Ideals and Formal Morita Equivalence of Algebras

Abstract

Motivated by deformation quantization, we introduced in an earlier work the notion of formal Morita equivalence in the category of ${}^{\ast }$-algebras over a ring $\ring C$ which is the quadratic extension by $\im$ of an ordered ring $\ring R$. The goal of the present paper is twofold. First, we clarify the relationship between formal Morita equivalence, Ara's notion of Morita ${}^{\ast }$-equivalence of rings with involution, and strong Morita equivalence of $C^{\ast }$-algebras. Second, in the general setting of ${}^{\ast }$-algebras over $\ring C$, we define `closed' ${}^{\ast }$-ideals as the ones occuring as kernels of ${}^{\ast }$-representations of these algebras on pre-Hilbert spaces. These ideals form a lattice which we show is invariant under formal Morita equivalence. This result, when applied to Pedersen ideals of $C^{\ast }$-algebras, recovers the so-called Rieffel correspondence theorem. The triviality of the minimal element in the lattice of closed ideals, called the `minimal ideal', is also a formal Morita invariant and this fact can be used to describe a large class of examples of ${}^{\ast }$-algebras over $\ring C$ with equivalent representation theory but which are not formally Morita equivalent. We finally compute the closed ${}^{\ast }$-ideals of some ${}^{\ast }$-algebras arising in differential geometry.

LaTeX2e, 19 pages