ORDER COMPLETENESS IN LIPSCHITZ ALGEBRAS

The algebraic properties of Lipschitz spaces have received much attent ion. This has led to a good understanding of such things as complex ho momorphisms and ideals (but not subalgebras) when the underlying metri c space is compact. Taking a cue from the recent observation that Lips chitz spaces are order-complete (N. Weaver, Pacific J. Math. 164 (1994 ), 179-193), we here investigate these topics under the hypothesis of order continuity or order closure in place of norm continuity or norm closure. We obtain simple characterizations of order-continuous comple x homomorphisms and order-complete subalgebras and ideals, even when t he underlying metric space is not compact. In particular, we show that order-complete subalgebras and quotients by order-complete ideals are themselves Lipschitz spaces. (C) 1995 Academic Press, Inc.