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Let be an operator ideal on LCS’s. A continuous seminorm p of a LCS X is said to be - continuous if $Q{̃}_p{\in }^{inj}(X,X{̃}_p)$, where $X{̃}_p$ is the completion of the normed space $X_p=X/p^{-1}\left(0\right)$ and $Q{̃}_p$ is the canonical map. p is said to be a Groth()- seminorm if there is a continuous seminorm q of X such that p ≤ q and the canonical map $Q{̃}_{pq}:X{̃}_q\to X{̃}_p$ belongs to $(X{̃}_q,X{̃}_p)$. It is well known that when is the ideal of absolutely summing (resp. precompact, weakly compact) operators, a LCS X is a nuclear (resp. Schwartz, infra-Schwartz) space if and only if every continuous...

Let X and Y be locally compact Hausdorff spaces, let E and F be Banach spaces, and let T be a linear isometry from C₀(X,E) into C₀(Y,F). We provide three new answers to the Banach-Stone problem: (1) T can always be written as a generalized weighted composition operator if and only if F is strictly convex; (2) if T is onto then T can be written as a weighted composition operator in a weak sense; and (3) if T is onto and F does not contain a copy of $\ell {₂}^{\infty }$ then T can be written as a weighted composition...

Let L be a norm closed left ideal of a C*-algebra A. Then the left quotient A/L is a left A-module. In this paper, we shall implement Tomita’s idea about representing elements of A as left multiplications: ${\pi }_p\left(a\right)(b+L)=ab+L$. A complete characterization of bounded endomorphisms of the A-module A/L is given. The double commutant ${\pi }_p{\left(A\right)}^{\text{'}\text{'}}$ of ${\pi }_p\left(A\right)$ in B(A/L) is described. Density theorems of von Neumann and Kaplansky type are obtained. Finally, a comprehensive study of relative multipliers of A is carried out.

Let ₁, ₂ be (not necessarily unital or closed) standard operator algebras on locally convex spaces X₁, X₂, respectively. For k ≥ 2, consider different products $T₁\ast \cdots \ast T_k$ on elements in ${}_i$, which covers the usual product $T₁\ast \cdots \ast T_k=T₁\cdots T_k$ and the Jordan triple product T₁ ∗ T₂ = T₂T₁T₂. Let Φ: ₁ → ₂ be a (not necessarily linear) map satisfying $\sigma \left(\Phi \left(A₁\right)\ast \cdots \ast \Phi \left(A_k\right)\right)=\sigma \left(A₁\ast \cdots \ast A_k\right)$ whenever any one of $A_i$’s has rank at most one. It is shown that if the range of Φ contains all rank one and rank two operators then Φ must be a Jordan isomorphism multiplied by a root...

Being expected as a Banach space substitute of the orthogonal projections on Hilbert spaces, generalized n-circular projections also extend the notion of generalized bicontractive projections on JB*-triples. In this paper, we study some geometric properties of JB*-triples related to them. In particular, we provide some structure theorems of generalized n-circular projections on an often mentioned special case of JB*-triples, i.e., Hilbert C*-modules over abelian C*-algebras C0(Ω).

The existence results for an abstract Cauchy problem involving a higher order differential inclusion with infinite delay in a Banach space are obtained. We use the concept of the existence family to express the mild solutions and impose the suitable conditions on the nonlinearity via the measure of noncompactness in order to apply the theory of condensing multimaps for the demonstration of our results. An application to some classes of partial differential equations is given.