Some weighted group algebras are operator algebras
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<α<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.