Open partial isometries and positivity in operator spaces

Tom 182 / 2007

David P. Blecher, Matthew Neal Studia Mathematica 182(2007), 227-262 MSC: Primary 46L08, 46A40, 47L07; Secondary 46B40, 46L07, 47B60, 47L05. DOI: 10.4064/sm182-3-4


We first study positivity in $C^*$-modules using tripotents (= partial isometries) which are what we call open. This is then used to study ordered operator spaces via an “ordered noncommutative Shilov boundary” which we introduce. This boundary satisfies the usual universal diagram/property of the noncommutative Shilov boundary, but with all the arrows completely positive. Because of their independent interest, we also systematically study open tripotents and their properties.


  • David P. BlecherDepartment of Mathematics
    University of Houston
    Houston, TX 77204-3008, U.S.A.
  • Matthew NealDepartment of Mathematics
    Denison University
    Granville, OH 43023, U.S.A.

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