Compact tripotents in bi-dual JB*-triples

Edwards, Charles; Rüttimann, Gottfried T.

doi:10.1017/s0305004100074740

Cited by 62 publications

(91 citation statements)

References 35 publications

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“…With respect to the separately weak Ã -continuous product aY b U 3 a b fa u bg and the norm-preserving involution a U 3 a y fu a ug, A 2 u is a JBW Ã -algebra with unit u. For details the reader is referred to [1], [3], [4], [5], [7], [10], [12], [13], [14], [15], [16], [17], [19], [20], [23].…”

Section: Preliminariesmentioning

confidence: 99%

Exposed faces of the unit ball in a JBW*-triple

Edwards¹,

Rüttiman²

1998

MATH. SCAND.

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Section: Preliminariesmentioning

confidence: 99%

Exposed faces of the unit ball in a JBW*-triple

Edwards¹,

Rüttiman²

1998

MATH. SCAND.

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“…The justification is essentially due to the following result established by L. Cheng, Y. Dong and R. Tanaka. Accordingly to the notation in [11] and [13], a tripotent e in the second dual, E * * , of a JB * -triple E is said to be compact-G δ if there exists a norm one element a in E such that e is the support tripotent of a. A tripotent e in E * * is called compact if e = 0 or it is the infimum of a decreasing net of compact-G δ tripotents in E * * .…”

Section: A Jbwmentioning

confidence: 99%

Tingley's problem through the facial structure of an atomic JBW⁎-triple

Fernández–Polo

Peralta

2017

Journal of Mathematical Analysis and Applications

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Abstract. We prove that every surjective isometry between the unit spheres of two atomic JBW * -triples E and B admits a unit extension to a surjective real linear isometry from E into B. This result constitutes a new positive answer to Tignley's problem in the Jordan setting. IntroductionIn a recent contribution we establish that every surjective isometry between the unit spheres of two B(H)-spaces extends uniquely to a surjective complex linear or conjugate linear surjective isometry between the corresponding spaces (see [15]). This result constitutes a positive answer to Tingley's isometric extension problem [31] in the setting of B(H)-spaces and atomic von Neumann algebras. Solutions to Tingley's problem for compact operators, compact C * -algebras and weakly compact JB * -triples have been previously obtained in [26,14]. For additional information on the historic background and the state of the art of Tingley's problem the reader is referred to the introduction of [15] and to the monograph [32].Problems in C * -algebras, von Neumann algebras and operator algebras are often considered in the context of Banach Jordan algebras and Jordan triple systems. Such studies widen the scope and often introduce new ideas and techniques not present in the associative case. The class of JB * -triples have a rich interaction with Banach Space Theory. The spaces in this class enjoy a unique geometry which makes more interesting the study of certain geometric problems in a wider setting. This paper is devoted to extend the recent results in [15] to the context of atomic JBW * -triples (i.e. ℓ ∞ -sums of Cartan factors).We recall that a JB * -triple is a complex Banach space E which can be equipped with a continuous triple product {., ., .} : E × E × E → E, which is symmetric and linear in the first and third variables, conjugate linear in the second variable and satisfies the following axioms , a) is an hermitian operator with non-negative spectrum; (c) L(a, a) = a 2 .2010 Mathematics Subject Classification. Primary 47B49, Secondary 46A22, 46B20, 46B04, 46A16, 46E40.

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“…For every element x in the pre-dual Banach space A. there exists a least tripotent e(x) in A, the support tripotent of x, such that e(x)(x) is equal to Ilxll [12]. For every element a in A, there exists a least tripotent r(a) in A, referred to as the support tripotent of a, such that a is a positive element in the JBW*-algebra A2(r(a)) [12,16].…”

Section: Pq( A)p = Q( Pa)mentioning

confidence: 99%