Extending addition in Elliott’s local semigroup. (English) Zbl 0799.46077

Mundici, Daniele; Panti, Giovanni

J. Funct. Anal. 117, No. 2, 461-472 (1993).

Summary: We study the unique extendability of Elliott’s partial addition of Murray-von Neumann equivalence classes of projections in AF \(C^*\)- algebras. We prove that there is at most one commutative associative monotone extension satisfying the natural residuation condition that for each projection \(p\) the class of \(1-p\) is the smallest one whose sum with the class of \(p\) equals 1. We prove that for every AF \(C^*\)-algebra \(A\) this associative commutative monotone residual extension exists if, and only if, the Murray-von Neumann order on equivalence classes of projections in \(A\) is a lattice order. By Elliott’s classification theorem, the resulting monoid uniquely characterizes \(A\). We give a simple equational characterization of the monoids arising as classifiers.

MSC:

46L55	Noncommutative dynamical systems
46L05	General theory of \(C^*\)-algebras

Keywords:

Elliott’s partial addition of Murray-von Neumann equivalence classes of projections in AF \(C^*\)-algebras; unique extendability; commutative associative monotone extension; Murray-von Neumann order on equivalence classes of projections; lattice order; equational characterization of the monoids arising as classifiers