zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
$C^*$-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.

46L30States of $C^*$-algebras
46L60Applications of selfadjoint operator algebras to physics
81R15Operator algebra methods (quantum theory)
Full Text: DOI