Suppose that $A$ is a semi-simple and commutative Banach algebra. In this paper we try to characterize the character space of the Banach algebra $C_{rm{BSE}}(Delta(A))$ consisting of all  BSE-functions on $Delta(A)$ where $Delta(A)$ denotes the character space of $A$. Indeed, in the case that $A=C_0(X)$ where $X$ is a non-empty locally compact Hausdroff space, we give a complete characterizatio...

We offer a new definition of $varepsilon$-orthogonality in normed spaces, and we try to explain some properties of which. Also we introduce some types of $varepsilon$-orthogonality in an arbitrary  $C^ast$-algebra $mathcal{A}$, as a Hilbert $C^ast$-module over itself, and investigate some of its properties in such spaces. We state some results relating range-kernel orthogonality in $C^*$-algebras.

Suppose $pi:mathcal{A}rightarrow mathcal{B}$ is a surjective unital $ast$-homomorphism between C*-algebras $mathcal{A}$ and $mathcal{B}$, and $0leq aleq1$ with $ain  mathcal{A}$. We give a sufficient condition that ensures there is a proection $pin mathcal{A}$ such that $pi left( pright) =pi left( aright) $. An easy consequence is a result of [L. G. Brown and G. k. Pedersen, C*-algebras of real...