Translation invariant pure state on \(\bigotimes _{\mathbb {Z}}M_d(\mathbb C)\) and Haag duality. (English) Zbl 1320.46045
Summary: We prove the Haag duality property of any translation invariant pure state on \(\mathcal{B}= \bigotimes_{\mathbb {Z}}M_d(\mathbb C)\), \(d \geq 2\), where \(M_d(\mathbb C)\) is the set of \(d \times d\) dimensional matrices over the field of complex numbers. We also prove a necessary and sufficient condition for a translation invariant factor state to be pure on \(\mathcal{B}\).
MSC:
| 46L30 | States of selfadjoint operator algebras |
Keywords:
uniformly hyperfinite factors; Cuntz algebra; Popescu dilation; Kolmogorov’s property; Arveson’s spectrum; Haag dualityReferences:
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