Primitive Triangular UHF Algebras

Abstract

We prove that a large class of triangular UHF algebras are primitive. We use two avenues to obtain our results: a direct approach in which we explicitly construct a faithful, algebraically irreducible representation of the algebra on a separable Hilbert space, as well as an indirect, algebraic approach which utilizes the prime ideal structure of the algebra. Using these results, we completely characterize the primitive ideal spaces of all lexicographic algebras: An ideal there is primitive if and only if it is closed prime. Specializing on the algebrasA(Q,ν), we obtain a complete classification of their algebraic isomorphisms and epimorphisms through the use of a new invariant involving the primitive ideal space. Finally, we characterize the primitive ideal spaces of Z-analytic and order-preserving algebras, and obtain information about their epimorphisms.