**M. Megrelishvili**

## Generalized Heisenberg Groups and Shtern's
Question

**Abstract:**

Let $H(X):=(\R \times X) \leftthreetimes X^*$ be the generalized Heisenberg
group induced by a normed space $X$. We prove that $X$

and $X^*$ are relatively minimal subgroups 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 the question of A. Shtern.

**Keywords:**

Heisenberg group, unitary representation, minimal topological group, relatively
minimal subgroup, weakly almost periodic, positive definite, reflexive space.

**MSC 2000:** 22A05, 43A60, 54H10