On unitary representability of topological groups

Abstract

We prove that the additive group $(E^\ast,\tau_k(E))$ of an $\mathscr{L}_\infty$-Banach space $E$, with the topology $\tau_k(E)$ of uniform convergence on compact subsets of $E$, is topologically isomorphic to a subgroup of the unitary group of some Hilbert space (is \emph{unitarily representable}). This is the same as proving that the topological group $(E^\ast,\tau_k(E))$ is uniformly homeomorphic to a subset of $\ell_2^\kappa$ for some $\kappa$. As an immediate consequence, preduals of commutative von Neumann algebras or duals of commutative $C^\ast$-algebras are unitarily representable in the topology of uniform convergence on compact subsets. The unitary representability of free locally convex spaces (and thus of free Abelian topological groups) on compact spaces, follows as well. The above facts cannot be extended to noncommutative von Neumann algebras or general Schwartz spaces.