arXiv subjects
Publications and source records attributed to Gerardo M. Escolano.
Let $\mathfrak{J}$ be a JB$^*$-algebra with no quotients isomorphic to $S_2(\mathbb{C})$. Let $μ$ be a local quasi-linear Jordan functional on $\mathfrak{J}_{sa}$. We show that $μ$ is a linear functional on $\mathfrak{J}_{sa}$ if and only if the restriction of $μ$ to the closed unit ball of $\mathfrak{J}_{sa}$ is uniformly weakly continuous.
Let $\mathfrak{A}$ and $\mathfrak{B}$ be JBW$^*$-algebras whose sets of unitaries are denoted by $\mathcal{U}(\mathfrak{A})$ and $\mathcal{U}(\mathfrak{B})$, respectively. We show that $\mathcal{U}(\mathfrak{A})$ is closed for Jordan products of operator commuting pairs inside itself. Assuming that $\mathfrak{A}$ and $\mathfrak{B}$ are JBW$^*$-algebras without direct summands of type $I_1$ or $I_2$, we prove that for each bicontinuous bijection $Φ: \mathcal{U}(\mathfrak{A}) \rightarrow \mathcal{U}(\mathfrak{B})$ satisfying $Φ(u \circ v) = Φ(u)\circ Φ(v),$ whenever $u$ and $v$ are operator commuting unitaries in $\mathfrak{A}$, there exist a linear Jordan $^*$-isomorphism $θ: \mathfrak{A} \rightarrow \mathfrak{B}$, a real linear mapping $β: \mathfrak{A_{sa}}\rightarrow Z(\mathfrak{B}_{sa})$, and an invertible central element $c \in \mathfrak{B}_{sa}$ such that $$ Φ(e^{i a}) = e^{i β(a)}\circ e^{i c\circθ(a)} = e^{i β(a)} \circ θ\left( e^{i θ^{-1}( c )\circ a}\right),$$ for all $a\in \mathfrak{A}_{sa}$. The conclusion improves when $\mathfrak{A}$ is a JBW$^*$-algebra factor not of type $I_2$.
Let $\mathcal{P} (\mathfrak{J})$ denote the lattice of projections of a JBW$^*$-algebra $\mathfrak{J}$, and let $X$ be a Banach space. A bounded finitely additive $X$-valued measure on $\mathcal{P}(\mathfrak{J})$ is a mapping $μ: \mathcal{P}(\mathfrak{J}) \rightarrow X$ satisfying: $(a)$ $μ(p +q) = μ(p) + μ(q)$, whenever $p \circ q = 0$ in $\mathcal{P} (\mathfrak{J})$, $(b)$ $\sup \{ \| μ(p)\| \, : \, p \in \mathcal{P} (\mathfrak{J})\} < \infty$. In this paper we establish a Mackey-Gleason-Bunce-Wright theorem by showing that if $\mathfrak{J}$ contains no type $I_2$ direct summand, every bounded finitely additive measure $μ: \mathcal{P}(\mathfrak{J}) \rightarrow X$ admits an extension to a bounded linear operator from $\mathfrak{J}$ to $X$. This solves a long-standing open conjecture.
Let $\mathfrak{M}$ and $\mathfrak{J}$ be JBW$^*$-algebras admitting no central summands of type $I_1$ and $I_2,$ and let $Φ: \mathfrak{M} \rightarrow \mathfrak{J}$ be a linear bijection preserving operator commutativity in both directions, that is, $$[x,\mathfrak{M},y] = 0 \Leftrightarrow [Φ(x),\mathfrak{J},Φ(y)] = 0,$$ for all $x,y\in \mathfrak{M}$, where the associator of three elements $a,b,c$ in $\mathfrak{M}$ is defined by $[a,b,c]:=(a\circ b)\circ c - (c\circ b)\circ a$. We prove that under these conditions there exist a unique invertible central element $z_0$ in $\mathfrak{J}$, a unique Jordan isomorphism $J: \mathfrak{M} \rightarrow \mathfrak{J}$, and a unique linear mapping $β$ from $\mathfrak{M}$ to the centre of $\mathfrak{J}$ satisfying $$ Φ(x) = z_0 \circ J(x) + β(x), $$ for all $x\in \mathfrak{M}.$ Furthermore, if $Φ$ is a symmetric mapping (i.e., $Φ(x^*) = Φ(x)^*$ for all $x\in \mathfrak{M}$), the element $z_0$ is self-adjoint, $J$ is a Jordan $^*$-isomorphism, and $β$ is a symmetric mapping too. In case that $\mathfrak{J}$ is a JBW$^*$-algebra admitting no central summands of type $I_1$, we also address the problem of describing the form of all symmetric bilinear mappings $B : \mathfrak{J}\times \mathfrak{J}\to \mathfrak{J}$ whose trace is associating (i.e., $[B(a,a),b,a] = 0,$ for all $a, b \in \mathfrak{J})$ providing a complete solution to it. We also determine the form of all associating linear maps on $\mathfrak{J}$.
We establish some Lie--Trotter formulae for unital complex Jordan--Banach algebras, showing that for each couple of elements $a,b$ in a unital complex Jordan--Banach algebra $\mathfrak{A}$ the identities $$ \lim_{n\to \infty} \left(e^{\frac{a}{n}}\circ e^{\frac{b}{n}} \right)^{n} = e^{a+b},\ \lim_{n\to \infty} \left(U_{e^{\frac{a}{n}}} \left( e^{\frac{b}{n}}\right) \right)^{n} = e^{2 a+b}, \hbox{ and }$$ $$ \lim_{n\to \infty} \left(U_{e^{\frac{a}{n}},e^{\frac{c}{n}}} \left( e^{\frac{b}{n}}\right) \right)^{n} = e^{a+b + c}$$ hold. These formulae are actually deduced from a more general result involving holomorphic functions with values in $\mathfrak{A}$. These formulae are employed in the study of spectral-valued (non-necessarily linear) functionals $f:\mathfrak{A}\to \mathbb{C}$ satisfying $f(U_x (y))=U_{f(x)}f(y),$ for all $x,y\in \mathfrak{A}$. We prove that for any such a functional $f,$ there exists a unique continuous (Jordan-)multiplicative linear functional $ψ\colon \mathfrak{A}\to\mathbb{C}$ such that $ f(x)=ψ(x),$ for every $x$ in the connected component of set of all invertible elements of $\mathfrak{A}$ containing the unit element. If we additionally assume that $\mathfrak{A}$ is a JB$^*$-algebra and $f$ is continuous, then $f$ is a linear multiplicative functional on $\mathfrak{A}$. The new conclusions are appropriate Jordan versions of results by Maouche, Brits, Mabrouk, Shulz, and Tour{é}.