A Remark on W*-Tensor Products of W*-Algebras

  • Corneliu Constantinescu
Part of the Springer Optimization and Its Applications book series (SOIA, volume 35)


Let E be a W*-algebra, T a hyperstonian compact space, C(T) the W*-algebra of continuous scalar valued functions on T, and F(T,E) the set of bounded maps x : TE such that for every element a of the predual of E the function
$$T \to {\rm{IK,}}\,\,\,\,\,\,\,\,\,t \mapsto \langle x_t ,a\rangle $$
is continuous. We define for every xF(T,E) an element \(\tilde x\)C(T)\(\bar \otimes \)E such that the map
$$f(T,E) \to b(T)\bar \otimes E,\,\,\,\,\,\,\,x \mapsto \tilde x$$
is a bijective isometry of ordered involutive Banach spaces (where this structure on F(T,E) is defined pointwise). In general F(T,E) is not an algebra for the pointwise multiplication, but for x,y,zF(T,E) we characterize the case when \(\tilde x\tilde y = \tilde z\).


Hilbert Space Tensor Product Orthogonal Projection Pairwise Disjoint Dual Norm 
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    Constantinescu, C., C*-algebras, Elsevir, 2001.Google Scholar
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    Takesaki, M., Theory of Operator Algebra I, Springer, 2002.Google Scholar
  3. 3.
    Wegge-Olsen, N. E., K-theory and C*-algebras, Oxford University Press, 1993.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.BenglenSwitzerland

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