The extension of Banach Lie-Poisson spaces is studied and linked to the extension of a special class of Banach Lie algebras. The case of W -algebras is given particular attention. Semidirect products and the extension of the restricted Banach Lie-Poisson space by the Banach Lie-Poisson space of compact operators are given as examples.

We use the Poisson integral formula in order to establish harmonic functional calculus for quotient involutive Banach algebras. We give several properties of this functional calculus. L.Waelbroeck’s holomorphic functional calculus is a special case of ours. We use our harmonic functional calculus to extend result of J.W.M.Ford to quotient involutive Banach algebras. L.WAELBROECK a montré dans [...

We study a noncommutative (operator space) version of the ‘boundary’, and in particular the Shilov boundary, of a function space. The main idea is that Hilbert C∗−modules and their properties, which we studied earlier in the operator space framework, replace certain topological tools. We include some general notes on the ‘commutative case’ of some of the topics we discuss, coming in part from j...

Let X be a non-degenerate left Banach module over a normed algebra A having a bounded approximate left identity. We show that, if A is a left ideal of a larger algebra, then this representation can be extended to a representation of the larger algebra. Based on this result, we study in detail the existence and properties of representations of the various centralizer algebras of A which are comp...

$r$-module. in this paper, we explore more properties of $max$-injective modules and we study some conditions under which the maximal spectrum of $m$ is a $max$-spectral space for its zariski topology.

Given a locally compact abelian group $G$ and closed subgroup $\Lambda$ in $G\times\widehat{G}$, Rieffel associated to Hilbert $C^*$-module $\mathcal{E}$, known as Heisenberg module. He proved that $\mathcal{E}$ is an equivalence bimodule between the twisted $C^*$-algebra $C^*(\Lambda,\textsf{c})$ $C^*(\Lambda^\circ,\bar{\textsf{c}})$, where $\Lambda^{\circ}$ denotes adjoint of $\Lambda$. Our m...

let $r$ be a ring and $m$ a right $r$-module with $s=end_r(m)$. a module $m$ is called semi-projective if for any epimorphism $f:mrightarrow n$, where $n$ is a submodule of $m$, and for any homomorphism $g: mrightarrow n$, there exists $h:mrightarrow m$ such that $fh=g$. in this paper, we study sgq-projective and$pi$-semi-projective modules as two generalizations of semi-projective modules. a m...

let $r$ be a commutative ring with identity and $m$ be a unitary$r$-module. the primary-like spectrum $spec_l(m)$ is thecollection of all primary-like submodules $q$ such that $m/q$ is aprimeful $r$-module. here, $m$ is defined to be rsp if $rad(q)$ isa prime submodule for all $qin spec_l(m)$. this class containsthe family of multiplication modules properly. the purpose of thispaper is to intro...

We will show that Feichtinger’s algebra S0(R ) (also known as modulation space M 0 (R ) and well defined for general locally compact Abelian groups) is a natural Hilbert module for quantum tori and that a result by V. Losert can be interpreted as implying that the construction of such Hilbert modules gives Feichtinger’s algebra, if one considers the category of time-frequency invariant homogeno...

Let X be a non-degenerate left Banach module over a normed algebra A having a bounded approximate left identity. We show that, if A is a left ideal of a larger algebra, then this representation can be extended to a representation of the larger algebra. Based on this result, we study in detail the existence and properties of representations of the various centralizer algebras of A which are comp...