International Journal of Mathematics vol:12 issue:5 pages:555-577
Motivated by deformation quantization, we introduced in an earlier work the notion of formal Morita equivalence in the category of algebras over a ring C which is the quadratic extension by i of an ordered ring R. The goal of the present paper is twofold. First, we clarify the relationship between formal Morita equivalence, Ara's notion of Morita I-equivalence of rings with involution, and strong Morita equivalence of C-algebras. Second, in the general setting of algebras over C, we define "closed" ideals as the ones occurring as kernels of 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-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 algebras over C with equivalent representation theory but which are not formally Morita equivalent. We finally compute the closed *-ideals of some algebras arising in differential geometry.