On the growth of Hermitian groups

Abstract

A locally compact group $G$ is said to be Hermitian if every selfadjoint element of $L^1(G)$ has real spectrum. Using Halmos’ notion of capacity in Banach algebras and a result of Jenkins, Fountain, Ramsay and Williamson we will put a bound on the growth of Hermitian groups. In other words, we will show that if $G$ has a subset that grows faster than a certain constant, then $G$ cannot be Hermitian. Our result allows us to give new examples of non-Hermitian groups which could not tackled by the existing theory. The examples include certain infinite free Burnside groups, automorphism groups of trees, and $p$-adic general and special linear groups.