Involutory automorphisms of operator algebras

Davies, E. B.

doi:10.1090/S0002-9947-1971-0284818-0

Involutory automorphisms of operator algebras
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Trans. Amer. Math. Soc. 158 (1971), 115-142 Request permission

Abstract:

We develop the mathematical machinery necessary in order to describe systematically the commutation and anticommutation relations of the field algebras of an algebraic quantum field theory of the fermion type. In this context it is possible to construct a skew tensor product of two von Neumann algebras and completely describe its type in terms of the types of the constituent algebras. Mathematically the paper is a study of involutory automorphisms of ${W^\ast }$-algebras, of particular importance to quantum field theory being the outer involutory automorphisms of the type III factors. It is shown that each of the hyperfinite type III factors studied by Powers has at least two outer involutory automorphisms not conjugate under the group of all automorphisms of the factor.

References

M. F. Atiyah, R. Bott, and A. Shapiro, Clifford modules, Topology 3 (1964), no. suppl, suppl. 1, 3–38. MR 167985, DOI 10.1016/0040-9383(64)90003-5
Claude Chevalley, The construction and study of certain important algebras, Mathematical Society of Japan, Tokyo, 1955. MR 0072867
E. B. Davies, On the Borel structure of $C^{\ast }$-algebras, Comm. Math. Phys. 8 (1968), 147–163. MR 231209
E. B. Davies, The structure of $\sum ^{\ast }$-algebras, Quart. J. Math. Oxford Ser. (2) 20 (1969), 351–366. MR 256178, DOI 10.1093/qmath/20.1.351
Jacques Dixmier, Les $C^{\ast }$-algèbres et leurs représentations, Cahiers Scientifiques, Fasc. XXIX, Gauthier-Villars Éditeur, Paris, 1969 (French). Deuxième édition. MR 0246136

Les algèbres d’opérateurs dans l’espace Hilbertien

Sergio Doplicher, Rudolf Haag, and John E. Roberts, Fields, observables and gauge transformations. I, Comm. Math. Phys. 13 (1969), 1–23. MR 258394
Edward G. Effros, The canonical measures for a separable $C^{\ast }$-algebra, Amer. J. Math. 92 (1970), 56–60. MR 259623, DOI 10.2307/2373497
James G. Glimm, On a certain class of operator algebras, Trans. Amer. Math. Soc. 95 (1960), 318–340. MR 112057, DOI 10.1090/S0002-9947-1960-0112057-5
Rudolf Haag and Daniel Kastler, An algebraic approach to quantum field theory, J. Mathematical Phys. 5 (1964), 848–861. MR 165864, DOI 10.1063/1.1704187
Richard V. Kadison, Unitary invariants for representations of operator algebras, Ann. of Math. (2) 66 (1957), 304–379. MR 89378, DOI 10.2307/1970002
Masahiro Nakamura and Zirô Takeda, On outer automorphisms of certain finite factors, Proc. Japan Acad. 37 (1961), 215–216. MR 126733
Robert T. Powers, Representations of uniformly hyperfinite algebras and their associated von Neumann rings, Ann. of Math. (2) 86 (1967), 138–171. MR 218905, DOI 10.2307/1970364
Shôichirô Sakai, On topological properties of $W^*$-algebras, Proc. Japan Acad. 33 (1957), 439–444. MR 98992
I. E. Segal, Tensor algebras over Hilbert spaces. II, Ann. of Math. (2) 63 (1956), 160–175. MR 77908, DOI 10.2307/1969994
Erling Størmer, On anti-automorphisms of von Neumann algebras, Pacific J. Math. 21 (1967), 349–370. MR 212584
Erling Størmer, The even $\textrm {CAR}$-algebra, Comm. Math. Phys. 16 (1970), 136–137. MR 269232