In this article we study two different generalizations of von Neumann regularity, namely strong topological regularity and weak regularity, in the Banach algebra context. We show that both are hereditary properties and under certain assumptions, weak regularity implies strong topological regularity. Then we consider strong topological regularity of certain concrete algebras. Moreover we obtain ...

We prove that if a unital Banach algebra A is the dual of a Banach space A♯, then the set of weak* continuous states is weak* dense in the set of all states on A. Further, weak* continuous states linearly span A♯.

We show that the solid hull of every weakly precompact set a Banach lattice E is if and only order interval in precompact, or equivalently, disjoint compact precompact. Some results on domination property for positive operators are obtained. Among other things, we that, pair lattices F with $$\sigma $$ -Dedekind complete, operator from to dominated by either norm $$E^{\prime }$$ continuous else

Using the game approach to fragmentability, we give new and simpler proofs of the following known results: (a) If the Banach space admits an equivalent Kadec norm, then its weak topology is fragmented by a metric which is stronger than the norm topology. (b) If the Banach space admits an equivalent rotund norm, then its weak topology is fragmented by a metric. (c) If the Banach space is weakly ...

In this paper, we deal with the problem for finding a common element of finite sets in a Banach space. We first prove that an operator given by a convex combination of sunny generalized nonexpansive retractions in a Banach space is asymptotically regular. Using this result, we obtain a weak convergence theorem which is connected with the problem of image recovery. Further, using another weak co...

In this paper, we first introduce a class of nonlinear mappings called generic generalized nonspreading which contains the class of generalized nonspreading mappings in a Banach space and then prove an attractive point theorem for such mappings in a Banach space. Furthermore, we prove a mean convergence theorem of Baillon’s type and a weak convergence theorem of Mann’s type for such nonlinear m...