A weak∗‐topological dichotomy with applications in operator theory. (5th May 2014)

• Record Type: Journal Article
• Authors: Kania, Tomasz; Koszmider, Piotr; Laustsen, Niels Jakob
• Is Part Of: Transactions of the London Mathematical Society. Volume 1:Number 1(2014)
• Page Start: 1
• Page End: 28
• Publication Date: 2014-05-05
• Subjects: Mathematics -- Periodicals; 510.5
• DOI: 10.1112/tlms/tlu001
• Languages: English
• ISSNs: 2052-4986

Abstract

Abstract : Denote by[ 0, ω 1 ) the locally compact Hausdorff space consisting of all countable ordinals, equipped with the order topology, and letC 0 [ 0, ω 1 ) be the Banach space of scalar‐valued, continuous functions which are defined on[ 0, ω 1 ) and vanish eventually. We show that a weak * ‐compact subset of the dual space ofC 0 [ 0, ω 1 ) is either uniformly Eberlein compact, or it contains a homeomorphic copy of a particular form of the ordinal interval[ 0, ω 1 ] . This dichotomy yields a unifying approach to most of the existing studies of the Banach spaceC 0 [ 0, ω 1 ) and the Banach algebraB ( C 0 [ 0, ω 1 ) ) of bounded, linear operators acting on it, and it leads to several new results, as well as to stronger versions of known ones. Specifically, we deduce that a Banach space which is a quotient ofC 0 [ 0, ω 1 ) can either be embedded in a Hilbert‐generated Banach space, or it is isomorphic to the direct sum ofC 0 [ 0, ω 1 ) and a subspace of a Hilbert‐generated Banach space; and we obtain several equivalent conditions describing the Loy–Willis idealM, which is the unique maximal ideal ofB ( C 0 [ 0, ω 1 ) ), including the following: an operator belongs toM if and only if it factors through the Banach space( ⨁ α < ω 1 C [ 0, α ] ) c 0 . Among the consequences of these characterizations ofM is thatM has a bounded left approximate identity; this resolves a problem left open by Loy and Willis.