Vol. 48, No. 1, 1973

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Multiplicative and extreme positive operators

M. Solveig Espelie

Vol. 48 (1973), No. 1, 57–66

Let A and B denote complex Banach -algebras and L(A,B) the space of continuous linear operators from A into B. Let P L(A,B) be the convex set of positive linear operators of norm 1. If A has an identity, and if B is semi-simple and symmetric, the multiplicative operators of P are shown to be extreme points of P. If, on the other hand, it is assumed that, T= Tefor τ P, then any extreme point T of P satisfies TeTab = TaTb for all a,b A. With A as above and B a B-algebra, the extreme points of P are multiplicative. Thus we characterize the extreme points of P L(A,C(X)) as the multiplicative operators. The results are extended to include the case when A has an approximate identity.

Mathematical Subject Classification 2000
Primary: 47B55
Secondary: 46K05
Received: 31 August 1971
Revised: 18 April 1973
Published: 1 September 1973
M. Solveig Espelie