Extreme operators into C(K)

Robert McCallum Blumenthal, Joram Lindenstrauss and Robert Ralph Phelps

Vol. 15 (1965), No. 3, 747–756

DOI: 10.2140/pjm.1965.15.747

Abstract

If X and Y are real Banach spaces let S(X,Y ) denote the convex set of all linear operators from X into Y having norm less than or equal to 1. The main theorem is this: If K₁ and K₂ are compact Hausdorff spaces with K₁ metrizable and if T is an extreme point of S(C(K₁),C(K₂)), then there are continuous functions ϕ : K₂ → K₁ and λ in C(K₂) with |λ| = 1 such that (Tf)(k) = λ(k)f(ϕ(k)) for all k in K₂ and f in C(K₁). There are several additional theorems which discuss the possibility of replacing C(K₁) in this theorem by an arbitrary Banach space.

Mathematical Subject Classification

Primary: 47.25

Milestones

Received: 11 February 1965

Revised: 23 April 1965

Published: 1 September 1965