The structure of the isometric representations of unital discrete semi-groups with the left cancellation property and no divisors of the identity is studied. We obtain Wold decompositions and prove that the left regular representation of such semigroups is irreducible.

Let $b_{\ell;3}(n)$ denote the number of $\ell$-regular partitions $n$ in 3 colours. In this paper, we find some general generating functions and new infinite families congruences modulo arbitrary powers $3$ when $\ell\in\{9,27\}$. For instance, for positive integers $k$, have\begin{align*}b_{9;3}\left(3^k\cdot n+3^k-1\right)&\equiv0~\left(\mathrm{mod}~3^{2k}\right),\\b_{27;3}\left(3^{2k+3}...

‎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...

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...

It is shown that a ring R is (von Neumann) regular if and only if every cyclic left R-module is GP -injective if and only if R is left PP and left GP -injective. In addition, two examples are given to answer two open questions in the negative. 2000 Mathematics Subject Classification: 16E50, 16D60

We show that the quantum Heisenberg group Hq(1) can be obtained by means of contraction from quantum SUq(2) group. Its dual Hopf algebra is the quantum Heisenberg algebra Uq(h(1)). We derive left and right regular representations for Uq(h(1)) as acting on its dual Hq(1). Imposing conditions on the right representation the left representation is reduced to an irreducible holomorphic representati...

A generalization of injective modules (noted GI-modules), distinct from p-injective modules, is introduced. Rings whose p-injective modules are GI are characterized. If M is a left GI-module, E = End(AM), then E/J(E) is von Neumann regular, where J(E) is the Jacobson radical of the ring E. A is semisimple Artinian if, and only if, every left A-module is GI. If A is a left p. p., left GI-ring su...

For any natural number n, the group $$G_n$$ of all invertible affine transformations n-dimensional Euclidean space has, up to equivalence, just one square-integrable representation and left regular is a multiple this representation. We provide concrete realization $$\sigma $$ distinguished in two-dimensional case. explicitly decompose Hilbert $$L^2(G_2)$$ as direct sum left-invariant closed sub...

we describe under various conditions abelian subgroups of the automorphism‎ ‎group $mathrm{aut}(t_{n})$ of the regular $n$-ary tree $t_{n}$‎, ‎which are‎ ‎normalized by the $n$-ary adding machine $tau =(e‎, ‎dots‎, ‎e,tau )sigma _{tau‎ ‎}$ where $sigma _{tau }$ is the $n$-cycle $left( 0,1‎, ‎dots‎, ‎n-1right) $‎. ‎as‎ ‎an application‎, ‎for $n=p$ a prime number‎, ‎and for $n=4$‎, ‎we prove that...