Generalized Heisenberg groups and Shtern's question

TL;DR

This paper investigates properties of generalized Heisenberg groups derived from normed spaces, proving minimality of certain subgroups and addressing a question about reflexive representability versus weakly continuous representations.

Contribution

It establishes that X is a relatively minimal subgroup of H(X) and provides a counterexample showing G is reflexively representable but not distinguished by weakly continuous unitary representations.

Findings

X is a relatively minimal subgroup of H(X)

G = H(L_4[0,1]) is reflexively representable

Weakly continuous unitary representations of G do not separate points

Abstract

Let H(X) be the generalized Heisenberg group induced by a normed space X. We prove that X is a relatively minimal subgroup of H(X). We show that the group $G:=H(L_4[0,1])$ is reflexively representable but weakly continuous unitary representations of G in Hilbert spaces do not separate points of G. This answers a question of A. Shtern.