The Positive Fixed Points of Banach Lattices

Let Z be a Banach lattice endowed with positive cone C and an order-continuous norm j.j . Let G be a left-amenable semigroup of positive linear endomorphisms of Z . Then the positive fixed points Co of Z under G form a lattice cone, and their linear span Z0 is a Banach lattice under an order-continuous norm ||.||0 which agrees with |.| on Co. A counterexample shows that under the given conditions Z0 need not contain all the fixed points of Z under G , and need not be a sublattice of (Z, C). The paper concludes with a discussion of some related results. Let G be a semigroup. We denote by m(C7) the Banach space of all bounded linear functions from G into the real numbers R, under the supremum norm. We denote by m'iG) the Banach dual of w(C7). With each T g G we associate an endomorphism Tm of /n(C7) defined by iTmb)iU) = biTU) for U G G and b G m(G) where TU denotes the composite of T and U under the semigroup operation. An element p G m*iG) is called a mean for G iff inf biT) < pib) < sup biT) for all b G G T€G T€G and left-invariant for G iff T'mp=p forallTGG, where T'm denotes the adjoint of Tm . Following M. Day [1, p. 108] we call the semigroup G left-amenable iff there exists a left-invariant mean for G. In particular, any Abelian semigroup is left-amenable [1, Theorem 4, p. 108]. A Banach lattice is said to have order-continuous norm iff every decreasing sequence of positive elements is norm convergent [3, 5.12, p. 92; 5.10(d), p. 89]. Theorem. Let (Z,C, ||.||) be a Banach lattice with order-continuous norm. Let G be a left-amenable semigroup of positive linear operators from Z into Z. Define C0 = {x G C : Tx = x for all T gG} , Z0 = C0 CQ. Received by the editors November 3, 1988; and in revised form March 21, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 46B30, 52A43. ©1989 American Mathematical Society 0002-9939/89 $1.00+ $.25 per page