Facial Structure in Operator Algebra Theory

Akemann, Charles A.; Pedersen, Gert K.

doi:10.1112/plms/s3-64.2.418

Cited by 49 publications

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“…The intersection of the members of each finite subfamily of this net is nonempty. Since A x is weak* compact it follows that the net has a non-empty intersection which is a non-empty weak* closed face of A 1 …”

Section: Tripotents In Jbw*-triplesmentioning

confidence: 99%

“…LEMMA Proof. By Lemma 3-1 (ii), ({u } },') is a decreasing net of non-empty weak* closed faces of A 1 . The intersection of the members of each finite subfamily of this net is nonempty.…”

Section: Tripotents In Jbw*-triplesmentioning

confidence: 99%

“…Since u is compact {u}, A is weak* compact and therefore so also is A*(u) sa x . Let (x^) be a net in the unit ball A*(u) 1 converging in the weak* topology to an element x in^4*.Using lemma 21 and theorem 32 of [7], it is clear that the nets ((\/2)(x j + x])) and ((i/2i){x } -x])) lie in the weak* compact set A%{u) saA . Therefore there exists a subnet (x r ) of the net (a^) such that the nets (({/2)(xy + x},)) and {{\/2i)(x r -x}.))…”

Section: -8 Let a Be A Complex Banach Space The Second Dual Of Whicmentioning

confidence: 99%

“…The set £? (0>) of central elements of the complete orthomodular lattice 0 contains 0 and 1, and if z is contained in ^(0) then so also is z 1 . With the restricted order and orthocomplementation 2^(0 > ) forms a subcomplete complete Boolean orthomodular sublattice of SP which is said to be the centre of 2P.…”

Section: Preliminariesmentioning

confidence: 99%

“…Recall that, for each element u in tfl(A), the set {u}, is a norm-exposed face of A+ v Define {w A }, to the set A+ 1 . The following result was proved in [8].…”

Section: Tripotents In Jbw*-triplesmentioning

confidence: 99%

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Compact tripotents in bi-dual JB*-triples

Edwards¹,

Rüttimann

1996

Math. Proc. Camb. Phil. Soc.

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The set %(C)~ consisting of the partially ordered set °U(G) of tripotents in a JBW*-triple C with a greatest element adjoined forms a complete lattice. This paper is mainly concerned with the situation in which C is the second dual ^4** of a complex Banach space A and, more particularly, when A is itself a JB* -triple. A subset %(A)õ f ^l(A**)~ consisting of the set %(A) of tripotents compact relative to A (denned in Section 4) with a greatest element adjoined is studied. It is shown to be an atomic complete lattice with the properties that the infimum of an arbitrary family of elements of %{A)~ is the same whether taken in % C (A)~ or in <%(A**)~ and that every decreasing net of non-zero elements of %(A)~ has a non-zero infimum. The relationship between the complete lattice %(A)~ and the complete lattice %(B)~, where B is a Banach space such that B** is a weak*-closed subtriple of A** is also investigated. When applied to the special case in which A is a C*-algebra the results provide information about the set of compact partial isometries relative to A and are closely related to those recently obtained by Akemann and Pedersen. In particular it is shown that a partial isometry is compact relative to A if and only if, in their terminology, it belongs locally to A. The main results are applied to this and other examples.

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Section: Tripotents In Jbw*-triplesmentioning

confidence: 99%

“…LEMMA Proof. By Lemma 3-1 (ii), ({u } },') is a decreasing net of non-empty weak* closed faces of A 1 . The intersection of the members of each finite subfamily of this net is nonempty.…”

Section: Tripotents In Jbw*-triplesmentioning

confidence: 99%

Section: -8 Let a Be A Complex Banach Space The Second Dual Of Whicmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

“…Recall that, for each element u in tfl(A), the set {u}, is a norm-exposed face of A+ v Define {w A }, to the set A+ 1 . The following result was proved in [8].…”

Section: Tripotents In Jbw*-triplesmentioning

confidence: 99%

See 3 more Smart Citations

Compact tripotents in bi-dual JB*-triples

Edwards¹,

Rüttimann

1996

Math. Proc. Camb. Phil. Soc.

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Non‐associativeC*‐algebras

Palacios

García

2021

Algebra and Applications 1

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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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Facial Structure in Operator Algebra Theory

Cited by 49 publications

References 0 publications

Compact tripotents in bi-dual JB*-triples

Compact tripotents in bi-dual JB*-triples

Non‐associativeC*‐algebras

Noncommutative topology and Jordan operator algebras

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