Acta mathematica scientia,Series B ›› 2015, Vol. 35 ›› Issue (5): 995-1002.doi: 10.1016/S0252-9602(15)30033-3

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Lp-CONTINUITY OF NONCOMMUTATIVE CONDITIONAL EXPECTATIONS

Jian HU, Congbian MA, Youliang HOU   

  1. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
  • Received:2014-07-25 Revised:2015-01-26 Online:2015-09-01 Published:2015-09-01
  • Contact: Congbian MA E-mail:macongbian@126.com
  • Supported by:

    This work was supported by National Natural Science Foundation of China (11271293, 11471251) and the Research Fund for the Doctoral Program of Higher Education of China (2014201020205).

Abstract:

Let (M, τ) be a noncommutative probability space, (Mn)n≥1 a sequence of von Neumann subalgebras of M and N a von Neumann subalgebra of M. We introduce the notions of μ-approach and orthogonal approach for (Mn)n≥1 and prove that ε(x|Mn)ε(x|N) for any xLp(M) (1 ≤ p < ∞) if and only if (Mn)n≥1 τ-approaches and orthogonally approaches N.

Key words: von Neumann algebra, trace, conditional expectation, orthogonal approach

CLC Number: 

  • 46L51
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