Automorphism of von Neumann algebras as point transformations
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- by Charles Radin
- Proc. Amer. Math. Soc. 39 (1973), 343-346
- DOI: https://doi.org/10.1090/S0002-9939-1973-0313829-9
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Abstract:
Given a concrete separable ${C^ \ast }$-algebra $\mathfrak {A}$ with unit and a faithful normal finite trace $\tau$ on $\mathfrak {A}''$, we introduce a notion of $\tau$-almost every state on $\mathfrak {A}$ which has the proper relationship with Segal’s noncommutative integration theory. We then prove that any $\ast$-automorphism of $\mathfrak {A}''$ is implemented by some point transformation in the state space of $\mathfrak {A}$, defined $\tau$-almost everywhere. This generalizes the classical result of von Neumann-Maharam.References
- Dorothy Maharam, Automorphisms of products of measure spaces, Proc. Amer. Math. Soc. 9 (1958), 702–707. MR 97494, DOI 10.1090/S0002-9939-1958-0097494-6
- Shôichirô Sakai, $C^*$-algebras and $W^*$-algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 60, Springer-Verlag, New York-Heidelberg, 1971. MR 0442701 I. E. Segal, A noncommutative extension of abstract integration, Ann. of Math. (2) 57 (1953), 401-457; ibid. (2) 58 (1953), 595-596. MR 14, 991 ; MR 15, 204.
- Minoru Tomita, Spectral theory of operator algebras. I, Math. J. Okayama Univ. 9 (1959/60), 63–98. MR 117581
Bibliographic Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 39 (1973), 343-346
- MSC: Primary 46L10
- DOI: https://doi.org/10.1090/S0002-9939-1973-0313829-9
- MathSciNet review: 0313829