We prove that every surjective isometry between the unit spheres of two trace class spaces admits a unique extension to complex linear or conjugate spaces. This provides positive solution Tingley's problem in new operator algebras.

We obtain a complete characterization of all orthogonality preserving operators from JB *-algebra to *-triple. If T : J → E is bounded linear operator (respectively, C *-algebra) *-triple and h denotes the element T**(1), then preserving, if only if, preserves zero-triple-products, there exists Jordan *-homomorphism [Formula: see text] such that S(x) commute T(x) = h• r(h) S(x), for every x ∈ J, where range tripotent h, Peirce-2 subspace associated • natural product making *-algebra. This...

We introduce the notion of Banach Jordan triple modules and determine precise conditions under which every derivation from a JB⁎-triple E into (Jordan) E-module is continuous. In particular, real or complex its dual space automatically continuous, motivating study (which we have carried out elsewhere) weakly amenable JB⁎-triples. Specializing to C⁎-algebras leads unified treatment derivations modules, shedding light on celebrated theorem Barry Johnson.

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We establish spherical variants of the Gleason-Kahane-Żelazko and Kowalski-S lodkowski theorems, we apply them to prove that every weak-2-local isometry between two uniform algebras is a linear map.Among consequences, solve couple problems posed by O.

Abstract There are numerous cases of discrepancies between results obtained in the setting real Banach spaces and those complex context. This article is a modern exposition subtle differences key theories for corresponding linear operators them. We deeply discuss some aspects complexification give several examples showing how drastically different can be behavior versus their counterparts.

Abstract In a recent paper, we showed that subspace of real ‐triple is an ‐summand if and only it ‐closed triple ideal. As consequence, ‐ideals ‐triples, including ‐algebras, ‐algebras TROs, correspond to norm‐closed ideals. this extend result by identifying the in (possibly non‐self‐adjoint) operator algebras Jordan algebras. The argument for necessarily different. We also give simple characterizations one‐sided algebras, some applications theory.

In a first result, we prove that every continuous local triple derivation on JB∗-triple is derivation. We also give an automatic continuity is, show derivations are even if not assumed priori to be so. These results provide positive answers the conjectures posed by Mackey (Bull. London Math. Soc. 45 (2013) 811–824). particular, C∗-algebra explore connections between (bounded local) and generalized (Jordan) C∗-algebra.

We prove that every weak-local derivation on a C*-algebra is continuous, and the same conclusion remains valid for weak*-local derivations von Neumann algebras. further show C*-algebras algebras are derivations. also study connections between bilocal bilocal*-automorphism with our notions of extreme-strong-local automorphisms.

At the regional conference held at University of California, Irvine, in 1985 [24], Harald Upmeier posed three basic questions regarding derivations on JB*-triples: (1) Are automatically bounded? (2) When are all bounded inner? (3) Can be approximated by inner derivations? These had been answered binary cases. Question 1 was affirmatively Sakai [17] for C*-algebras and [23] JB-algebras. 2 [18] Kadison [12] von Neumann algebras JW-algebras. 3 JB-algebras, it follows trivially from...

It is shown that every norm-closed face of the closed unit ball A1 in a JB*-triple A norm-semi-exposed, thereby completing description facial structure A1.

Abstract We prove that every bounded local triple derivation on a unital C*-algebra is derivation. A similar statement established in the category of JB*-algebras. Key Words: Generalised derivationGeneralised Jordan derivationLocal derivationTriple derivationUnital C*-algebra2010 Mathematics Subject Classification: Primary: 47B47, 46L57Secondary: 17C65, 46L05, 46L08 ACKNOWLEDGMENTS Authors partially supported by Spanish Ministry Economy and Competitiveness, D.G.I. project no. MTM2011-23843,...

We prove that, if M > 4 ( 12 3 ) and ɛ 0, V W are complex JBW*-triples (with preduals V* W*, respectively), U is a separately weak*-continuous bilinear form on × W, then there exist norm-one functionals ϕ1, ϕ2 ∈ ψ1, ψ2 W* satisfying | x , y ⩽ ∥ φ 2 ε 1 ψ for all (x, y) W. Here, functional ϕ JB*-triple V, |·|ϕ stands the prehilbertian seminorm associated to given by : = { z } where V** satisfies |z| 1. arrive at this of 'Grothendieck's inequality' through results C.-H. Chu, B. Iochum, G....