1. Martingale convergence in von Neumann algebras
- Author
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E. Christopher Lance
- Subjects
Discrete mathematics ,Pure mathematics ,Jordan algebra ,General Mathematics ,Subalgebra ,C*-algebra ,Neumann series ,Von Neumann's theorem ,symbols.namesake ,Von Neumann algebra ,symbols ,Abelian von Neumann algebra ,Affiliated operator ,Mathematics - Abstract
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
- Published
- 1978
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