A week-operator closed subalgebra

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

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Let H be a Hilbert space of dimension greater than 1, x be a unit vector in H and B be the subalgebra of B(H) consisting of those operators for which x is an eigenvector. Let P be the projection with range [x] (If K Í H, we denote the closed linear span of K by [K]). Then T Î B iff PTP = TP. B(H) with weak-operator topology is Hausdorff and the mappings TŽ PTP and TŽ TP are weak-operator continuous, hence B is weak-operator closed in B(H).
Choose a unit vector h Î H orthogonal to x. Suppose that Q is the projection onto [{x, h}] and S is the operator defined by Sh = x, Sx = 0 and S(I - Q) = 0. Then P, Q and S are in B. Thus if T˘ Î B˘( the commutant of B), then x and h are eigenvectors for T˘, say T˘x = ax and T˘h = bh. Since T˘S = ST˘, bx = bS h = ST˘h = T˘x = ax and a = b. But h is an arbitrary element orthogonal to x; therefore T˘ = aI. Thus B˘ = {aI;a Î C}. (Here I denotes the identity operator on H.)

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On 22 Feb 2001, 00:20.