  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.
