C⁎-algebras, group algebras, and the algebra A(X) of approximable operators on a Banach space X having bounded approximation property are known to be zero product determined. In this paper we give quantitative estimate by showing that, for A, there exists constant α with that every continuous bilinear functional φ:A×A→C linear ξ A such thatsup‖a‖=‖b‖=1⁡|φ(a,b)−ξ(ab)|≤αsup‖a‖=‖b‖=1,ab=0⁡|φ(a,b)|...

An operator is said to be strongly irreducible if its commutant has no nontrivial idempotent. This paper first shows that if an operator is not strongly irreducible then the set of idempotents in its commutant is either finite or uncountable. The second part of the paper focuses on the Jordan block which is a well-known class of irreducible operators, and determines when a Jordan block is stron...

‎Let $mathcal{A}$ be a unital Banach algebra‎, ‎$mathcal{M}$ be a left $mathcal{A}$-module‎, ‎and $W$ in $mathcal{Z}(mathcal{A})$ be a left separating point of $mathcal{M}$‎. ‎We show that if $mathcal{M}$ is a unital left $mathcal{A}$-module and $delta$ is a linear mapping from $mathcal{A}$ into $mathcal{M}$‎, ‎then the following four conditions are equivalent‎: ‎(i) $delta$ is a Jordan left de...

An accidental degeneracy of resonances gives rise to a double pole in the scattering matrix, a double zero in the Jost function and a Jordan chain of length two of generalized Gamow-Jordan eigenfunctions of the radial Schrödinger equation. The generalized Gamow-Jordan eigenfunctions are basis elements of an expansion in bound and resonant energy eigenfunctions plus a continuum of scattering wav...

let $mathcal {a} $ and $mathcal {b} $ be c$^*$-algebras. assume that $mathcal {a}$ is of real rank zero and unital with unit $i$ and $k>0$ is a real number. it is shown that if $phi:mathcal{a} tomathcal{b}$ is an additive map preserving $|cdot|^k$ for all normal elements; that is, $phi(|a|^k)=|phi(a)|^k $ for all normal elements $ainmathcal a$, $phi(i)$ is a projection, and there exists a posit...

Let G = [ A M N B ] be a generalized matrix algebra defined by the Morita context (A,B,A MB,B NA,ΦMN ,ΨNM) . In this article we mainly study the question of whether there exist the so-called “proper” Jordan derivations for the generalized matrix algebra G . It is shown that if one of the bilinear pairings ΦMN and ΨNM is nondegenerate, then every antiderivation of G is zero. Furthermore, if the ...

We present an example of a zero-dimensional F-space that is not strongly zero-dimensional.

Let $mathcal {A} $ and $mathcal {B} $ be C$^*$-algebras. Assume that $mathcal {A}$ is of real rank zero and unital with unit $I$ and $k>0$ is a real number. It is shown that if $Phi:mathcal{A} tomathcal{B}$ is an additive map preserving $|cdot|^k$ for all normal elements; that is, $Phi(|A|^k)=|Phi(A)|^k $ for all normal elements $Ainmathcal A$, $Phi(I)$ is a projection, and there exists a posit...