The Banach Principle for ideal convergence in the classical and noncommutative context

Abstract

Versions of the Banach Principle for different types of convergence 'with respect to an ideal' are established both in the commutative and noncommutative (von Neumann algebraic) context.

Achievement sets and sum ranges with ideal supports

“…Since the convergence is a basic notion in Analysis, most of them deal with ideal convergence of sequences [2], [17], [22]. The following list of topics and related papers is far from being complete and it gives only a flavor of these matters: ideal convergence of sequences of functions [1]; ideal convergence of series [8], [18]; ideal convergence in measure [16], [20]; ideal versions of combinatorial theorems [6]; ideal versions of the Riemann rearrangement theorem and the Levy-Steinitz theorem [7], [16]; ideal version of the Banach principle [11].…”

Modes of ideal continuity of (\ell)-group-valued measures

“…Ideal convergence was introduced in [42], though a primitive version of [41] is also included in the references of [42], and independently in [49] under the name of "cofilter convergence", and was recently developed and investigated in several papers in the context of normed and/or metric spaces. Among the related literature, we quote [4,5,8,9,22,30,31,35,38,41,50,53,54,55]. This concept has been studied also in topological spaces ( [28,29,43,44]) and ( )-groups ( [7,11,12,13,14,15,16,17,18,19,20,23,24]).…”