Research PaperNo Access

(No) Wigner Theorem for C*-algebras

Klaas Landsman

Institute for Mathematics, Astrophysics, and Particle Physics (IMAPP), Radboud University, Nijmegen, Netherlands

Dutch Institute for Emergent Phenomena (DIEP), Netherlands

E-mail Address: landsman@math.ru.nl

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and

Kitty Rang

De Nederlandsche Bank (DNB), Westeinde 1, 1017 ZN Amsterdam, Netherlands

E-mail Address: kitty.r.rang@gmail.com

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https://doi.org/10.1142/S0129055X20500191Cited by:0 (Source: Crossref)

Abstract

Wigner’s Theorem states that bijections of the set $𝒫_{1} (H)$ of one-dimensional projections on a Hilbert space $H$ that preserve transition probabilities are induced by either a unitary or an anti-unitary operator on $H$ (which is uniquely determined up to a phase). Since elements of $𝒫_{1} (H)$ define pure states on the C*-algebra $B (H)$ of all bounded operators on $H$ (though typically not producing all of them), this suggests possible generalizations to arbitrary C*-algebras. This paper is a detailed study of this problem, based on earlier results by R. V. Kadison (1965), F. W. Shultz (1982), K. Thomsen (1982), and others. Perhaps surprisingly, the sharpest known version of Wigner’s Theorem for C*-algebras (which is a variation on a result from Shultz, with considerably simplified proof) generalizes the equivalence between the hypotheses in the original theorem and those in an analogous result on (anti-)unitary implementability of Jordan automorphisms of $B (H)$ , and does not yield (anti-)unitary implementability itself, far from it: abstract existence results that do give such implementability seem arbitrary and practically useless. As such, it would be fair to say that there is no Wigner Theorem for C*-algebras.

Keywords:

AMSC: 81P16, 46L60

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