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1-amenability of 𝒜(X) for Banach spaces with 1-unconditional bases

A. Blanco (2012)

Studia Mathematica

The main result of the note is a characterization of 1-amenability of Banach algebras of approximable operators for a class of Banach spaces with 1-unconditional bases in terms of a new basis property. It is also shown that amenability and symmetric amenability are equivalent concepts for Banach algebras of approximable operators, and that a type of Banach space that was long suspected to lack property 𝔸 has in fact the property. Some further ideas on the problem of whether or not amenability (in...

2-local Jordan automorphisms on operator algebras

Ajda Fošner (2012)

Studia Mathematica

We investigate 2-local Jordan automorphisms on operator algebras. In particular, we show that every 2-local Jordan automorphism of the algebra of all n× n real or complex matrices is either an automorphism or an anti-automorphism. The same is true for 2-local Jordan automorphisms of any subalgebra of ℬ which contains the ideal of all compact operators on X, where X is a real or complex separable Banach spaces and ℬ is the algebra of all bounded linear operators on X.

2-summing multiplication operators

Dumitru Popa (2013)

Studia Mathematica

Let 1 ≤ p < ∞, = ( X ₙ ) n ∈ ℕ be a sequence of Banach spaces and l p ( ) the coresponding vector valued sequence space. Let = ( X ₙ ) n ∈ ℕ , = ( Y ₙ ) n ∈ ℕ be two sequences of Banach spaces, = ( V ₙ ) n ∈ ℕ , Vₙ: Xₙ → Yₙ, a sequence of bounded linear operators and 1 ≤ p,q < ∞. We define the multiplication operator M : l p ( ) → l q ( ) by M ( ( x ₙ ) n ∈ ℕ ) : = ( V ₙ ( x ₙ ) ) n ∈ ℕ . We give necessary and sufficient conditions for M to be 2-summing when (p,q) is one of the couples (1,2), (2,1), (2,2), (1,1), (p,1), (p,2), (2,p), (1,p), (p,q); in the last case 1 < p < 2, 1 < q < ∞.

A generalization of peripherally-multiplicative surjections between standard operator algebras

Takeshi Miura, Dai Honma (2009)

Open Mathematics

Let A and B be standard operator algebras on Banach spaces X and Y, respectively. The peripheral spectrum σπ (T) of T is defined by σπ (T) = z ∈ σ(T): |z| = maxw∈σ(T) |w|. If surjective (not necessarily linear nor continuous) maps φ, ϕ: A → B satisfy σπ (φ(S)ϕ(T)) = σπ (ST) for all S; T ∈ A, then φ and ϕ are either of the form φ(T) = A 1 TA 2 −1 and ϕ(T) = A 2 TA 1 −1 for some bijective bounded linear operators A 1; A 2 of X onto Y, or of the form φ(T) = B 1 T*B 2 −1 and ϕ(T) = B 2 T*B −1 for some...

A note on embedding into product spaces

M. A. Sofi (2006)

Czechoslovak Mathematical Journal

Using factorization properties of an operator ideal over a Banach space, it is shown how to embed a locally convex space from the corresponding Grothendieck space ideal into a suitable power of E , thus achieving a unified treatment of several embedding theorems involving certain classes of locally convex spaces.

A note on the hyperreflexivity constant for certain reflexive algebras

Satoru Tosaka (1999)

Studia Mathematica

Using results on the reflexive algebra with two invariant subspaces, we calculate the hyperreflexivity constant for this algebra when the Hilbert space is two-dimensional. Then by the continuity of the angle for two subspaces, there exists a non-CSL hyperreflexive algebra with hyperreflexivity constant C for every C>1. This result leads to a kind of continuity for the hyperreflexivity constant.

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