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Let H be a separable complex Hilbert space, A a von Neumann algebra in L(H), [phi] a faithful, normal state on A, and B a commutative von Neumann subalgebra of A. Given a sequence (X[n] : n is less than or equal to 1) of operators in B, we examine the relations between bundle convergence in B and bundle convergence in A.
The notion of bundle convergence for sequences of operators in a von Neumann algebra A equipped with a faithful and normal state phi as well as for sequences of vectors in their [L_2]-spaces were introduced by Hensz, Jajte and Paszkiewicz in 1996 as an appropriate substitute for almost everywhere convergence in the commutative setting. First, we prove that if (B_k : k = l, 2,...) is a sequence in...
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