Given a finite (associative, unital) ring R, let K(R) denote the set of polynomials in R[x] that send each element of R to 0 under evaluation. We study K(R) and its elements. We conjecture that K(R) is a two-sided ideal of R[x] for any finite ring R, and prove the conjecture for several classes of finite rings (including commutative rings, semisimple rings, local rings, and all finite rings of ...

A C-metric algebra consists of a unital C-algebra and a Leibniz Lip-norm on the C-algebra. We show that if the Lip-norms concerned are lower semicontinuous, then any unital -homomorphism from a C-metric algebra to another one is necessarily Lipschitz. It results that the free product of two Lipschitz unital -homomorphisms between C-metric algebras coming from -filtrations is still a Lipschitz u...

Inspired by the recent work of Bekka, we study two reasonable analogues of property (T ) for not necessarily unital C∗-algebras. The stronger one of the two is called “property (T )” and the weaker one is called “property (Te)”. It is shown that all non-unital C*-algebras do not have property (T ) (neither do their unitalizations). Moreover, all non-unital σ-unital C*-algebras do not have prope...

Let A be a unital separable C∗-algebra and B = C ⊗ K, where C is a unital C∗-algebra. Let τ : A → M(B)/B be a weakly unital full essential extensions of A by B. We show that there is a bijection between a quotient group of K0(B) onto the set of strong unitary equivalence classes of weakly unital full essential extensions σ such that [σ] = [τ ] in KK(A,B). Consequently, when this group is zero, ...

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...

Let A be a unital AH-algebra and let α ∈ Aut(A) be an automorphism. A necessary condition for A ⋊α Z being embedded into a unital simple AF-algebra is the existence of a faithful tracial state. If in addition, there is an automorphism κ with κ∗1 = −idK1(A) such that α ◦ κ and κ ◦ α are asymptotically unitarily equivalent, then A⋊α Z can be embedded into a unital simple AF-algebra. Consequently,...

We consider the ideal structure of a reduced crossed product unital C∗-algebra equipped with an action discrete group. More specifically we find sufficient and necessary conditions for group to have intersection property, meaning that non-zero ideals in restrict underlying C∗-algebra. show property on is equivalent equivariant injective envelope. also centre envelope always contains C∗-algebrai...

Parallel to the first two authors’ earlier classification of separable unital oneparameter continuous fields of Kirchberg algebras with torsion free K-groups supported in one dimension, one-parameter separable unital continuous fields of AF-algebras are classified by their ordered K0-sheaves. We prove Effros-Handelman-Shen type theorems for separable unital one-parameter continuous fields of AF...

and Applied Analysis 3 Lemma 2.2 see 5, Theorem 3.3 , 6, Theorem 3.3 . Let A be a simple unital C∗-algebra. If TRR A 0, then RR A 0. If Tsr A 1 and has the (SP)-property, then tsr A 1. Definition 2.3 see 13, Definition 1.2 . Let A be an infinite dimensional finite simple separable unital C∗-algebra, and let α : G → Aut A be an action of a finite group G on A. We say that α has the tracial Rokhl...