### A characterization of commutativity for non-associative normed algebras.

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Jordan ${H}^{*}$-pairs appear, in a natural way, in the study of Lie ${H}^{*}$-triple systems ([3]). Indeed, it is shown in [4, Th. 3.1] that the problem of the classification of Lie ${H}^{*}$-triple systems is reduced to prove the existence of certain ${L}^{*}$-algebra envelopes, and it is also shown in [3] that we can associate topologically simple nonquadratic Jordan ${H}^{*}$-pairs to a wide class of Lie ${H}^{*}$-triple systems and then the above envelopes can be obtained from a suitable classification, in terms of associative ${H}^{*}$-pairs, of...

By investigating the extent to which variation in the coefficients of a convex combination of unitaries in a unital $J{B}^{*}$-algebra permits that combination to be expressed as convex combination of fewer unitaries of the same algebra, we generalise various results of R. V. Kadison and G. K. Pedersen. In the sequel, we shall give a couple of characterisations of $J{B}^{*}$-algebras of $tsr\phantom{\rule{4pt}{0ex}}1$.

We prove that every biorthogonality preserving linear surjection from a weakly compact JB*-triple containing no infinite-dimensional rank-one summands onto another JB*-triple is automatically continuous. We also show that every biorthogonality preserving linear surjection between atomic JBW*-triples containing no infinite-dimensional rank-one summands is automatically continuous. Consequently, two atomic JBW*-triples containing no rank-one summands are isomorphic if and only if there exists a (not...

Given a complex Hilbert space H, we study the manifold $$\mathcal{A}$$ of algebraic elements in $$Z=\mathcal{L}\left(H\right)$$ . We represent $$\mathcal{A}$$ as a disjoint union of closed connected subsets M of Z each of which is an orbit under the action of G, the group of all C*-algebra automorphisms of Z. Those orbits M consisting of hermitian algebraic elements with a fixed finite rank r, (0< r<∞) are real-analytic direct submanifolds of Z. Using the C*-algebra structure of Z, a Banach-manifold structure and a G-invariant torsionfree affine...

In recent papers, the Right and the Strong* topologies have been introduced and studied on general Banach spaces. We characterize different types of continuity for multilinear operators (joint, uniform, etc.) with respect to the above topologies. We also study the relations between them. Finally, in the last section, we relate the joint Strong*-to-norm continuity of a multilinear operator T defined on C*-algebras (respectively, JB*-triples) to C*-summability (respectively, JB*-triple-summability)....