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  1. 3 Studia Mathematica

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  1. 3 Blecher D.

  2. 3 Neal M.

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  1. 2 2012

  2. 1 2007

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Open projections in operator algebras II: Compact projections

100%
Blecher D., Neal M.
Studia Mathematica
|
2012
|
tom 209
|
nr 3
203-224
EN
We generalize some aspects of the theory of compact projections relative to a C*-algebra, to the setting of more general algebras. Our main result is that compact projections are the decreasing limits of 'peak projections', and in the separable case compact projections are just the peak projections. We also establish new forms of the noncommutative Urysohn lemma relative to an operator algebra, and we show that a projection is compact iff the associated face in the state space of the algebra is weak* closed.
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Open projections in operator algebras I: Comparison theory

100%
Blecher D., Neal M.
Studia Mathematica
|
2012
|
tom 208
|
nr 2
117-150
EN
We begin a program of generalizing basic elements of the theory of comparison, equivalence, and subequivalence, of elements in C*-algebras, to the setting of more general algebras. In particular, we follow the recent lead of Lin, Ortega, Rørdam, and Thiel of studying these equivalences, etc., in terms of open projections or module isomorphisms. We also define and characterize a new class of inner ideals in operator algebras, and develop a matching theory of open partial isometries in operator ideals which simultaneously generalize the open projections in operator algebras (in the sense of the authors and Hay), and the open partial isometries (tripotents) introduced by the authors.
3
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Open partial isometries and positivity in operator spaces

100%
Blecher D., Neal M.
Studia Mathematica
|
2007
|
tom 182
|
nr 3
227-262
EN
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.
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