If X and Y are Banach spaces, then subalgebras ⊂ B(X) and ⊂ B(Y), not necessarily unital nor complete, are called standard operator algebras if they contain all finite rank operators on X and Y respectively. The peripheral spectrum of A ∈ is the set ${\sigma }_{\pi }\left(A\right)=\lambda \in \sigma \left(A\right):\left|\lambda \right|=ma{x}_{z\in \sigma \left(A\right)}\left|z\right|$ of spectral values of A of maximum modulus, and a map φ: → is called peripherally-multiplicative if it satisfies the equation ${\sigma }_{\pi }\left(\phi \left(A\right)\circ \phi \left(B\right)\right)={\sigma }_{\pi }\left(AB\right)$ for all A,B ∈ . We show that any peripherally-multiplicative and surjective map φ: → , neither assumed to be linear nor...

It is shown that every algebraic isomorphism between standard subalgebras of 𝒥-subspace lattice algebras is quasi-spatial and every Jordan derivation of standard subalgebras of 𝒥-subspace lattice algebras is an additive derivation. Also, it is proved that every finite rank operator in a 𝒥-subspace lattice algebra can be written as a finite sum of rank one operators each belonging to that algebra. As an additional result, a multiplicative bijection of a 𝒥-subspace lattice algebra onto an arbitrary...

A systematic investigation of algebras of holomorphic functions endowed with the Hadamard product is given. For example we show that the set of all non-invertible elements is dense and that each multiplicative functional is continuous, answering some questions in the literature.

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...

We prove that a biseparating map between spaces of vector-valued continuous functions is usually automatically continuous. However, we also discuss special cases when this is not true.

Let ${\tau }_X$ and ${\tau }_Y$ be representations of a topological group G on Banach spaces X and Y, respectively. We investigate the continuity of the linear operators Φ: X → Y with the property that $\Phi \circ {\tau }_X\left(t\right)={\tau }_Y\left(t\right)\circ \Phi$ for each t ∈ G in terms of the invariant vectors in Y and the automatic continuity of the invariant linear functionals on X.