On the growth of Hermitian groups

  • Rui Palma

    University of Oslo, Norway

Abstract

A locally compact group GG is said to be Hermitian if every selfadjoint element of L1(G)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 GG has a subset that grows faster than a certain constant, then GG 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 pp-adic general and special linear groups.

Cite this article

Rui Palma, On the growth of Hermitian groups. Groups Geom. Dyn. 9 (2015), no. 1, pp. 29–53

DOI 10.4171/GGD/304