SUBCENTRALITY OF RESTRICTIONS OF BOUNDARY MEASURES ON STATE-SPACES OF C-STAR-ALGEBRAS
Let F be a closed face of the weak* compact convex state space of a unital C*-algebra A. The author has already shown that F is a Choquet simplex if and only if pφFπφ(A)″pφF is abelian for any φ in F with associated cyclic representation (Hφ,πφ,ξφ), where pφF is the orthogonal projection of Hφ onto...
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| Format: | Journal article |
| Language: | English |
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1981
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| _version_ | 1797092109215334400 |
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| author | Batty, C |
| author_facet | Batty, C |
| author_sort | Batty, C |
| collection | OXFORD |
| description | Let F be a closed face of the weak* compact convex state space of a unital C*-algebra A. The author has already shown that F is a Choquet simplex if and only if pφFπφ(A)″pφF is abelian for any φ in F with associated cyclic representation (Hφ,πφ,ξφ), where pφF is the orthogonal projection of Hφ onto the subspace spanned by vectors η defining vector states a → 〈πφ(a)η, η)〉 lying in F. It is shown here that if B is a C*-subalgebra of A containing the unit and such that ξφ is cyclic in Hφ for πφ(B) for any φ in F, then the boundary measures on F are subcentral as measures on the state space of B if and only if pφF(πφ(A), πφ(B)′)″pφF is abelian for all φ in F. If A is separable, this is equivalent to the condition that any state in F with (πφ(A)′ ∩ πφ(B)″) one-dimensional is pure. Taking A to be the crossed product of a discrete C*-dynamical system (B, G, α), these results generalise known criteria for the system to be G-central. © 1981. |
| first_indexed | 2024-03-07T03:41:40Z |
| format | Journal article |
| id | oxford-uuid:be1b33f7-9cd1-4998-8de1-2212d8f7ea95 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T03:41:40Z |
| publishDate | 1981 |
| record_format | dspace |
| spelling | oxford-uuid:be1b33f7-9cd1-4998-8de1-2212d8f7ea952022-03-27T05:36:46ZSUBCENTRALITY OF RESTRICTIONS OF BOUNDARY MEASURES ON STATE-SPACES OF C-STAR-ALGEBRASJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:be1b33f7-9cd1-4998-8de1-2212d8f7ea95EnglishSymplectic Elements at Oxford1981Batty, CLet F be a closed face of the weak* compact convex state space of a unital C*-algebra A. The author has already shown that F is a Choquet simplex if and only if pφFπφ(A)″pφF is abelian for any φ in F with associated cyclic representation (Hφ,πφ,ξφ), where pφF is the orthogonal projection of Hφ onto the subspace spanned by vectors η defining vector states a → 〈πφ(a)η, η)〉 lying in F. It is shown here that if B is a C*-subalgebra of A containing the unit and such that ξφ is cyclic in Hφ for πφ(B) for any φ in F, then the boundary measures on F are subcentral as measures on the state space of B if and only if pφF(πφ(A), πφ(B)′)″pφF is abelian for all φ in F. If A is separable, this is equivalent to the condition that any state in F with (πφ(A)′ ∩ πφ(B)″) one-dimensional is pure. Taking A to be the crossed product of a discrete C*-dynamical system (B, G, α), these results generalise known criteria for the system to be G-central. © 1981. |
| spellingShingle | Batty, C SUBCENTRALITY OF RESTRICTIONS OF BOUNDARY MEASURES ON STATE-SPACES OF C-STAR-ALGEBRAS |
| title | SUBCENTRALITY OF RESTRICTIONS OF BOUNDARY MEASURES ON STATE-SPACES OF C-STAR-ALGEBRAS |
| title_full | SUBCENTRALITY OF RESTRICTIONS OF BOUNDARY MEASURES ON STATE-SPACES OF C-STAR-ALGEBRAS |
| title_fullStr | SUBCENTRALITY OF RESTRICTIONS OF BOUNDARY MEASURES ON STATE-SPACES OF C-STAR-ALGEBRAS |
| title_full_unstemmed | SUBCENTRALITY OF RESTRICTIONS OF BOUNDARY MEASURES ON STATE-SPACES OF C-STAR-ALGEBRAS |
| title_short | SUBCENTRALITY OF RESTRICTIONS OF BOUNDARY MEASURES ON STATE-SPACES OF C-STAR-ALGEBRAS |
| title_sort | subcentrality of restrictions of boundary measures on state spaces of c star algebras |
| work_keys_str_mv | AT battyc subcentralityofrestrictionsofboundarymeasuresonstatespacesofcstaralgebras |