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\(*\)-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.)

Citations:

Zbl 1039.46052
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