[Paper Review] Generalized Heisenberg groups and Shtern's question

Michael Megrelishvili|ArXiv.org|Nov 6, 2004

Advanced Banach Space Theory8 references3 citations

TL;DR

This paper establishes that for any normed space $X$, the subgroups $X$ and $X^*$ are relatively minimal in the generalized Heisenberg group $H(X) = (\mathbb{R} \times X) \ltimes X^*$. It further proves that $G = H(L_4[0,1])$ is reflexively representable, yet its weakly continuous unitary representations in Hilbert spaces fail to separate points—answering a question posed by A. Shtern regarding the existence of such groups.

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.

Motivation & Objective

• To resolve A. Shtern's question concerning the existence of a reflexively representable topological group whose weakly continuous unitary representations in Hilbert spaces do not separate points.
• To establish the relative minimality of $X$ and $X^*$ as subgroups within the generalized Heisenberg group $H(X)$ for any normed space $X$.
• To analyze the double limit property (DLP) of functions on $L_p$-spaces to verify the weak almost periodicity of specific functions on $H(L_4[0,1])$.
• To demonstrate that the group $G = H(L_4[0,1])$ is reflexively representable via a bounded continuous function that is weakly almost periodic but not separating points.
• To show that the failure of point separation in unitary representations is tied to the structure of $L_4$ and $L_{4/3}$ dual spaces via the DLP and Eberlein-Šmulian theorem.

Proposed method

• Define the generalized Heisenberg group $H(X) = (\mathbb{R} \times X) \ltimes X^*$ using the canonical duality pairing $w: X \times X^* \to \mathbb{R}$, with group operation $(r_1,x_1,f_1) \cdot (r_2,x_2,f_2) = (r_1 + r_2 + f_1(x_2), x_1 + x_2, f_1 + f_2)$.
• Prove relative minimality of $X$ and $X^*$ in $H(X)$ by showing that any coarser Hausdorff group topology on $H(X)$ induces the original topology on $X$ and $X^*$, using the continuity of the quotient map $q: (H(X), \sigma) \to X \times X^*$.
• Construct a bounded continuous function $\phi(r,x,f) = \frac{1}{1 + |r| + \|x\| + \|f\|}$ on $G = H(L_4[0,1])$ that separates the identity from closed sets.
• Establish that $\phi$ is weakly almost periodic (wap) by verifying it satisfies the double limit property (DLP) for all component functions: $\|f\|$, $\|x\|$, and $f(x)$.
• Use Shoenberg's result that $f \mapsto e^{-\|f\|^p}$ is positive definite on $L_p$ for $1 \leq p \leq 2$, implying such functions are wap, to confirm the DLP for $\|f\|$ on $L_{4/3}$.
• Apply the Eberlein-Šmulian theorem to extract weakly convergent subsequences from bounded sequences in $L_4$ and $L_{4/3}$, and use separate continuity of the duality pairing to equate double limits $\lim_m \lim_n f_n(y_m) = \lim_n \lim_m f_n(y_m)$.

Experimental results

Research questions

• RQ1 Does there exist a reflexively representable topological group whose weakly continuous unitary representations in Hilbert spaces fail to separate points?
• RQ2 Are the subgroups $X$ and $X^*$ relatively minimal in the generalized Heisenberg group $H(X) = (\mathbb{R} \times X) \ltimes X^*$ for every normed space $X$?
• RQ3 Does the function $\phi(r,x,f) = \frac{1}{1 + |r| + \|x\| + \|f\|}$ on $H(L_4[0,1])$ satisfy the double limit property (DLP), implying it is weakly almost periodic?
• RQ4 Can the double limits $\lim_m \lim_n f_n(y_m)$ and $\lim_n \lim_m f_n(y_m)$ be shown to be equal for bounded sequences in $L_4$ and $L_{4/3}$, under the assumption of existence?
• RQ5 Is the norm on $c_0$ the only $L_p$-space norm that fails the DLP, and how does this relate to the failure of point separation in unitary representations?

Key findings

• For every normed space $X$, the subgroups $X$ and $X^*$ are relatively minimal in the generalized Heisenberg group $H(X) = (\mathbb{R} \times X) \ltimes X^*$, as any coarser Hausdorff group topology on $H(X)$ induces the original topology on $X$ and $X^*$.
• The group $G = H(L_4[0,1])$ is reflexively representable, as it admits a continuous embedding into the isometry group of a reflexive Banach space with the strong operator topology.
• The function $\phi(r,x,f) = \frac{1}{1 + |r| + \|x\| + \|f\|}$ on $G$ is weakly almost periodic, as it satisfies the double limit property (DLP) for all component functions.
• The DLP holds for $\|f\|$ on $L_{4/3}$ because $f \mapsto e^{-\|f\|^{4/3}}$ is positive definite on $L_{4/3}$, and positive definite functions are weakly almost periodic.
• The DLP holds for $\|x\|$ on $L_4$ due to a result in the paper's Lemma 3.3.5, which establishes the DLP for the norm on $L_p$ for $1 \leq p \leq 2$.
• The equality $\lim_m \lim_n f_n(y_m) = \lim_n \lim_m f_n(y_m)$ holds for bounded sequences in $L_4$ and $L_{4/3}$ due to weak sequential compactness and separate continuity of the duality pairing, resolving the DLP for the pairing term.