Primitive Limit Algebras and C*-Envelopes,

Abstract

In this paper, we study irreducible representations of regular limit subalgebras of AF-algebras. The main result is twofold: every closed prime ideal of a limit of direct sums of nest algebras (NSAF) is primitive, and every prime regular limit algebra is primitive. A key step is that the quotient of an NSAF algebra by any closed ideal has an AF C * -envelope, and this algebra is exhibited as a quotient of a concretely represented AF-algebra. When the ideal is prime, the C * -envelope is primitive. The GNS construction is used to produce algebraically irreducible (in fact n -transitive for all n ⩾1) representations for quotients of NSAF algebras by closed prime ideals. Thus the closed prime ideals of NSAF algebras coincide with the primitive ideals. Moreover, these representations extend to *-representations of the C * -envelope of the quotient, so that a fortiori these algebras are also operator primitive. The same holds true for arbitrary limit algebras and the {0} ideal.