Projections in weakly compactly generated Banach spaces and Chang’s conjecture

Abstract. Classical results on weakly compactly generated (WCG) Banach spaces imply the existence of projectional resolutions of iden-tity (PRI) and the existence of many projections on separable sub-spaces (SCP). We address the questions if these can be the only pro-jections 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 modifica-tion of Shelah’s and Steprāns ’ construction provides an unconditional example for λ = ω1. We note that it is impossible for λ> ω2. 2000 Mathematics Subject Classification. Primary 46B26, 03E35. Key words and phrases. Weakly compactly generated Banach spaces, Chang’s conjec