Representations of groups on Banach spaces

We establish a general framework for representability of a metric group on a (well-behaved) class of Banach spaces. More precisely, let G G be a topological group, and A A a unital symmetric C ∗ C^* -subalgebra of U C (G) UC (G), the algebra of bounded uniformly continuous functions on G G. Generalizing the notion of a stable metric, we study A A -metrics δ, i. e. , the function δ (e, ⋅) (e, ) belongs to A A ; the case A = W A P (G) A=W -0. 7mm A -0. 2mm P (G), the algebra of weakly almost periodic functions on G G, recovers stability. If the topology of G G is induced by a left invariant metric d d, we prove that A A determines the topology of G G if and only if d d is uniformly equivalent to a left invariant A A -metric. As an application, we show that the additive group of C 0, 1 C0, 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 G G be a metric group, and assume A ⊆ L U C (G) A LUC (G), the algebra of bounded left uniformly continuous functions on G G, is a unital C ∗ C^* -algebra which is the uniform closure of coefficients of representations of G G on members of F F, where F F is a class of Banach spaces closed under ℓ 2 ₂ -direct sums. We prove that A A determines the topology of G G if and only if G G embeds into the isometry group of a member of F F, equipped with the weak operator topology. As applications, we obtain characterizations of unitary and reflexive representability.