two vectors with bounded orbit. Moreover, we show that the norm of the representation of an element g from G is at most exponential in terms of the metric distance from g to the identity element of G.