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ABSTRACT. We give an example of a locally compact group G, for which Loo(G) has a closed infinite dimensional reflexive left invariant subspace.In a recent paper I. Glicksberg [2] showed that if G is a compact or abelian group then every closed left invariant reflexive subspace of L^G) must be finite dimensional.The purpose of this note is to exhibit a nonabelian, noncompact group for which L^G) has an infinite dimensional closed left translation invariant reflexive subspace.Let H be an infinite dimensional Hilbert space and B(H) the bounded operators on H. Let G be the group of unitary operators in B(H) with the discrete topology.