Operator Product States on Tensor Powers of $C^\ast$-Algebras

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Abstract: The program of matrix product states on an infinite tensor product

m a t h c a l A^{o t i m e s m a t h b b Z}

of

C^{a} s t

-algebras, initiated by Fannes, Nachtergaele and Werner in their seminal paper Commun. Math. Phys. {�f 144}, 443-490 (1992), is re-assessed in a generic context where

m a t h c a l A

is an infinite nuclear

C^{a} s t

-algebra. This covers, for example, two and higher dimensional spin lattices. The program consists of three major blocks: reduction, factorization and reconstruction. We use the Archimedeanization process of Kavruk, Paulsen, Todorov and Tomforde to produce an operator system structure on the quotient of

m a t h c a l A^{o t i m e s m a t h b b Z}

by the so called entanglement kernel. Then we argue that the Haagerup tensor product accommodates naturally the factorization block and demonstrate how its representations on free products of algebras lead to operator product presentations of the states.

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