On unitary representability of topological groups

Abstract

Abstract. We prove that the additive group (E * , τ k (E)) of an L∞-Banach space E, with the topology τ 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 unitarily representable). This is the same as proving that the topological group (E * , τ k (E)) is uniformly homeomorphic to a subset of ℓ κ 2 for some κ. As an immediate consequence, preduals of commutative von Neumann algebras or duals of commutative C * -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.

Embedding a topological group into its WAP-compactification

“…While it is proved in [8] that S(c 0 ) is not unitarily representable, we prove below that it is reflexively representable. We shall derive this fact from the following factorization theorem of Fonf, Johnson, Plichko, and Shevchyk that uses Johnson's space R defined in [12].…”

“…Among stable spaces we find all L p (µ)-spaces for 1 ≤ p < ∞. It is known on the other hand that L p (µ) is not unitarily representable for p > 2 (this depends on results of Schoenberg and Megrelishvili, see the discussion in [8]). For a proof of the reflexive representability of some L p (µ)-spaces that does not refer to the stability of the norm, see [17].…”

“…Proof. The proof is as that of [8,Theorem 3.3]. Let K( 1 ) denote any set that is cofinal in the family of all compact subsets of 1 (ordered by inclusion).…”

“…In the last section we deal with Schwartz locally convex spaces. In [8] it was shown that Schwartz spaces need not be representable as groups of isometries of any Hilbert space. We see here in contrast that Schwartz spaces are always representable as groups of isometries on some reflexive Banach space.…”

“…There are some other interesting classes of unitarily representable groups, such as the additive groups of some Banach spaces, like L p (µ), 1 ≤ p ≤ 2, or free Abelian topological groups [24]. The question of unitary representability is treated in more detail in [8].…”