Antonio M. Peralta

Derivations on Real and Complex JB*-Triples

These three questions had all been answered in the binary cases. Question 1 was answered affirmatively by Sakai [17] for C∗-algebras and by Upmeier [23] for JB -algebras. Question 2 was answered by Sakai [18] and Kadison [12] for von Neumann algebras and by Upmeier [23] for JW -algebras. Question 3 was answered by Upmeier [23] for JB -algebras, and it follows trivially from the Kadison–Sakai answer to question 2 in the case of C∗-algebras. In the ternary case, both question 1 and question 3 were answered by Barton and Friedman in [3] for complex JB ∗-triples. In this paper, we consider question 2 for real and complex JBW ∗-triples and question 1 and question 3 for real JB ∗-triples. A real or complex JB ∗-triple is said to have the inner derivation property if every derivation on it is inner. By pure algebra, every finite-dimensional JB ∗-triple has the inner derivation property. Our main results, Theorems 2, 3 and 4 and Corollaries 2 and 3 determine which of the infinite-dimensional real or complex Cartan factors have the inner derivation property.

ORTHOGONALITY PRESERVERS REVISITED

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

Weak-local derivations and homomorphisms on C*-algebras

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

Local triple derivations on C*-algebras and JB*-triples

In a first result we prove that every continuous local triple derivation on a JB

A Kowalski–Słodkowski theorem for 2-local \(^*\)-homomorphisms on von Neumann algebras

^*

Local Triple Derivations on C*-Algebras†

-triple is a triple derivation. We also give an automatic continuity result, that is, we show that local triple derivations on a JB

Surjective isometries between real JB*-triples

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On the facial structure of the unit ball in a JB*-triple

-triple are continuous even if not assumed a priori to be so. In particular every local triple derivation on a C

The alternative Dunford-Pettis property in C*-algebras and von Neumann preduals

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Little Grothendieck's theorem for real JB^*-triples

-algebra is a triple derivation. We also explore the connections between (bounded local) triple derivations and generalised (Jordan) derivations on a C