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Product states of the gauge invariant and rotationally invariant CAR algebras. (English) Zbl 0594.46053

Let \({\mathfrak A}\) be a UHF \(C^*\)-algebra of type \(2^{\infty}\). Let \({\mathfrak A}^ T\) and \({\mathfrak A}^ G\) denote the gauge invariant and rotationally invariant subalgebras of \({\mathfrak A}\). Suppose a product state \(\omega\) on \({\mathfrak A}\) induces a *-representation \(\pi_ T\). Then are given sufficient conditions to have
(1) \(\pi\) (\({\mathfrak A}^ T)''=\pi ({\mathfrak A})''\) and \(\omega^ T=\omega ({\mathfrak A}^ T)\) is a factor state of the same type as \(\omega\) ; and
(2) \(\pi\) (\({\mathfrak A}^ G)''=\pi ({\mathfrak A})''\) with \(\omega^ G=\omega | {\mathfrak A}^ G\), a factor state of the same type as \(\omega\).
Also a characterization is given of those states \(\omega\) for which \(\omega^ T\) (respectively, \(\omega^ G)\) is a factor state and in this case, the type of \(\omega^ T\) (respectively, of \(\omega^ G)\) is studied. Besides, necessary and sufficient conditions are obtained for the quasi-equivalence of two factor states \(\omega^ T_ 1\) and \(\omega^ T_ 2\) (respectively, \(\omega^ G_ 1\) and \(\omega^ G_ 2)\).
Reviewer: T.V.Panchapagesan

MSC:

46L30 States of selfadjoint operator algebras
46L35 Classifications of \(C^*\)-algebras
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