Let $mathcal{R}$ be a  commutative ring with identity, let $A$ and $B$ be two $mathcal{R}$-algebras and $varphi:Blongrightarrow A$ be an $mathcal{R}$-additive algebra homomorphism. We introduce a new algebra $Atimes_varphi B$, and give some basic properties of this algebra. Generalized $2$-cocycle derivations on $Atimes_varphi B$ are studied. Accordingly, $Atimes_varphi B$ is considered from th...

The origin of Korovkin Approximation theory is the classical theorem of P.P. Korovkin (1953),which says that for a sequence (Tn) of positive linear operators on C[a, b], in order to conclude the uniform convergence of Tnf to f for all f ∈ C[a, b], it suffices to check the uniform convergence only for the three functions f ∈ {1, x, x2}. Starting from this beautiful result many mathematicians hav...

We show that if T is an isometry (as metric spaces) between the invertible groups of unital Banach algebras, then T is extended to a surjective real-linear isometry up to translation between the two Banach algebras. Furthermore if the underling algebras are closed unital standard operator algebras, (T (eA)) −1 T is extended to a surjective real algebra isomorphism; if T is a surjective isometry...

Let A be a Banach algebra. We call a pair (G,A) a Gelfand theory for A if the following axioms are satisfied: (G 1) A is a C∗-algebra, and G : A → A is a homomorphism; (G 2) the assignment L 7→ G−1(L) is a bijection between the sets of maximal modular left ideals of A and A, respectively; (G 3) for each maximal modular left ideal L of A, the linear map GL : A/G−1(L) → A/L induced by G has dense...

The Lipschitz function algebras were first defined in the 1960s by some mathematicians, including Schubert. Initially, the Lipschitz real-value and complex-value functions are defined and quantitative properties of these algebras are investigated. Over time these algebras have been studied and generalized by many mathematicians such as Cao, Zhang, Xu, Weaver, and others. Let  be a non-emp...

1 Fundamentals of the Theory of Banach Algebras 2 1.1 Basic definitions and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 The spectrum and the resolvent set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Properties of the inverse function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Characters and the Gelfand Transform . . . . . ....

We give continuity properties of bitraces on (possibly non-commutative) Banach ∗-algebras based on the Closed Graph Theorem, leading to a simplified proof of the Theorem of Varopoulos in the commutative case.

We establish a necessary condition for a commutative Banach algebra A so that there exists a homomorphism θ from A into another Banach algebra such that the prime radical of the continuity ideal of θ is not a finite intersection of prime ideals in A. We prove that the prime radical of the continuity ideal of an epimorphism from A onto another Banach algebra (or of a derivation from A into a Ban...

We present here a quite unexpected result: Apart from already known commutative C∗-algebras generated by Toeplitz operators on the unit ball, there are many other Banach algebras generated by Toeplitz operators which are commutative on each weighted Bergman space. These last algebras are non conjugated via biholomorphisms of the unit ball, non of them is a C∗-algebra, and for n = 1 all of them ...