Equivalent norms on Banach Jordan algebras

Youngson, Martin Alexander

doi:10.1017/s0305004100056085

Cited by 5 publications

(2 citation statements)

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“…This is proved using the characterisation of the NON-UNTTAL BANACH JORDAN ALGEBRAS 21 complex unital Banach Jordan algebras which are the homeomorphic images of unital JB*-algebras obtained in (30). We deduce from this some extensions of the theory of unital JB*-algebras to non-unital JB*-algebras which we shall require.…”

Section: Adjoining a Unit To A Jb *-Algebramentioning

confidence: 87%

Non-unital Banach Jordan algebras and C*-triple systems

Youngson¹

1981

Proceedings of the Edinburgh Mathematical Society

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The definition of a suitable Jordan analogue of C*-algebras (which we call JB*-algebras in this paper) was recently suggested by Kaplansky (see (26)). The theory of unital JB*-algebras is now comparatively well understood due to the work of Alfsen, Shultz and Størmer (1) from which a Gelfand-Neumark theorem for unital JB*-algebras can be obtained (26). Independently, from work on simply connected symmetric complex Banach manifolds with base point, Kaup introduced the definition of C*-triple systems in (14) and subsequently in (7) it was shown that every unital JB*-algebra is a C*-triple system. In this paper, we wish to extend this result to show that every JB*-algebra is a C*-triple system.

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Section: Adjoining a Unit To A Jb *-Algebramentioning

confidence: 87%

Non-unital Banach Jordan algebras and C*-triple systems

Youngson¹

1981

Proceedings of the Edinburgh Mathematical Society

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“…Moreover the set of positive elements is closed in the set of selfadjoint elements. Using these results we can improve the characterization of JB'-algebras in an equivalent norm given in [8]. We recall that A is a JB*-algebra if for every x in A we have ||C/x(x*)|| = ||x||3.…”

Section: Theorem the Following Are Equivalent: (I) A Is Hermitian Tmentioning

confidence: 97%

On Symmetry of Banach Jordan Algebras

Aupetit¹,

Youngson²

1984

Proceedings of the American Mathematical Society

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ABSTRACT.Using a very simple subharmonic argument we prove that a Banach Jordan algebra is Hermitian if and only if the sum of two positive elements is positive. We apply this result to give a characterization of Banach Jordan algebras with involution which are JB*-algebras for an equivalent norm.If B is a complex Banach algebra with involution, the Shirali-Ford theorem states that every selfadjoint element of B has real spectrum if and only if x*x has positive spectrum for every x in A. In particular this theorem implies that such an algebra has many Hilbert space representations.The corresponding problem in Banach Jordan algebras is also important as if a Hermitian algebra is symmetric it is possible to make use of the representation theory for JB-algebras given by Alfsen, Shultz and St0rmer in [1]. A solution to this problem was given by Behncke in [5] but the argument he used, while containing a nice idea, is rather long. The aim of this paper is to give a simple proof of Behncke's result using a subharmonic argument and to give an application to the characterization of JB*-algebras. Instead of working with real Banach Jordan algebras throughout this paper we shall let A denote a complex unital Banach Jordan algebra with involution and S shall denote the real linear subspace of selfadjoint elements of A. As for Banach algebras, if o G A one can define the spectrum and spectral radius of a; these will be denoted by Sp(a) and p(a), respectively. Some standard properties of Sp(o)

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Non‐associativeC*‐algebras

Palacios

García

2021

Algebra and Applications 1

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Equivalent norms on Banach Jordan algebras

Cited by 5 publications

References 10 publications

Non-unital Banach Jordan algebras and C*-triple systems

Non-unital Banach Jordan algebras and C*-triple systems

On Symmetry of Banach Jordan Algebras

Non‐associativeC*‐algebras

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