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 algebras A(Q, &), 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.
1998 Academic Press
Representation theory has played a fundamental role in the study of algebras since the beginning of the subject. Of particular interest are the irreducible representations, i.e., those whose representation spaces have no nontrivial invariant manifolds. Naturally, then, ideals which are the kernels of the irreducible representations, the primitive ideals, are a key component of representation theory. Of fundamental importance is the intersection of the primitive ideals, i.e., the Jacobson radical, especially since this ideal admits several equivalent characterizations which do not involve representations [10, Theorem IX.2.3].
In the present paper, we study the representation theory of triangular UHF algebras. A norm-closed subalgebra A of a UHF C*-algebra B is said to be a triangular UHF algebra if A & A* is a canonical masa in B.
A large class of these algebras was previously shown by Donsig to be semisimple [2] (zero Jacobson radical); his proof used the spectral radius characterization of the radical, which does not require any knowledge of article no. FU983336