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On the Lebesgue decomposition of the normal states of a JBW-algebra

Jacques Dubois, Brahim Hadjou (1992)

Mathematica Bohemica

In this article, a theorem is proved asserting that any linear functional defined on a JBW-algebra admits a Lebesque decomposition with respect to any normal state defined on the algebra. Then we show that the positivity (and the unicity) of this decomposition is insured for the trace states defined on the algebra. In fact, this property can be used to give a new characterization of the trace states amoungst all the normal states.

On unitary convex decompositions of vectors in a J B * -algebra

Akhlaq A. Siddiqui (2013)

Archivum Mathematicum

By exploiting his recent results, the author further investigates the extent to which variation in the coefficients of a unitary convex decomposition of a vector in a unital J B * -algebra permits the vector decomposable as convex combination of fewer unitaries; certain C * -algebra results due to M. Rørdam have been extended to the general setting of J B * -algebras.

On weak sequential convergence in JB*-triple duals

Leslie J. Bunce, Antonio M. Peralta (2004)

Studia Mathematica

We study various Banach space properties of the dual space E* of a homogeneous Banach space (alias, a JB*-triple) E. For example, if all primitive M-ideals of E are maximal, we show that E* has the Alternative Dunford-Pettis property (respectively, the Kadec-Klee property) if and only if all biholomorphic automorphisms of the open unit ball of E are sequentially weakly continuous (respectively, weakly continuous). Those E for which E* has the weak* Kadec-Klee property are characterised by a compactness...

Range tripotents and order in JBW*-triples

Lina Oliveira (2010)

Banach Center Publications

In a JBW*-triple, i.e., a symmetric complex Banach space possessing a predual, the set of tripotents is naturally endowed with a partial order relation. This work is mainly concerned with this partial order relation when restricted to the subset 𝓡(A) of tripotents in a JBW*-triple B formed by the range tripotents of the elements of a JB*-subtriple A of B. The aim is to present recent developments obtained for the poset 𝓡(A) of the range tripotents relative to A, whilst also providing the necessary...

Supporting sequences of pure states on JB algebras

Jan Hamhalter (1999)

Studia Mathematica

We show that any sequence ( φ n ) of mutually orthogonal pure states on a JB algebra A such that ( φ n ) forms an almost discrete sequence in the relative topology induced by the primitive ideal space of A admits a sequence ( a n ) consisting of positive, norm one, elements of A with pairwise orthogonal supports which is supporting for ( φ n ) in the sense of φ n ( a n ) = 1 for all n. Moreover, if A is separable then ( a n ) can be taken such that ( φ n ) is uniquely determined by the biorthogonality condition φ n ( a n ) = 1 . Consequences of this result improving...

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