Inner Ideals and Facial Structure of the Quasi-State Space of a JB-Algebra

Neal, Matthew D.

doi:10.1006/jfan.2000.3577

Cited by 15 publications

(8 citation statements)

References 20 publications

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“…□ As in [16] (above Proposition 2.14 there), the correspondence between a HSA and its support projection is a bijective order embedding from the lattice of HSA's of a Jordan operator algebra and the lattice of -open projections in * * (see e.g. [32] for a JB-algebra variant of this). Write ( ) for the quasistate space of , that is the set of states multiplied by numbers in [0, 1].…”

Section: Support Projections and Hsa'smentioning

confidence: 96%

Jordan operator algebras: basic theory

Blecher

Wang

2018

Mathematische Nachrichten

171

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Section: Support Projections and Hsa'smentioning

confidence: 96%

Jordan operator algebras: basic theory

Blecher

Wang

2018

Mathematische Nachrichten

171

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“…See also [23] for a related result, with a different proof strategy (which relies on results of the second author [33]). …”

Section: Proof (I)⇒(ii)mentioning

confidence: 99%

Open partial isometries and positivity in operator spaces

Blecher

Neal

2007

Studia Math.

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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.1. Introduction. We are interested here in cones of positive operators X + = {x ∈ X : x ≥ 0}, for a space X of bounded linear operators on a Hilbert space, where ≥ denotes the usual ordering of such operators. Besides the intrinsic interest of such objects (for example, operator positivity plays a central role in many areas of mathematical physics today), our work is a sequel to [13], which was a first step in a new approach to positivity in an operator space X, namely studying it in terms of the "noncommutative Shilov boundary" of X (see [6,26,12]). The latter object is a Hilbert C * -module, or, equivalently, a ternary ring of operators (or TRO for short), by which we will mean a closed subspace Z of a C * -algebra A such that ZZ * Z ⊂ Z. If X contains positive operators, then so will any containing TRO. The starting point of the present investigation and [13] was the question of whether, in this case, all morphisms in the universal property of the noncommutative Shilov boundary can also be chosen to be positive (allowing this boundary to be used as a new tool in the study of ordered operator spaces). To answer this, one is led immediately to study positivity in TROs, and we

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“…Monotonicity of x λ follows from (3.5) and Lemma 3.3. Note that this fact is also in [22,Lemma 3.1]. By [17, Proposition 3.5.3], x −1 is operator monotone decreasing.…”

Section: Operator Concavity and Monotonicitymentioning

confidence: 70%