Meet irreducible ideals in direct limit algebras

Abstract

We study the meet irreducible ideals (ideals I so that I = J ∩ K implies I = J or I = K) in certain direct limit algebras. The direct limit algebras will generally be strongly maximal triangular subalgebras of AF C * -algebras, or briefly, strongly maximal TAF algebras. Of course, all ideals are closed and two-sided.These ideals have a description in terms of the coordinates, or spectrum, that is a natural extension of one description of meet irreducible ideals in the upper triangular matrices. Additional information is available if the limit algebra is an analytic subalgebra of its C * -envelope or if the analytic algebra is trivially analytic with an injective 0-cocycle. In the latter case, we obtain a complete description of the meet irreducible ideals, modeled on the description in the algebra of upper triangular matrices. This applies, in particular, to all full nest algebras.One reason for interest in the meet irreducible ideals of a strongly maximal TAF algebra is that each meet irreducible ideal is the kernel of a nest representation of the algebra (Theorem 2.4). A nest representation of an operator algebra A is a norm continuous representation of A acting on a Hilbert space with the property that the lattice of closed invariant subspaces for the representation is totally ordered. These representations were introduced in [L1] as analogues for a general operator algebra of the irreducible representations of a C * -algebra. The meet irreducible ideals seem analogous to the primitive ideals in a C * -algebra. Indeed, in a C * -algebra, the meet irreducible ideals are precisely the primitive ideals [L3, Theorem 2.1]. This analogy can be extended by noting that the meet irreducible ideals form a topological space under the hull-kernel topology and every ideal is the intersection of the meet * Partially supported by an NSF grant. † Partially supported by an NSERC grant.

Compact operators and nest representations of limit algebras

“…The completely meet irreducible ideals have played a fundamental role in the classification theory of strongly maximal TAF algebras [6,7,8]. Motivated by this, and their study of nest representations, the five authors of [5] attempted a characterization of the completely meet irreducible ideals of a strongly maximal TAF algebra in terms of its nest representations (their Corollary 5.7). In [5,Example 5.8] it was realized that the order type of the nest alone is not sufficient for such a characterization, and therefore the problem was left open.…”

“…Another problem that was left open in [5] was the characterization of completely meet irreducible ideals using representation theory. An ideal J of a Banach algebra A is called completely meet irreducible if, for any collection of ideals J a , a ∈ A, containing J , the relation J = a∈A J a implies J = J a , for some a ∈ A.…”

“…In [5] it was shown that for a strongly maximal TAF algebra every meet irreducible ideal is n-primitive. The converse question, whether every n-primitive ideal is meet irreducible, was left open in [5] and was answered affirmatively in [4].…”

“…Recently there has been an interest in the representation theory of non-selfadjoint algebras and in particular triangular subalgebras of AF C * -algebras [3,4,5,12,22]. In the case of non-selfadjoint operator algebras, the irreducible representations are not always sufficient to capture the structure of the algebra, and therefore appropriate substitutes have been introduced.…”

“…Motivated by this, and their study of nest representations, the five authors of [5] attempted a characterization of the completely meet irreducible ideals of a strongly maximal TAF algebra in terms of its nest representations (their Corollary 5.7). In [5,Example 5.8] it was realized that the order type of the nest alone is not sufficient for such a characterization, and therefore the problem was left open. The solution of this problem is contained in Theorem 2.3 and Corollary 2.5.…”