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On the convergence of operators. (English) Zbl 0663.60001
This paper is a contribution to generalized probability theory. A non- commutative version of a known theorem in conventional probability theory is proved.
Let m be a faithful state on the set of all closed linear subspaces of a complex separable Hilbert space H(dim \(H\geq 3)\) and let \(\{A_ n\}\) be a sequence of uniformly bounded self-adjoint operators on H such that the sequence \(\{A_ n\}\) converges on A in the square mean with respect to the state m. Then \(A_ n\) converges on A in distribution (with respect to m). The assumption of uniform boundedness is necessary.
Reviewer: B.Stehlíková
60A99 Foundations of probability theory
46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
47D99 Groups and semigroups of linear operators, their generalizations and applications
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