Open partial isometries and positivity in operator spaces
Volume 182 / 2007
Studia Mathematica 182 (2007), 227-262
MSC: Primary 46L08, 46A40, 47L07; Secondary 46B40, 46L07, 47B60, 47L05.
DOI: 10.4064/sm182-3-4
Abstract
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.
