Abstract
Let\(\mathfrak{A}\) be a von Neumann algebra with the vector ω cyclic and separating for\(\mathfrak{A}\). Let\(\mathfrak{B}_G\) be a group of unitary operators under which both ω and\(\mathfrak{A}\) are invariant. Let\(\mathfrak{B}\) (resp. ℜ′) be the fixed point algebra in 21 (resp. in\(\mathfrak{A}\)′). LetF o be an orthogonal projection onto the subspace of all vectors invariant under\(\mathfrak{B}_G\). It is shown that ℜ=(\(\mathfrak{A}\) ν {F o})″ and that the irreducibility of ℜ implies thatF o is one-dimentional. Other consequences of the Theorem ofKovács andSzücs are also derived. In sec. 3. the spectrum properties of the group\(\mathfrak{B}_G\) are studied. It is proved that the point spectrum of\(\mathfrak{B}_G\) is symmetric and that it is a group provided ℜ is irreducible. In this case there exists a homomorphism χ→\(\hat \chi\) (resp. χ →\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\chi }\)) of the point spectrum of\(\mathfrak{B}_G\) into the group of unitary operators in\(\mathfrak{A}\) (resp. in\(\mathfrak{A}\)′) uniquely (up to the phase) defined by\(\hat \chi\) V g=χ(g)V g \(\hat \chi\) (resp. the same for\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\chi }\)). In sec. 4. the application of the foregoing results to the KMS-Algebra is given.
Similar content being viewed by others
References
Haag, R., N. M. Hugenholtz, andM. Winnink: Commun. Math. Phys.5, 215 (1967).
Winnink, M.: An application of C*-algebras to quantum statistical mechanics of systems in equilibrium. Thesis, Groningen 1968.
Robinson, D. W.: Commun. Math. Phys.7 337 (1968).
Kovacs, I., andJ. Szücs: Acta Sc. Math.27 233 (1966).
Doplicher, S., D. Kastler, andE. Stormer: Invariant states and Asymptotic Abelianness, preprint 1968.
—— —— Commun. Math. Phys.7 1 (1968).
Stormer, E.: Commun. Math. Phys.5 1 (1967).
Dixmier, J.: Les algebres d'opérateurs dans l'espace Hilbertien. Paris 1957.
Araki, H., andH. Miyata: On KMS Boundary Condition, preprint 1968.
Doplicher, S., R. W. Kadison, D. Kastler, andD. W. Robinson: Commun. Math. Phys.6 101 (1967).
Robinson, D. W., andD. Ruelle: Ann. Inst. Henri Poincaré Vol. VI, no 4, 299 (1967).
Jadczyk, A. Z.: Commun. Math. Phys.12 58 (1969).
Jadczyk, A. Z., andL. Nikolova: Internal symmetries and observables, preprint (1969).
Hugenholtz, N. M., andJ. D. Wieringa: On locally normal states in quantum statistical mechanics, preprint (1968).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Jadczyk, A.Z. On some groups of automorphisms of von Neumann algebras with cyclic and separating vector. Commun.Math. Phys. 13, 142–153 (1969). https://doi.org/10.1007/BF01649873
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01649873