Introduction

This appendix contains sections on: ultrafilters and characters of l∞(ℕ); a discussion of maximal ideals in finite von Neumann algebras; the construction of the ultrapowers and ℕω; property Γ and relative commutants in ℕω.

In these notes ultrafilter is used for free ultrafilter in ℕ as these are the only ultrafilters discussed. See the article by Ge and Hadwin [78] for a detailed discussion of ultrafilters and ultraproducts directed at operator algebras, or the books [34, 90, 107, 204] for a discussion of filters and ultrafilters in set theory and general topology. It is convenient to think of ultrafilters as characters ω of l∞(ℕ) induced by points in βℕ\ℕ so this relationship is discussed briefly in the second section.

Section A.3 on maximal ideals in a finite von Neumann algebra contains a theorem due to Wright [211] that the quotient of a finite von Neumann algebra by a maximal two-sided ideal is a finite factor with trace arising from the original algebra and maximal ideal; Wright actually proved this for AW*-algebras. A theorem for AW*-algebras that yields this was rediscovered by Feldman [68], though he does not state this exact result or examine the norm closed ideals as Wright does. This result for von Neumann algebras appears in Sakai's Yale notes [165] with no reference, and there is an account by Takesaki [187, p. 357].