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In this paper we construct a representation isometric lattice isomorphism for the peripherical spectrum of a positive operator on a Banach lattice. By a representation lattice homomorphism, we mean that the peripherical spectrum of the operator is identified with the spectrum of the induced isometric lattice homomorphism. A simple proof of a "zero-two" law follows easily from our representation technique. 1. THE REPRESENTATION LATTICE HOMOMORPHISM We develop our technique in the context of Banach lattices with p-additive norm. Following Zaanen [Zaa], 1.1. Definition. Let 1 ? IIxIIP + IYIP Vx, Y ?0. As a straightforward consequence of the p-superadditivity of the norm, we get the following basic property of ALp-spaces: (1) If T is a contraction on E that verifies 0 < x < Tx for some x E E , then Tx = x. Now let E be an ALp-space (1 < p < 00), and let 9Y denote a free ultrafilter on N. The ST-product Esis actually an ALp-space (see [S1, Chapter V, ? 1]). We denote by P the following isometric lattice isomorphism on Es: P((xwn, X2 v i...s)e+c c -(E)) = (x2ch X3,r ...i)s+gci (E)b whose inverse isometric lattice homomorphism is given by Q((x . x , .. _ l cg-E) = (O , x , x2 , . )+cq( Received by the editors February 13, 1992. 1991 Mathematics Subject Classification. Primary 47B55. The author was supported by an F.P.I. grant from the Spanish Ministerio de Educaci6n y Ciencia. ? 1993 American Mathematical Society 0002-9939/93 $1.00 + $.25 per page

A representation lattice isomorphism for the peripherical spectrum