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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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