## $$*$$-ideals and formal Morita equivalence of $$*$$-algebras.(English)Zbl 1111.46303

Summary: Motivated by deformation quantization, we introduced in [H. Bursztyn and S. Waldmann, J. Geom. Phys. 37, No. 4, 307–364 (2001; Zbl 1039.46052)] 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 $$*$$-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.

### MSC:

 46K05 General theory of topological algebras with involution 46L08 $$C^*$$-modules 46L99 Selfadjoint operator algebras ($$C^*$$-algebras, von Neumann ($$W^*$$-) algebras, etc.)

Zbl 1039.46052
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