Positive definite hermitian mappings associated to tripotent elements

We give a simple proof of a meaningful result established by Y. Friedman and B. Russo in 1985, whose proof was originally based on strong holomorphic results. We provide a simple proof which is directly deduced from the axioms of JB*-triples with techniques of Functional Analysis

A note on 2-local representations of C∗^*-algebras

We survey the results on linear local and 2-local homomorphisms and zero products preserving operators between C$^*$-algebras, and we incorporate some new precise observations and results to prove that every bounded linear 2-local homomorphism between C$^*$-algebras is a homomorphism. Consequently, every linear 2-local $^*$-homomorphism between C$^*$-algebras is a $^*$-homomorphism

On the unit sphere of positive operators

Given a C$^*$-algebra $A$, let $S(A^+)$ denote the set of those positive elements in the unit sphere of $A$. Let $H_1$, $H_2,$ $H_3$ and $H_4$ be complex Hilbert spaces, where $H_3$ and $H_4$ are infinite-dimensional and separable. In this note we prove a variant of Tingley's problem by showing that every surjective isometry $\Delta : S(B(H_1)^+)\to S(B(H_2)^+)$ or (respectively, $\Delta : S(K(H_3)^+)\to S(K(H_4)^+)$) admits a unique extension to a surjective complex linear isometry from $B(H_1)$ onto $B(H_2))$ (respectively, from $K(H_3)$ onto $B(H_4)$). This provides a positive answer to a conjecture posed by G. Nagy [\emph{Publ. Math. Debrecen}, 2018]

Weak Banach-Saks property and Komlos theorem for preduals of JBW∗^*-triples

We show that the predual of a JBW$^*$-triple has the weak Banach-Saks property, that is, reflexive subspaces of a JBW$^*$-triple predual are super-reflexive. We also prove that JBW$^*$-triple preduals satisfy the Koml\'os property (which can be considered an abstract version of the weak law of large numbers). The results rely on two previous papers from which we infer the fact that, like in the classical case of $L^1$, a subspace of a JBW$^*$-triple predual contains $\ell_1$ as soon as it contains uniform copies of $\ell_1^n$

Weak 2-local derivations on Mn\mathbb{M}_n

We introduce the notion of weak-2-local derivation (respectively, $^*$-derivation) on a C$^*$-algebra $A$ as a (non-necessarily linear) map $\Delta : A\to A$ satisfying that for every $a,b\in A$ and $\phi\in A^*$ there exists a derivation (respectively, a $^*$-derivation) $D_{a,b,\phi}: A\to A$, depending on $a$, $b$ and $\phi$, such that $\phi \Delta (a) = \phi D_{a,b,\phi} (a)$ and $\phi \Delta (b) = \phi D_{a,b,\phi} (b)$. We prove that every weak-2-local $^*$-derivation on $M_n$ is a linear derivation. We also show that the same conclusion remains true for weak-2-local $^*$-derivations on finite dimensional C$^*$-algebras

Linear isometries between real JB*-triples and C*-algebras

Let $T: A\to B$ be a (not necessarily surjective) linear isometry between two real JB$^*$-triples. Then for each $a\in A$ there exists a tripotent $u_a$ in the bidual, $B'',$ of $B$ such that \begin{enumerate}[$(a)$] \item $\{u_a,T(\{f,g,h\}),u_a\}=\{u_a,\{T(f),T(g),T(h)\},u_a\}$, for all $f,g,h$ in the real JB$^*$-subtriple, $A_a,$ generated by $a$; \item The mapping $\{u_a,T(\cdot),u_a\} :A_a\rightarrow B''$ is a linear isometry. \end{enumerate} Furthermore, when $B$ is a real C$^*$-algebra, the projection $p=p_a= u_a^* u_a$ satisfies that $T(\cdot)p :A_a\rightarrow B''$ is an isometric triple homomorphism. When $A$ and $B$ are real C$^*$-algebras and $A$ is abelian of real type, then there exists a partial isometry $u\in B''$ such that the mapping $T(\cdot)u^*u :A\rightarrow B''$ is an isometric triple homomorphism. These results generalise, to the real setting, some previous contributions due to C.-H. Chu and N.-C. Wong, and C.-H. Chu and M. Mackey in 2004 and 2005. We give an example of a non-surjective real linear isometry which cannot be complexified to a complex isometry, showing that the results in the real setting can not be derived by a mere complexification argument.Comment: to appear in Quart. J. Mat

Quasi-linear functionals determined by weak-2-local ∗^*-derivations on B(H)B(H)

We prove that, for every separable complex Hilbert space $H$, every weak-2-local $^*$-derivation on $B(H)$ is a linear $^*$-derivation. We also establish that every (non-necessarily linear nor continuous) weak-2-local derivation on a finite dimensional C$^*$-algebra is a linear derivation

On the Mazur--Ulam property for the space of Hilbert-space-valued continuous functions

Let $K$ be a compact Hausdorff space and let $H$ be a real or complex Hilbert space with dim$(H_\mathbb{R})\geq 2$. We prove that the space $C(K,H)$ of all $H$-valued continuous functions on $K$, equipped with the supremum norm, satisfies the Mazur--Ulam property, that is, if $Y$ is any real Banach space, every surjective isometry $\Delta$ from the unit sphere of $C(K,H)$ onto the unit sphere of $Y$ admits a unique extension to a surjective real linear isometry from $C(K,H)$ onto $Y$. Our strategy relies on the structure of $C(K)$-module of $C(K,H)$ and several results in JB$^*$-triple theory. For this purpose we determine the facial structure of the closed unit ball of a real JB$^*$-triple and its dual space

von Neumann algebra preduals satisfy the linear biholomorphic property

We prove that for every JBW$^*$-triple $E$ of rank $>1$, the symmetric part of its predual reduces to zero. Consequently, the predual of every infinite dimensional von Neumann algebra $A$ satisfies the linear biholomorphic property, that is, the symmetric part of $A_*$ is zero. This solves a problem posed by M. Neal and B. Russo in [Mathematica Scandinavica, to appear

The Mazur-Ulam property for commutative von Neumann algebras

Let $(\Omega,\mu)$ be a $\sigma$-finite measure space. Given a Banach space $X$, let the symbol $S(X)$ stand for the unit sphere of $X$. We prove that the space $L^{\infty} (\Omega,\mu)$ of all complex-valued measurable essentially bounded functions equipped with the essential supremum norm, satisfies the Mazur-Ulam property, that is, if $X$ is any complex Banach space, every surjective isometry $\Delta: S(L^{\infty} (\Omega,\mu))\to S(X)$ admits an extension to a surjective real linear isometry $T: L^{\infty} (\Omega,\mu)\to X$. This conclusion is derived from a more general statement which assures that every surjective isometry $\Delta : S(C(K))\to S(X),$ where $K$ is a Stonean space, admits an extension to a surjective real linear isometry from $C(K)$ onto $X$