Positive Contractions on L 1 -Spaces

Abstract

Let T be a positive linear operator on L1 of a probability space such that II T1j i: 1. In this note we consider the following question: Under what condition is T multiplicative on Let (X, F, m) be a probability space and L1=L1(X, F, m) the Banach space of equivalence classes of integrable complex-valued functions on X. Let T be a positive linear operator on L1 such that II TIIl< 1, and define G(T)={f E L1; If I = I Tf I = 1). The main purpose of this note is to prove the following: THEOREM. If G(T) is total in L1, i.e., the linear manifold generated by G(T) is dense in L1 in the norm topology, then for any bounded functions f and g we have T(fg)= (Tf)(Tg). As a corollary of the Theorem we have the following result; essentially the same idea has been used by Halmos [2, p. 45] to find a necessary and sufficient condition for a unitary operator on L2=L2(X, Y, m) to be induced by an automorphism of the measure algebra (F(m), m) associated with (X, F, m). COROLLARY. A necessary and sufficient condition that T be induced by a homomorphism of the measure algebra (f(m), m) into itself is that G(T) be total in L1. For the proof of the theorem we need some lemmas. LEMMA 1. Let K be a compact Hausdorff space and C(K) the Banach algebra of all complex-valued continuous functions f on K wvith norm lIf IIU=sup{If(s)I; s e K}. Let U be a positive linear operator on C(K) such that U1=1. Let f,geC(K), IfI=lgi=1, and IUfI=IUgi=1. Then U(fg)=(Uf)(Ug)Received by the editors February 28, 1973 and, in revised form, May 10, 1973. AMS (MOS) subject classifications (1970). Primary 47B55, 47A35; Secondary 28A65, 46J10.