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Normal states are determined by their facial distances
Author(s) -
Lau Anthony ToMing,
Ng ChiKeung,
Wong NgaiChing
Publication year - 2020
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms.12344
Subject(s) - mathematics , von neumann algebra , centralizer and normalizer , bijection, injection and surjection , pure mathematics , tomita–takesaki theory , center (category theory) , affiliated operator , von neumann architecture , space (punctuation) , metric (unit) , state (computer science) , jordan algebra , algebra over a field , combinatorics , algebra representation , bijection , linguistics , chemistry , philosophy , operations management , algorithm , economics , crystallography
Let M be a semi‐finite von Neumann algebra with normal state space S ( M ) . For any ϕ ∈ S ( M ) , letM ϕ : = { x ∈ M : x ϕ = ϕ x }be the centralizer of ϕ with center Z ( M ϕ ) . We show that for ϕ , ψ ∈ S ( M ) , the following are equivalent.ϕ = ψ .Z ( M ψ ) ⊆ Z ( M ϕ )andϕ | Z ( M ϕ )= ψ | Z ( M ϕ ).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.