Revisiting the Transfinite Christensen-Pedersen Argument

Abstract

Christensen and Pedersen proved that every properly infinite ${\mathrm{AW}}^{\ast }$-algebra is monotone sequentially complete, and Saitô and Wright developed a transfinite form of their dilation argument. We revisit the transfinite construction using normality of ${\mathrm{AW}}^{\ast }$-algebras. Normality simplifies the limit stages by turning suprema into compressions of joins, so the construction only needs a supply of fresh orthogonal projections large enough to contain the supports of the summands at successor stages. We use this simplified proof to show that a $\ast$-homomorphism between ${\mathrm{AW}}^{\ast }$-algebras that preserves only the joins needed to encode such a sum preserves the sum itself. We also use it to deduce order-continuity facts about $\kappa$-join-preserving $\ast$-homomorphisms. We also show that a finite ${\mathrm{AW}}^{\ast }$-algebra has suprema for all bounded positive families whose supports have bounded total center-valued dimension.

arXiv comment18 pages, rewritten to acknowledge precedents and emphasize novel contributions