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Translation Invariant Pure State on \(\otimes _{{\mathbb {Z}}}\!M_d({\mathbb {C}})\) and Haag Duality

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Abstract

We prove Haag duality property of any translation invariant pure state on \({\mathcal B}= \otimes _{{\mathbb {Z}}}\!M_d({\mathbb {C}}), \;d \ge 2\), 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 \({\mathcal B}\).

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  1. The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai, 600113, India

    Anilesh Mohari

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Correspondence to Anilesh Mohari.

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Communicated by Palle Jorgensen.

This paper has grown over the years starting with initial work in the middle of 2005. The author gratefully acknowledge discussion with Ola Bratteli and Palle E. T. Jorgensen for inspiring participation in sharing the intricacy of the present problem. Finally the author is indebted to Taku Matsui for valuable comments on an earlier draft of the present problem where the author made an attempt to prove Haag duality.

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Mohari, A. Translation Invariant Pure State on \(\otimes _{{\mathbb {Z}}}\!M_d({\mathbb {C}})\) and Haag Duality. Complex Anal. Oper. Theory 8, 745–789 (2014). https://doi.org/10.1007/s11785-013-0336-0

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