For a frame $L$, consider the $f$-ring $ mathcal{F}_{mathcal P}L=Frm(mathcal{P}(mathbb R), L)$. In this paper, first we show that each minimal ideal of $ mathcal{F}_{mathcal P}L$ is a principal ideal generated by $f_a$, where $a$ is an atom of $L$. Then we show that if $L$ is an $mathcal{F}_{mathcal P}$-completely regular frame, then the socle of $ mathcal{F}_{mathcal P}L$ consists of those $f$...

‎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...

let $l$ be a completely regular frame and $mathcal{r}l$ be the ‎ring of continuous real-valued functions on $l$‎. ‎we show that the‎ ‎lattice $zid(mathcal{r}l)$ of $z$-ideals of $mathcal{r}l$ is a‎ ‎normal coherent yosida frame‎, ‎which extends the corresponding $c(x)$‎ ‎result of mart'{i}nez and zenk‎. ‎this we do by exhibiting‎ ‎$zid(mathcal{r}l)$ as a quotient of $rad(mathcal{r}l)$‎, ‎the‎ ‎...

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...

in this work‎, ‎an iterative method based on a matrix form of lsqr algorithm is constructed for solving the linear operator equation $mathcal{a}(x)=b$‎ ‎and the minimum frobenius norm residual problem $||mathcal{a}(x)-b||_f$‎ ‎where $xin mathcal{s}:={xin textsf{r}^{ntimes n}~|~x=mathcal{g}(x)}$‎, ‎$mathcal{f}$ is the linear operator from $textsf{r}^{ntimes n}$ onto $textsf{r}^{rtimes s}$‎, ‎$ma...

Let $mathcal{A}$ be a Banach algebra and $mathcal{M}$ be a Banach $mathcal{A}$-bimodule. We say that a linear mapping $delta:mathcal{A} rightarrow mathcal{M}$ is a generalized $sigma$-derivation whenever there exists a $sigma$-derivation $d:mathcal{A} rightarrow mathcal{M}$ such that $delta(ab) = delta(a)sigma(b) + sigma(a)d(b)$, for all $a,b in mathcal{A}$. Giving some facts concerning general...

In this paper, we will study the theory of cyclic homology for regular multiplier Hopf algebras. We associate a cyclic module to a triple $(mathcal{R},mathcal{H},mathcal{X})$ consisting of a regular multiplier Hopf algebra $mathcal{H}$, a left $mathcal{H}$-comodule algebra $mathcal{R}$, and a unital left $mathcal{H}$-module $mathcal{X}$ which is also a unital algebra. First, we construct a para...

‎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...

In this paper, we introduce $(mathcal{C},mathcal{C}')$-controlled continuous $g$-Bessel families and their multipliers in Hilbert spaces and investigate some of their properties. We show that under some conditions sum of two $(mathcal{C},mathcal{C}')$-controlled continuous $g$-frames is a $(mathcal{C},mathcal{C}')$-controlled continuous $g$-frame. Also, we investigate when a $(mathcal{C},mathca...