In this paper we study the concept of ph-biatness ofa Banach algebra A, where ph is a continuous homomorphism on A.We prove that if ph is a continuous epimorphism on A and A hasa bounded approximate identity and A is ph- biat, then A is ph-amenable. In the case where ph is an isomorphism on A we showthat the ph- amenability of A implies its ph-biatness.

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 $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 $mathcal{a}$ and $mathcal{b}$ be two $c^{*}$-algebras such that $mathcal{b}$ is prime. in this paper, we investigate the additivity of maps $phi$ from $mathcal{a}$ onto $mathcal{b}$ that are bijective, unital and satisfy $phi(ap+eta pa^{*})=phi(a)phi(p)+eta phi(p)phi(a)^{*},$ for all $ainmathcal{a}$ and $pin{p_{1},i_{mathcal{a}}-p_{1}}$ where $p_{1}$ is a nontrivial projection in $mathcal{a...

Let $mathcal{A}$ and $mathcal{B}$ be two $C^{*}$-algebras such that $mathcal{B}$ is prime. In this paper, we investigate the additivity of maps $Phi$ from $mathcal{A}$ onto $mathcal{B}$ that are bijective, unital and satisfy $Phi(AP+eta PA^{*})=Phi(A)Phi(P)+eta Phi(P)Phi(A)^{*},$ for all $Ainmathcal{A}$ and $Pin{P_{1},I_{mathcal{A}}-P_{1}}$ where $P_{1}$ is a nontrivial projection in $mathcal{A...

The purpose of this paper is to introduce some new classes of rings and modules that are closely related to the classes of almost Dedekind domains and almost Dedekind modules. We introduce the concepts of $\phi$-almost Dedekind rings and $\Phi$-almost Dedekind modules and study some properties of this classes. In this paper we get some equivalent conditions for $\phi$-almost Dedekind rings and ...

in this paper we study the concept of ph-biatness ofa banach algebra a, where ph is a continuous homomorphism on a.we prove that if ph is a continuous epimorphism on a and a hasa bounded approximate identity and a is ph- biat, then a is ph-amenable. in the case where ph is an isomorphism on a we showthat the ph- amenability of a implies its ph-biatness.

‎in this paper, we study some kinds of majorizations on $textbf{m}_{n}$ and their linear or strong linear preservers. also, we find the structure of linear or strong linear preservers which are multiplicative, i.e.  linear or strong linear preservers like $phi $ with the property $phi (ab)=phi (a)phi (b)$ for every $a,bin textbf{m}_{n}$.