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A remark on the deformation of GNS representations of *-algebras. (English) Zbl 1016.46034

The present article is one in a series of papers of the author (together with various coauthors) on \(*\)-representations of deformed \(*\)-algebras. Let \(R\) be an ordered ring (the main examples are \(\mathbb{R}\) and \(\mathbb{R}[[\lambda]]\)) and \(C := R(i)\) the corresponding extension by an element \(i\) with \(i^2 = -1\). Then there are natural notions of pre-Hilbert spaces over \(C\), \(*\)-algebras, \(*\)-representations and positive functionals on \(*\)-algebras. Let \({\mathcal A}\) be a \(*\)-algebra over \(C\) and \({\mathcal A}[[\lambda]]\) a formal deformation of \({\mathcal A}\), which is a \(*\)-algebra over \(C[[\lambda]]\). If \({\mathcal H}\) is a \(C[[\lambda]]\) pre-Hilbert space, one can associate in a natural way a \(C\)-pre-Hilbert space \({\mathcal CH}\), called the classical limit of \({\mathcal H}\). Applying this process to \(*\)-representations of \({\mathcal A}[[\lambda]]\), we get \(*\)-representations of \({\mathcal A}\).
The main result of the present paper deals with this process for GNS representations: If \(\omega\) is a positive functional on \({\mathcal A}[[\lambda]]\) which is a deformation of a positive functional \(\omega_0\) on \({\mathcal A}\), then the classical limit of the GNS representation associated to \(\omega\) is the GNS representation of \({\mathcal A}\) associated to \(\omega_0\). The paper concludes with a discussion of some basic physical examples.

MSC:

46K05 General theory of topological algebras with involution
81S10 Geometry and quantization, symplectic methods
46L05 General theory of \(C^*\)-algebras
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