The g-Drazin inverse involving power commutativity

Let A be a complex Banach algebra. An element a ? A has g-Drazin inverse if there exists b ? A such that b = bab, ab = ba, a-a2b ? A qnil. Let a, b ? Ad. If a3b = ba, b3a = ab, and a2adb = aadba, we prove that a + b ? Ad if and only if 1 + adb ? Ad. We present explicit formula for (a + b)d under certain perturbations. These extend the main results of Wang, Zhou and Chen (Filomat, 30(2016), 1185-1193) and Liu, Xu and Yu (Applied Math. Comput., 216(2010), 3652-3661).

The Fixed Point Property of a Banach Algebra Generated by an Element with Infinite Spectrum

A Banach spaceXis said to have the fixed point property if for each nonexpansive mappingT:E→Eon a bounded closed convex subsetEofXhas a fixed point. LetXbe an infinite dimensional unital Abelian complex Banach algebra satisfying the following: (i) condition (A) in Fupinwong and Dhompongsa, 2010, (ii) ifx,y∈Xis such thatτx≤τy,for eachτ∈Ω(X),thenx≤y,and (iii)inf⁡{r(x):x∈X,x=1}>0.We prove that there exists an elementx0inXsuch that〈x0〉R=∑i=1kαix0i:k∈N,αi∈R¯does not have the fixed point property. Moreover, as a consequence of the proof, we have that, for each elementx0inXwith infinite spectrum andσ(x0)⊂R,the Banach algebra〈x0〉=∑i=1kαix0i:k∈N,αi∈C¯generated byx0does not have the fixed point property.

A note on (-1, 1) ring satisfying a.c.c

Suppose that a semiprime (-1, 1) ring \(R\) is associative, satisfies the ascending chain condition for the right annihilators of the form \(r(w)\), where $w$ belongs to the nucleus \(N(R)\) and \(R\) contains no infinite direct sums of nonzero right ideals. Then the right quotient ring of $R$ relative to the subset \(W = \lbrace w \in N(R) / w \) is regular in \(R\rbrace\) exist and it is semisimple and artinian. Also if \(A\) be a nonassociative complex Banach algebra which satisfies ascending chain condition on left ideals and assume that the center \(Z(A)\) of \(A\) consists of regular elements then \(Z(A)\cong \mathbb{C}\). As a result if \(A\) be a (-1, 1) noetherian complex Banach algebra then \(A\) is finite-dimensional.

Ring homomorphisms on real Banach algebras

LetBbe a strictly real commutative real Banach algebra with the carrier spaceΦB. IfAis a commutative real Banach algebra, then we give a representation of a ring homomorphismρ:A→B, which needs not be linear nor continuous. IfAis a commutative complex Banach algebra, thenρ(A)is contained in the radical ofB.

Holomorphic Framings for Projections in A Banach Algebra

Given a complex Banach algebra, we consider the Stiefel bundle relative to the similarity class of a fixed projection. In the holomorphic category the Stiefel bundle is a holomorphic locally trivial principal bundle over a certain Grassmann manifold. Our main application concerns the holomorphic parametrization of framings for projections. In the spatial case this amounts to a holomorphic parametrization of framings for a corresponding complex Banach space.

Essentially defined derivations on semisimple Banach algebras

We prove that every partially defined derivation on a semisimple complex Banach algebra whose domain is a (non necessarily closed) essential ideal is closable. In particular, we show that every derivation defined on any nonzero ideal of a prime C*-algebra is continuous.

Strictly real Banach algebras

A complex Banach algebra is a complexification of a real Banach algebra if and only if it carries a conjugation operator. We prove a uniqueness theorem concerning strictly real selfconjugate subalgebras of a given complex algebra. An example is given of a complex Banach algebra carrying two distinct but commuting conjugations, whose selfconjugate subalgebras are both strictly real. The class of strictly real Banach algebras is shown to be a variety, and the manner of their generation by suitable elements is proved. A corollary describes some strictly real subalgebras in Hermitian Banach star algebras, including C* algebras.