### A note on distribution spaces on manifolds.

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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.

The notion of a compressible operator on a Banach space, E, derives from automatic continuity arguments. It is related to the notion of a cartesian Banach space. The compressible operators on E form an ideal in ℬ(E) and the automatic continuity proofs depend on showing that this ideal is large. In particular, it is shown that each weakly compact operator on the James' space, J, is compressible, whence it follows that all homomorphisms from ℬ(J) are continuous.