Weakly compact sets in H¹.

Freddy Delbaen

Vol. 63 (1976), No. 2, 367–369

DOI: 10.2140/pjm.1976.63.367

Abstract

Suppose that A is a uniform algebra on a compact set X and that ϕ : A → C is a nonzero multiplicative linear functional on A. Let M_ϕ be the set of positive representing measures for ϕ. If M_ϕ is finite dimensional, let m be a core measure of M_ϕ. The space H¹ is the closure of A in L¹(m). The space H^∞ is the weak* (i.e. σ(L^∞,L¹)) closure of A in L^∞(m). The weakly compact sets R in H¹ are then those sets such that for all 𝜖 > 0 there is a bounded set in H^∞ which approximates R up to 𝜖.

Mathematical Subject Classification 2000

Primary: 46J15

Milestones

Received: 4 August 1975

Published: 1 April 1976