Jordan operator algebras revisited

Blecher, David P.; Wang, Zhenhua

doi:10.1002/mana.201800568

Cited by 5 publications

(9 citation statements)

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“…Remark (1)At the end of the proof we used ‘open‐ness’ with respect to A 1 or with respect to a containing

C^{*}

‐algebra B (where

A^{1} \subset B^{1}

). This raises a concern since A 1 is not uniquely defined up to complete isometry . The same concern arises at several later points in the present paper where A 1 is used.…”

Section: Jordan Variants Of Hay's Theorem and Some Variants Of Resultmentioning

confidence: 84%

“…If A is approximately unital the unitization A 1 is unique up completely isometric Jordan homomorphism by [, Proposition 2.12]. The latter is not true in general if A is not approximately unital (a two dimensional Hilbertian example is given in ), but fortunately in the present paper the isometric case suffices.…”

Section: Introductionmentioning

confidence: 71%

“…Again if one wants a unital example then simply unitize. A sometimes useful example (see ) is the Hilbertian operator space

Φ_{n}

(or

Φ (I)

, see [, Section 9.3]), the span of operators

(T_{k})

(with

k \leq n

k \in I

) satisfying the CAR (canonical anticommutator relations). This is a Jordan operator algebra with zero product, and is a JC*‐triple, so that this example falls into the above discussion with

y = 0

.…”

Section: Examples Of Jordan Operator Algebrasmentioning

confidence: 99%

“…The Urysohn lemma in [, Theorem 2.24] will be discussed in Section 7 below, and some generalizations of the results in Sections 3, 4, and 7 of will be given in . Concerning Section 6 from , the first result there is generalized in [, Proposition 3.28].…”

Section: Initial Consequencesmentioning

confidence: 99%

“…Several complementary facts and additional theory will be forthcoming in a work in progress and in the second authors Ph. D. thesis .…”

Section: Introductionmentioning

confidence: 97%

See 4 more Smart Citations

Noncommutative topology and Jordan operator algebras

Blecher

Neal

2018

Mathematische Nachrichten

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Jordan operator algebras are norm-closed spaces of operators on a Hilbert space with 2 ∈ for all ∈ . We study noncommutative topology, noncommutative peak sets and peak interpolation, and hereditary subalgebras of Jordan operator algebras. We show that Jordan operator algebras present perhaps the most general setting for a "full" noncommutative topology in the * -algebraic sense of Akemann, L. G. Brown, Pedersen, etc, and as modified for not necessarily selfadjoint algebras by the authors with Read, Hay and other coauthors. Our breakthrough relies in part on establishing several strong variants of * -algebraic results of Brown relating to hereditary subalgebras, proximinality, deeper facts about + * for a left ideal in a * -algebra, noncommutative Urysohn lemmas, etc. We also prove several other approximation results in * -algebras and various subspaces of * -algebras, related to open and closed projections and technical * -algebraic results of Brown. K E Y W O R D S approximate identity, * -envelope, hereditary subalgebra, JC*-algebra, Jordan Banach algebra, Jordan operator algebra, noncommutative topology, open projection, operator spaces, peak interpolation, peak set, real positivity, states M S C ( 2 0 1 0 )Primary: 17C65, 46L07, 46L70, 46L85, 47L05, 47L30; Secondary: 46H10, 46H70, 47L10, 47L75 * * ∩ 1 = * * ∩ is a hereditary subalgebra in 1 and hence also in . (Here we are using the fact that ⟂⟂ ∩ 1 = .)So is -open. □

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“…Remark (1)At the end of the proof we used ‘open‐ness’ with respect to A 1 or with respect to a containing

C^{*}

‐algebra B (where

A^{1} \subset B^{1}

). This raises a concern since A 1 is not uniquely defined up to complete isometry . The same concern arises at several later points in the present paper where A 1 is used.…”

Section: Jordan Variants Of Hay's Theorem and Some Variants Of Resultmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 71%

“…Again if one wants a unital example then simply unitize. A sometimes useful example (see ) is the Hilbertian operator space

Φ_{n}

(or

Φ (I)

, see [, Section 9.3]), the span of operators

(T_{k})

(with

k \leq n

k \in I

) satisfying the CAR (canonical anticommutator relations). This is a Jordan operator algebra with zero product, and is a JC*‐triple, so that this example falls into the above discussion with