Jordan operator algebras: basic theory

Blecher, David P.; Wang, Zhenhua

doi:10.1002/mana.201700178

Cited by 11 publications

(173 citation statements)

References 36 publications

(161 reference statements)

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“…Or equivalently, which is closed under the “Jordan product”

a \circ b = \frac{1}{2} false(a b + b a false)

. In two recent papers a theory for this class was developed. It was shown there that much of the theory of associative operator algebras, in particular, surprisingly much of the recent associative theory from e.g.…”

Section: Introductionmentioning

confidence: 99%

“…We now quickly summarize notation (for more details see the introductions to ). If A is a Jordan operator algebra acting on a Hilbert space H , let

C^{*} false(A false)

be the

C^{*}

‐algebra generated by A in

B (H)

.…”

Section: Introductionmentioning

confidence: 99%

“…If

x, y \in A

then

x y x \in A

(by e.g. (1.3) in ). If

x y

is in A for all

x, y

in A then A is called an (associative) operator algebra.…”

Section: Introductionmentioning

confidence: 99%

“…We recall that an element b in a Jordan operator algebra A commutes with a projection p if

p b = b p

in some

C^{*}

‐algebra containing A as a Jordan subalgebra. By (1.1) in this is equivalent to

b \circ p = p b p

. A Jordan homomorphism between Jordan operator algebras is a linear map satisfying

T (a b + b a) = T (a) T (b) + T (b) T (a)

for

a, b \in A

, or equivalently, that

T (a^{2}) = T {(a)}^{2}

for all

a \in A

(the equivalence follows by applying T to

{false(a + b false)}^{2}

).…”

Section: Introductionmentioning

confidence: 99%

“…A Jordan contractive approximate identity (or J‐cai for short) is a net

(e_{t})

of contractions with

e_{t} \circ a \to a

for all

a \in A

. It is shown in [, Lemma 2.6] that if A has a Jordan cai then it has a partial cai , that is a net that acts as a cai for the ordinary product in any containing generated

C^{*}

‐algebra. If a Jordan (or equivalently, a partial) cai for A exists then A is called approximately unital .…”

Section: Introductionmentioning

confidence: 99%

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Jordan operator algebras revisited

Blecher

Wang

2019

Mathematische Nachrichten

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Jordan operator algebras are norm‐closed spaces of operators on a Hilbert space with a2∈A for all a∈A. In two recent papers by the authors and Neal, a theory for these spaces was developed. It was shown there that much of the theory of associative operator algebras, in particular, surprisingly much of the associative theory from several recent papers of the first author and coauthors, generalizes to Jordan operator algebras. In the present paper we complete this task, giving several results which generalize the associative case in these papers, relating to unitizations, real positivity, hereditary subalgebras, and a couple of other topics. We also solve one of the three open problems stated at the end of our earlier joint paper on Jordan operator algebras.

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“…Or equivalently, which is closed under the “Jordan product”

a \circ b = \frac{1}{2} false(a b + b a false)

Section: Introductionmentioning

confidence: 99%

“…We now quickly summarize notation (for more details see the introductions to ). If A is a Jordan operator algebra acting on a Hilbert space H , let

C^{*} false(A false)

be the

C^{*}

‐algebra generated by A in

B (H)

.…”

Section: Introductionmentioning

confidence: 99%

“…If

x, y \in A

then

x y x \in A

(by e.g. (1.3) in ). If

x y

is in A for all

x, y

in A then A is called an (associative) operator algebra.…”

Section: Introductionmentioning

confidence: 99%

“…We recall that an element b in a Jordan operator algebra A commutes with a projection p if

p b = b p

in some

C^{*}

‐algebra containing A as a Jordan subalgebra. By (1.1) in this is equivalent to

b \circ p = p b p

. A Jordan homomorphism between Jordan operator algebras is a linear map satisfying

T (a b + b a) = T (a) T (b) + T (b) T (a)

for

a, b \in A

, or equivalently, that

T (a^{2}) = T {(a)}^{2}

for all

a \in A

(the equivalence follows by applying T to

{false(a + b false)}^{2}

).…”

Section: Introductionmentioning

confidence: 99%

“…A Jordan contractive approximate identity (or J‐cai for short) is a net

(e_{t})

of contractions with

e_{t} \circ a \to a

for all

a \in A

. It is shown in [, Lemma 2.6] that if A has a Jordan cai then it has a partial cai , that is a net that acts as a cai for the ordinary product in any containing generated

C^{*}

‐algebra. If a Jordan (or equivalently, a partial) cai for A exists then A is called approximately unital .…”

Section: Introductionmentioning

confidence: 99%

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Jordan operator algebras revisited

Blecher

Wang

2019

Mathematische Nachrichten

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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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Real Positive Maps and Conditional Expectations on Operator Algebras

Blecher

2021

Trends in Mathematics

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Jordan operator algebras: basic theory

Cited by 11 publications

References 36 publications

Jordan operator algebras revisited

Jordan operator algebras revisited

Noncommutative topology and Jordan operator algebras

Real Positive Maps and Conditional Expectations on Operator Algebras

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