Representations of groups on Banach spaces

Abstract:

We establish a general framework for representability of a metric group on a (well-behaved) class of Banach spaces. More precisely, let $\mathcal {G}$ be a topological group, and $\mathcal {A}$ a unital symmetric $C^*$-subalgebra of $\mathrm {UC}(\mathcal {G})$, the algebra of bounded uniformly continuous functions on $\mathcal {G}$. Generalizing the notion of a stable metric, we study $\mathcal {A}$-metrics $\delta$, i.e., the function $\delta (e, \cdot )$ belongs to $\mathcal {A}$; the case $\mathcal {A}=W\hskip -0.7mm A\hskip -0.2mm P(\mathcal {G})$, the algebra of weakly almost periodic functions on $\mathcal {G}$, recovers stability. If the topology of $G$ is induced by a left invariant metric $d$, we prove that $\mathcal {A}$ determines the topology of $\mathcal {G}$ if and only if $d$ is uniformly equivalent to a left invariant $\mathcal {A}$-metric. As an application, we show that the additive group of $C[0,1]$ is not reflexively representable; this is a new proof of Megrelishvili [Topological transformation groups: selected topics, Elsevier, 2007, Question 6.7] (the problem was already solved by Ferri and Galindo [Studia Math. 193 (2009), pp. 99–108] with different methods and later the results were generalized by Yaacov, Berenstein, and Ferri [Math. Z. 267 (2011), pp.129–138]). Let now $\mathcal {G}$ be a metric group, and assume $\mathcal {A}\subseteq \mathrm {LUC}(\mathcal {G})$, the algebra of bounded left uniformly continuous functions on $\mathcal {G}$, is a unital $C^*$-algebra which is the uniform closure of coefficients of representations of $\mathcal {G}$ on members of $\mathscr {F}$, where $\mathscr {F}$ is a class of Banach spaces closed under $\ell _2$-direct sums. We prove that $\mathcal {A}$ determines the topology of $\mathcal {G}$ if and only if $\mathcal {G}$ embeds into the isometry group of a member of $\mathscr {F}$, equipped with the weak operator topology. As applications, we obtain characterizations of unitary and reflexive representability.