A Transfinite Christensen--Pedersen Argument

Christensen and Pedersen proved that every properly infinite ${\mathrm{A}\mathrm{W}}^{\ast }$-algebra is monotone sequentially complete. We isolate the part of their argument that extends to an arbitrary cardinal. The property of properly infinite ${\mathrm{A}\mathrm{W}}^{\ast }$-algebras used in their proof is the existence of an orthogonal sequence of projections, each equivalent to $1$, with sum $1$. We generalize this by defining a notion of $\kappa$-homogeneity, with the case $\kappa ={\mathrm{\aleph }}_0$ recovering the aforementioned property. $\kappa$-homogeneity supplies the fresh orthogonal space needed at each successor stage of a transfinite projection dilation, and normality of ${\mathrm{A}\mathrm{W}}^{\ast }$-algebras supplies, at every limit stage and again at the end of the construction, the passage from a join of projections to a supremum in the self-adjoint order. This second point replaces both the addability theorem and the perturbation argument used in the countable case. We prove that every $\kappa$-homogeneous ${\mathrm{A}\mathrm{W}}^{\ast }$-algebra is $\kappa$-monotone complete, and that $\kappa$-monotone completeness together with $\kappa$ ordinary states separating positive elements from zero implies monotone completeness. As an application, an ${\mathrm{A}\mathrm{W}}^{\ast }$-factor with a corner admitting a faithful state is monotone complete.