A week-operator closed subalgebra B of bounded operators on a Hilbert space H such that B ≠ B^{”}, where B^{”} denotes the doubel commutant of B.

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Let H be a Hilbert space of dimension greater than 1, ξ be a unit vector in H and B be the subalgebra of B(H) consisting of those operators for which ξ is an eigenvector. Let P be the projection with range [ξ] (If K ⊆ H, we denote the closed linear span of K by [K]). Then T ∈ B iff P T P = T P. B(H) with weak-operator topology is Hausdorff and the mappings T ⟶ P T P and T ⟶ T P are weak-operator continuous, hence B is weak-operator closed in B(H). Choose a unit vector η ∈ H orthogonal to ξ. Suppose that Q is the projection onto [{ξ, η}] and S is the operator defined by Sη = ξ, Sξ = 0 and S(I −Q) = 0. Then P, Q and S are in B. Thus if T′ ∈ B′( the commutant of B), then ξ and η are eigenvectors for T′, say T′ξ = αξ and T′η = βη. Since T′S = ST′, βξ = βSη = ST′η = T′ξ = αξ and α = β. But η is an arbitrary element orthogonal to ξ; therefore T′ = αI. Thus B′ = {αI; α ∈ C}. (Here I denotes the identity operator on H.)