Order Completeness in Lipschitz Algebras

The algebraic properties of Lipschitz spaces have received much attention. This has led to a good understanding of such things as complex homomorphisms and ideals (but not subalgebras) when the underlying metric space is compact. Taking a cue from the recent observation that Lipschitz 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 complex homomorphisms and order-complete subalgebras and ideals, even when the underlying metric space is not compact. In particular, we show that order-complete subalgebras and quotients by order-complete ideals are themselves Lipschitz spaces.