Projections in Weakly Compactly Generated Banach Spaces and Chang's Conjecture

Abstract

Abstract. Classical results on weakly compactly generated (WCG) Banach spaces imply the existence of projectional resolutions of identity (PRI) and the existence of many projections on separable subspaces (SCP). We address the questions if these can be the only projections in a nonseparable WCG space, in the sense that there is a PRI (Pα : ω ≤ α ≤ λ) such that any projection is the sum of an operator in the closure of the linear span of countably many Pα's (in the strong operator topology) and a separable range operator. Wark's modification of Shelah's and Steprāns' construction provides an unconditional example for λ = ω1. We note that it is impossible for λ > ω2. . The main result of the paper is that for λ = ω2, the second uncountable cardinal, the question is logically undecidable and depends on additional axioms deciding the combinatorics on ω2; for example Chang's conjecture implies that there are other projections than the projections mentioned above. The full strength results concern all linear operators not just the projections.

Decompositions of preduals of JBW and JBW⁎ algebras

“…This set-theoretical method can be used in various branches of mathematics. The use in topology was illustrated by A. Dow in [9], in functional analysis it was used by P. Koszmider in [17]. This method was later used by W. Kubiś in [18] to construct projectional skeletons in certain Banach spaces.…”

On projectional skeletons in Vašák spaces

“…This set-theoretical method is useful in various branches of mathematics. In particular, Dow in [7] illustrated its use in topology, Koszmider in [12] used it in functional analysis. Later, inspired by [12], Kubiś in [13] gave a method of constructing retractional (resp.…”

Claude Ambrose Rogers. 1 November 1920 — 5 December 2005

“…(The theorem itself was later central to the 'continuum of agents' models in mathematical economics; see Hildenbrand (1974).) The book is a testament to the pioneering work of Rogers and his several collaborators in the field of functional analysis, building on all themes present in the first of the two books, except on the connections here with mathematical logic (for which see, for example, Koszmider (2005)). A central concept of the book is fragmentability, and particularly its generalization: σ-fragmentability, a term that Rogers (and Jayne) coined at a meeting of minds with Namioka and Phelps at the 23rd Semester (On Banach Spaces) of the Stefan Banach International Mathematical Centre in Warsaw (in the spring of 1984).…”