Metric Characterisation of Unitaries in $$\hbox {JB}^*$$-Algebras

Cueto-Avellaneda, María; Peralta, Antonio M.

doi:10.1007/s00009-020-01556-w

Cited by 3 publications

(5 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Clearly, Φ = U w0 • J : M → N is a Jordan * -isomorphism and the equality Remark 4.2. It should be remarked that Theorem 4.1 can be also deduced, via an alternative argument, from the conclusions in the recent papers [6,7]. Let ∆ : M → N be a surjective isometry in the conditions of the just quoted theorem.…”

Section: Fix An Arbitrarymentioning

confidence: 91%

“…The lacking of left and right multiplication operators in the Jordan setting increase the difficulties to get similar conclusions to those in the previous Remark 2. 6 for Jordan-Banach algebras. As in previous contributions, passing to an homotope is in some sense a substitute for the left multiplication by an invertible element in an associative Banach algebra.…”

Section: Surjective Isometries Between Subsets Of Invertible Elements Inmentioning

confidence: 99%

“…Let T 0 : M → N be the surjective real-linear isometry given by Theorem Let w be an extreme point of the closed unit ball of a unital JB * -algebra M ′ . Theorem 3.8 in [6] proves that w is a unitary if and only if the set…”

Section: Fix An Arbitrarymentioning

confidence: 99%

See 2 more Smart Citations

Surjective isometries between sets of invertible elements in unital Jordan-Banach algebras

Peralta¹

2020

Preprint

Self Cite

View full text Add to dashboard Cite

Let M and N be unital Jordan-Banach algebras, and let M −1 and N −1 denote the sets of invertible elements in M and N , respectively. Suppose that M ⊆ M −1 and N ⊆ N −1 are clopen subsets of M −1 and N −1 , respectively, which are closed for powers, inverses and products of the form Ua(b). In this paper we prove that for each surjective isometry ∆ : M → N there exists a surjective real-linear isometry T 0 : M → N and an element u 0 in the McCrimmon radical of N such that ∆(a) = T 0 (a) + u 0 for all a ∈ M.Assuming that M and N are unital JB * -algebras we establish that for each surjective isometry ∆ : M → N the element ∆(1) = u is a unitary element in N and there exist a central projection p ∈ M and a complex-linear Jordan * -isomorphism J from M onto the u * -homotopefor all a ∈ M. Under the additional hypothesis that there is a unitary element ω 0 in N satisfying Uω 0 (∆(1)) = 1, we show the existence of a central projection p ∈ M and a complex-linear Jordan * -isomorphism Φ from M onto N such that ∆(a) = U w * 0 (Φ(p • a) + Φ((1 − p) • a * )) , for all a ∈ M.

show abstract

Section: Fix An Arbitrarymentioning

confidence: 91%

Section: Surjective Isometries Between Subsets Of Invertible Elements Inmentioning

confidence: 99%

See 1 more Smart Citation

Surjective isometries between sets of invertible elements in unital Jordan-Banach algebras

Peralta¹

2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…for every u, v in {E h (t) : t ∈ R}. Taking v = 1 = E h (0) and u arbitrary in (12) and having in mind that ∆ 0 (1) = 1 we get (13) ∆ 0 (u) * = ∆ 0 (u * ), for all u ∈ {E h (t) : t ∈ R}.…”

Section: Lemma 33 Applied To ∆ and The Family {Umentioning

confidence: 99%

“…The interest remains very active, for example, we recently obtained a metric characterization of unitary elements in a unital JB * -algebra (cf. [12]).…”

Section: Introductionmentioning

confidence: 99%

Can one identify two unital JB*-algebras by the metric spaces determined by their sets of unitaries?

Cueto-Avellaneda

Peralta

2021

Linear and Multilinear Algebra

Self Cite

View full text Add to dashboard Cite

Let M and N be two unital JB * -algebras and let U (M ) and U (N ) denote the sets of all unitaries in M and N , respectively. We prove that the following statements are equivalent:(a) M and N are isometrically isomorphic as (complex) Banach spaces;(b) M and N are isometrically isomorphic as real Banach spaces; (c) There exists a surjective isometry ∆ : U (M ) → U (N ). We actually establish a more general statement asserting that, under some mild extra conditions, for each surjective isometry ∆ : U (M ) → U (N ) we can find a surjective real linear isometry Ψ : M → N which coincides with ∆ on the subset e iMsa . If we assume that M and N are JBW * -algebras, then every surjective isometry ∆ : U (M ) → U (N ) admits a (unique) extension to a surjective real linear isometry from M onto N . This is an extension of the Hatori-Molnár theorem to the setting of JB * -algebras.

show abstract

Surjective isometries between sets of invertible elements in unital Jordan-Banach algebras

Peralta

2021

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

Metric Characterisation of Unitaries in $$\hbox {JB}^*$$-Algebras

Cited by 3 publications

References 22 publications

Surjective isometries between sets of invertible elements in unital Jordan-Banach algebras

Surjective isometries between sets of invertible elements in unital Jordan-Banach algebras

Can one identify two unital JB*-algebras by the metric spaces determined by their sets of unitaries?

Surjective isometries between sets of invertible elements in unital Jordan-Banach algebras

Contact Info

Product

Resources

About