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Normal states are determined by their facial distances.
- Source :
- Bulletin of the London Mathematical Society; Jun2020, Vol. 52 Issue 3, p505-514, 10p
- Publication Year :
- 2020
-
Abstract
- Let M be a semi‐finite von Neumann algebra with normal state space S(M). For any ϕ∈S(M), let Mϕ:={x∈M:xϕ=ϕx} be the centralizer of ϕ with center Z(Mϕ). We show that for ϕ,ψ∈S(M), the following are equivalent. ϕ=ψ.Z(Mψ)⊆Z(Mϕ) and ϕ|Z(Mϕ)=ψ|Z(Mϕ).ϕ,ψ have the same distances to all the closed faces of S(M). As an application, we give an alternative proof of the fact that metric preserving surjections between normal state spaces of semi‐finite von Neumann algebras are induced by Jordan ∗‐isomorphisms between the underlying algebras. We then use it to verify some facts concerning F‐algebras and Fourier algebras of locally compact quantum groups. [ABSTRACT FROM AUTHOR]
- Subjects :
- VON Neumann algebras
QUANTUM groups
COMPACT groups
DISTANCES
ALGEBRA
SURJECTIONS
Subjects
Details
- Language :
- English
- ISSN :
- 00246093
- Volume :
- 52
- Issue :
- 3
- Database :
- Complementary Index
- Journal :
- Bulletin of the London Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 143547561
- Full Text :
- https://doi.org/10.1112/blms.12344