Vol. 15, No. 3, 1965

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Extreme operators into C(K)

Robert McCallum Blumenthal, Joram Lindenstrauss and Robert Ralph Phelps

Vol. 15 (1965), No. 3, 747–756

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
Received: 11 February 1965
Revised: 23 April 1965
Published: 1 September 1965
Robert McCallum Blumenthal
Joram Lindenstrauss
Einstein Institute of Mathematics
Hebrew University of Jerusalem
91905 Jerusalem
Robert Ralph Phelps