In this paper, first, we introduce the new concept of 2-inner product on Banach modules over a $C^*$-algebra. Next,  we present the concept of 2-linear operators over a $C^*$-algebra. Our result improve  the main result of the paper  Z. Lewandowska.  In the final of this paper, we define the notions 2-adjointable mappings between 2-pre Hilbert C*-modules and prove supperstability of them ...

In [1,2,3], A. C. Baker and J.W. Baker studied the subspace Ma(S) of the convolution measure algebra M, (S) of a locally compact semigroup. H. Dzinotyiweyi in [5,7] considers an analogous measure space on a large class of C-distinguished topological semigroups containing all completely regular topological semigroups. In this paper, we extend the definitions to study the weighted semigroup ...

Mathieu and Ruddy proved that if be a unital spectral isometry from a unital C*-algebra Aonto a unital type I C*-algebra B whose primitive ideal space is Hausdorff and totallydisconnected, then is Jordan isomorphism. The aim of this note is to show that if be asurjective spectrum preserving additive map, then is a Jordan isomorphism without the extraassumption totally disconnected.

we study topological von neumann regularity and principal von neumann regularity of banach algebras. our main objective is comparing these two types of banach algebras and some other known banach algebras with one another. in particular, we show that the class of topologically von neumann regular banach algebras contains all $c^*$-algebras, group algebras of compact abelian groups and cer...

We study certain function algebras and their operator algebra completions on r-discrete abelian groupoids, the corresponding conditional expectations, maximal abelian subalgebras (masa) and eigen-functionals. We give a semidirect product decomposition for an abelian groupoid. This is done through a matched pair and leads to a C*-diagonal (for a special case). We use this decomposition to study ...

In the paper we describe class of all solvable extensions an infinite-dimensional filiform Leibniz algebra. The algebra is taken as a maximal pro-nilpotent ideal residually It proven that second cohomology group extension trivial.

 For a Banach algebra $fA$, we introduce ~$c.c(fA)$, the set of all $phiin fA^*$ such that $theta_phi:fAto  fA^*$ is a completely continuous operator, where $theta_phi$ is defined by $theta_phi(a)=acdotphi$~~ for all $ain fA$. We call $fA$, a completely continuous Banach algebra if $c.c(fA)=fA^*$. We give some examples of completely continuous Banach algebras and a sufficient condition for an o...