Primitive Limit Algebras and C*-envelopes

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 a 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. The purpose of this paper is to construct algebraically irreducible (primitive) representations of quotients of regular subalgebras of AFalgebras. In a recent article [16], Hudson and the second author produced a variety of primitive TUHF algebras. Elaborating on a representation of Orr and Peters [22], it was shown that if the standard embedding appears infinitely many times in the presentation of a TUHF algebra A, then A is primitive. This led to a complete classification of the primitive ideals of Power’s lexicographic algebras, with applications to their epimorphic theory. The question of when a TUHF, or more generally a TAF algebra, is primitive was implicit in [16] and was raised explicitly in several conferences. Similar questions had also been raised and investigated in [21] in the broader context of subalgebras of groupoid C*-algebras algebras. A well-known necessary condition for an ideal to be primitive is closed and prime. However, the converse is generally false for Banach algebras. In this paper, we establish the converse for NSAF 1991 Mathematics Subject Classification. 47L80. August 21, 2000 version. First author partially supported by an NSERC grant. Second author’s research partially supported by a grant from ECU.