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Martingale convergence in von Neumann algebras

Published online by Cambridge University Press:  24 October 2008

E. Christopher Lance
E. Christopher Lance
Affiliation:
University of Manchester
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Extract

Let N be a von Neumann subalgebra of a von Neumann algebra M. A linear mapping π: M → N is called a retraction if it is idempotent and has norm one. By a result of Tomiyama(15) a retraction is a positive mapping and is a module homo-morphism over N. A retraction is normal if it is ultraweakly continuous, and faithful if it does not annihilate any nonzero positive element of M. Suppose that (Nn)n≥1 is an increasing sequence of von Neumann subalgebras of M whose union is weakly dense in M and that, for each n, πn: M → Nn is a faithful normal retraction. The sequence (πn) is called a martingale if, whenever m ≥ n,

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 84 , Issue 1 , July 1978 , pp. 47 - 56
Copyright
Copyright © Cambridge Philosophical Society 1978

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