The purpose of this article is to develop the notions of amenabilityfor vector valued group algebras. We prove that L1(G, A) is approximatelyweakly amenable where A is a unital separable Banach algebra. We givenecessary and sufficient conditions for the existence of a left invariant meanon L∞(G, A∗), LUC(G, A∗), WAP(G, A∗) and C0(G, A∗).

Let C be a unital AH-algebra and let A be a unital separable simple C∗-algebra with tracial rank zero. Suppose that φ1, φ2 : C → A are two unital monomorphisms. We show that there is a continuous path of unitaries {ut : t ∈ [0,∞)} of A such that lim t→∞ u∗tφ1(a)ut = φ2(a) for all a ∈ C if and only if [φ1] = [φ2] in KK(C,A), τ ◦ φ1 = τ ◦ φ2 for all τ ∈ T (A) and the rotation map η̃φ1,φ2 associate...

Let C and A be two unital separable amenable simple C-algebras with tracial rank no more than one. Suppose that C satisfies the Universal Coefficient Theorem and suppose that φ1, φ2 : C → A are two unital monomorphisms. We show that there is a continuous path of unitaries {ut : t ∈ [0,∞)} of A such that lim t→∞ u∗tφ1(c)ut = φ2(c) for all c ∈ C if and only if [φ1] = [φ2] in KK(C,A), φ ‡ 1 = φ 2 ...

Kadison’s theorem of 1951 describes the unital surjective isometries between unital C*-algebras as the Jordan *-isomorphisms. We propose a nonselfadjoint version of his theorem and discuss the cases in which this is known to be true.

Let M be a Banach C*-module over a C*-algebra A carrying two A-valued inner products 〈., .〉1, 〈., .〉2 which induce equivalent to the given one norms on M. Then the appropriate unital C*-algebras of adjointable bounded A-linear operators on the Hilbert A-modules {M, 〈., .〉1} and {M, 〈., .〉2} are shown to be ∗-isomorphic if and only if there exists a bounded A-linear isomorphism S of these two Hi...

In this brief note we would like to give the construction of a free commutative unital associative Nijenhuis algebra on a commutative unital associative algebra based on an augmented modified quasi-shuffle product. —————————————

Let $\mathcal{A}= \{A_t \}_{t \in G}$ and $\mathcal{B}= \{B_t \}_{t\in be $C^*$-algebraic bundles over a finite group $G$. $C=\oplus_{t G}A_t$ $D=\oplus_{t\in G}B_t$. Also, let $A=A_e$ $B=B_e$, where $e$ is the unit element in We suppose that $C$ $D$ are unital $A$ $B$ have elements $D$, respectively. In this paper, we shall show if there an equivalence $\mathcal{A}-\mathcal{B}$-bundle $G$ with...

It is shown that the matrix normed structure of a non-unital operator algebra determines that of its unitization. This makes the study of certain unital operator algebras much easier and provides several interesting