Some weighted group algebras are operator algebras

TL;DR

This paper investigates conditions under which weighted group algebras on finitely generated groups with polynomial growth are isomorphic to operator algebras, highlighting the influence of growth order and weight type.

Contribution

It characterizes when weighted group algebras are operator algebras based on weight type and group growth, and explores algebraic center properties and specific group cases.

Findings

Weighted group algebra isomorphism to operator algebra depends on weight degree and growth order.

Algebraic center of the weighted algebra is a Q-algebra with a von Neumann inequality.

Counterexample provided by free group with two generators for exponential growth groups.

Abstract

Let $G$ be a finitely generated group with polynomial growth, and let $\om$ be a weight, i.e. a sub-multiplicative function on $G$ with positive values. We study when the weighted group algebra $\ell^1(G,\om)$ is isomorphic to an operator algebra. We show that $\ell^1(G,\om)$ is isomorphic to an operator algebra if $\om$ is a polynomial weight with large enough degree or an exponential weight of order $0<\alpha<1$. We will demonstrate the order of growth of $G$ plays an important role in this question. Moreover, the algebraic centre of $\ell^1(G,\om)$ is isomorphic to a $Q$-algebra and hence satisfies a multi-variable von Neumann inequality. We also present a more detailed study of our results when $G$ is the $d$-dimensional integers $\Z^d$ and 3-dimensional discrete Heisenberg group $\mathbb{H}_3(\Z)$. The case of the free group with two generators will be considered as a counter example of groups with exponential growth.