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$C^*$-independence, product states and commutation. (English) Zbl 1078.46044
Summary: Let $D$ be a unital $C^*$-algebra generated by $C^*$-subalgebras $A$ and $B$ possessing the unit of $D$. Motivated by the commutation problem of $C^*$-independent algebras arising in quantum field theory, the interplay between commutation phenomena, product type extensions of pairs of states and tensor product structure is studied. Roos’s theorem [{\it H. Roos}, Commun. Math. Phys. 16, 238--246 (1970; Zbl 0197.26303)] is generalized in showing that the following conditions are equivalent: (i) every pair of states on $A$ and $B$ extends to an uncoupled product state on $D$; (ii) there is a representation $\pi$ of $D$ such that $\pi(A)$ and $\pi(B)$ commute and $\pi$ is faithful on both $A$ and $B$; (iii) $A \otimes_{\min} B$ is canonically isomorphic to a quotient of $D$. The main results involve unique common extensions of pairs of states. One consequence of a general theorem proved is that, in conjunction with the unique product state extension property, the existence of a faithful family of product states forces commutation. Another is that if $D$ is simple and has the unique product extension property across $A$ and $B$ then the latter $C^*$-algebras must commute and $D$ be their minimal tensor product.

##### MSC:
 46L30 States of $C^*$-algebras 46L60 Applications of selfadjoint operator algebras to physics 81R15 Operator algebra methods (quantum theory)
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