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We construct a natural linear isomorphism between the little Bloch space B0 and the Banach space c0 of complex null sequences. This paper is written for the special issue of Studia Universitatis Babes-Bolyai Mathematica in memory of Professor Gabriela Kohr. [ABSTRACT FROM AUTHOR]

Let Ω be an open set in ℝd (d > 1) and let h(Ω) be the Fréchet space of harmonic functions on Ω. Given a bounded linear operator L : h (Ω) → h(Ω), we show that its eigenvalues λn, arranged in decreasing order and counting multiplicities, satisfy |λn| ≤ K exp(−cn1/(d−1)), where K and c are two explicitly computable positive constants. [ABSTRACT FROM PUBLISHER]

Abstract.  We show that, in a JB-algebra, the projections form a Banach manifold and also, the rank-n projections in a JBW-factor form a Riemannian symmetric space of compact type, for $$n\, \in \,\user2{\mathbb{N}}\, \cup \, \left\{ 0 \right\}$$ . [ABSTRACT FROM AUTHOR]

Abstract Let Z and W be JB*-triples and let T be a linear isometry from Z into W. For any z ∈ Z with ||z||<1, we show that [ABSTRACT FROM AUTHOR]

A class of harmonic function spaces is introduced and studied, namely the spaces $H^p_\sigma(G)$ of $\sigma$-harmonic $L^p$ functions on a locally compact group $G$, for $1\leq p \leq \infty$ and a given complex measure $\sigma$ on $G$ of unit norm. It is shown that there is a contractive projection from $L^p(G)$ onto $H^p_\sigma(G)$, for $1\lt p\leq \infty$, and structural results for $H^p_\sigma (G)$ are deduced. Given an adapted probability measure $\sigma$ on $G$, a uniqueness result is proved, that the space $H^p_\sigma(G)$ contains only constant functions, for $1\leq p\,{\lt}\,\infty$. For any $\sigma$, a result on the dimension of $H^1_\sigma (G)$ is proved. [ABSTRACT FROM AUTHOR]

Let $\sigma$ be a non-degenerate positive $M_n$-valued measure on a locally compact group $G$ with $\|\sigma\|\,{=}\,1$. An $M_n$-valued Borel function $f$ on $G$ is called $\sigma$-harmonic if $f(x) = \int_G f(xy^{-1})\,d\sigma(y)$ for all $x\,{\in}\,G$. Given such a function $f$ which is bounded and left uniformly continuous on $G$, it is shown that every central element in $G$ is a period of $f$. Further, it is shown that $f$ is constant if $G$ is nilpotent or central. [ABSTRACT FROM AUTHOR]

Let $\sigma$ be a positive matrix-valued measure on a locally compact abelian group $G$ such that $\sigma(G)$ is the identity matrix. We give a necessary and sufficient condition on $\sigma$ for the absence of a bounded non-constant matrix-valued function $f$ on $G$ satisfying the convolution equation $ f *\sigma = f$. This extends Choquet and Deny's theorem for real-valued functions on $G$. [ABSTRACT FROM AUTHOR]

Given a compact holomorphic fixed-point-free self-map, $f$, of the open unit ball of a Hilbert space, we show that the sequence of iterates, $(f^n)$, converges locally uniformly to a constant map $\xi$ with $\Vert \xi\Vert = 1$. This extends results of Denjoy (1926), Wolff (1926), Hervé (1963) and MacCluer (1983). The result is false without the compactness assumption, nor is it true for all open balls of $J^{*}$-algebras. [ABSTRACT FROM AUTHOR]

We obtain some results on the unique extension of (factor) states of $C^*$-algebras which complement various existing results. Our results also lead to a class of $C^*$-algebras whose states are $\sigma$-convex sums of factor states. [ABSTRACT FROM AUTHOR]