Approaching central projections in AF-algebras

TL;DR

This paper characterizes central projections in unital AF-algebras with lattice-ordered Murray-von Neumann projections, introducing a transformation that moves projections towards the center and analyzing their properties via K-theory and logical methods.

Contribution

It provides a new characterization of central projections in AF-algebras using a minimality condition and introduces a centripetal transformation with monotonic properties.

Findings

Central projections correspond to minimal elements in a specific order.

In liminary AF-algebras, extremal states of K_0 are discrete and K_0 has general comparability.

A transformation moves projections towards the center with a monotonic number of steps.

Abstract

Let $A$ be a unital AF-algebra whose Murray-von Neumann order of projections is a lattice. For any two equivalence classes $[p]$ and $[q]$ of projections we write $[p]\sqsubseteq [q]$ iff for every primitive ideal $\mathfrak p$ of $A$ either $p/\mathfrak p\preceq q/\mathfrak p\preceq (1- q)/\mathfrak p$ or $p/\mathfrak p\succeq q/\mathfrak p \succeq (1-q)/\mathfrak p.$ We prove that $p$ is central iff $[p]$ is $\sqsubseteq$-minimal iff $[p]$ is a characteristic element in $K_0(A)$. If, in addition, $A$ is liminary, then each extremal state of $K_0(A)$ is discrete, $K_0(A)$ has general comparability, and $A$ comes equipped with a centripetal transformation $[p]\mapsto [p]^\Game$ that moves $p$ towards the center. The number $n(p) $ of $\Game$-steps needed by $[p]$ to reach the center has the monotonicity property $[p]\sqsubseteq [q]\Rightarrow n(p)\leq n(q).$ Our proofs combine the $K_0$-theoretic version of Elliott's classification, the categorical equivalence $\Gamma$ between MV-algebras and unital $\ell$-groups, and \L o\'s ultraproduct theorem for first-order logic.