Let A, B be two regular commutative unital Banach algebras such that B is integral over A. In 2003, Dawson and Feinstein showed that the topological stable rank tsr(B) = 1 whenever tsr(A) = 1. In this note, we investigate whether we will have tsr(A) = tsr(B) in general. For instance, when A is a commutative unital C∗-algebra, we show that tsr(A) ≤ tsr(B), and the equality holds at least when th...

We consider Fréchet algebras which are subalgebras of the algebra F = C [[X]] of formal power series in one variable and of Fn = C [[X1, . . . , Xn]] of formal power series in n variables where n ∈ N. In each case, these algebras are taken with the topology of coordinatewise convergence. We begin with some basic definitions about Fréchet algebras, (F )-algebras, and other topological algebras, ...

In this paper, we introduce the new concept of biprojectivity of a Banach algebra modulo an ideal, as a generalization of this notion in the classical case. By using it , we obtain some necessary and sufficient conditions for contractibility of Banach algebras modulo an ideal. In particular we characterize the contractibility of quotient Banach algebras. Also we study the relationship between t...

A theorem is presented on existence and uniqueness up to the topological *-isomorphism of universal locally C*-algebra for an arbitrary locally JB-algebra. The abstract Banach associative symmetrical *-algebras over C, so called C*algebras, were introduces first by Gelfand and Naimark in []. In the present time the theory of C*-algebras become a vast portion of Functional Analysis having connec...

We investigate two systematic constructions of inverse-closed subalgebras of a given Banach algebra or operator algebra A, both of which are inspired by classical principles of approximation theory. The first construction requires a closed derivation or a commutative automorphism group on A and yields a family of smooth inverse-closed subalgebras of A that resemble the usual Hölder-Zygmund spac...

We prove that every surjective isometry from the unit sphere of space K(H), all compact operators on an arbitrary complex Hilbert H, onto real Banach Y can be extended to a linear K(H) Y. This is probably first example infinite dimensional non-commutative C*-algebra containing no unitaries and satisfying Mazur-Ulam property. also C*-algebras weakly JB*-triples satisfy