Analytic Subalgebras of Von Neumann Algebras

Muhly, Paul S. ; Saito, Kichi-Suke

Canadian journal of mathematics, Tome 39 (1987) no. 1, pp. 74-99

Résumé

Let M be a von Neumann algebra and let {αt }t∊R be a σ-weakly continuous flow on M; i.e., suppose that {αt }t∊R is a one-parameter group of *-automorphisms of M such that for each ρ in the predual, M∗, of M and for each x ∊ M, the function of t, ρ(α t(x)), is continuous on R. In recent years, considerable attention has been focused on the subspace of M, H∞(α), which is defined to be where H ∞(R) is the classical Hardy space consisting of the boundary values of functions bounded analytic in the upper half-plane. In Theorem 3.15 of [8] it is proved that in fact H∞(α) is a σ-weakly closed subalgebra of M containing the identity operator such that is σ-weakly dense in M, and such that

Détail

DOI : 10.4153/CJM-1987-005-4

Bibliographie

[1] 1. Arveson, W. B., Analyticitv in operator algebras, Amer. J. Math. 89 (1967), 578–642. Google Scholar

[2] 2. Arveson, W. B., On groups of automorphisms of operator algebras, J. Funct. Anal. 15 (1974), 217–243. Google Scholar

[3] 3. Haagerup, U., The standard form of von Neumann algebras, Math. Scand. 37 (1975), 217–243. Google Scholar

[4] 4. Haagerup, U., Operator valued weights in von Neumann algebras I, J. Funct. Anal. 32 (1979), 175–206. Google Scholar

[5] 5. Haagerup, U., Operator valued weights in von Neumann algebras II, J. Funct. Anal. 33 (1979), 339–361. Google Scholar

[6] 6. Haagerup, U., If-spaces associated with an arbitrary von Neumann algebra, Algebres d'operateurs et leurs applications en physique mathématique (Colloques internationaux du CNRS, No. 274, Marseille 20-24 Juin 1977), 175–184; (Éditions du CNRS, Paris, 1979). Google Scholar

[7] 7. Kawamurya, S. and Tomiyama, J., On subdiagonal algebras associated with flows in operator algebras, J. Math. Soc. Japan 29 (1977), 73–90. Google Scholar

[8] 8. Loebl, R. I. and Muhly, P. S., Analyticity and flows in von Neumann algebras, J. Funct. Anal. 29(1978), 214–252. Google Scholar

[9] 9. McAsey, M. J. and Muhly, P. S., Representations of non-self-adjoint crossed products., Proceedings of the London Mathematical Society, Third Series 47 (1983), 128–144. Google Scholar

[10] 10. McAsey, M., Muhly, P. S. and Saito, K.-S., Nonself adjoint crossed products (Invariant subspaces and maximality), Trans. Amer. Math. Soc. 248 (1979), 381–409. Google Scholar

[11] 11. McAsey, M., Muhly, P. S. and Saito, K.-S., Nonself adjoint crossed products II, J. Math. Soc. Japan 33 (1981), 485–495 . Google Scholar

[12] 12. McAsey, M., Muhly, P. S. and Saito, K.-S., Nonself adjoint crossed products III (Infinite algebras), J. Operator Theory 12 (1984), 3–22. Google Scholar

[13] 13. Muhly, P. S., Maximal weak*-Dirichlet algebras, Proc. Amer. Math. Soc. 36 (1972), 515–518. Google Scholar

[14] 14. Muhly, P. S., Function algebras and flows, Acta Sci. Math. (Szeged) 35 (1973), 111–121. Google Scholar

[15] 15. Saito, K.-S., On noncommutative Hardy spaces associated with flows infinite von Neumann algebras, Tôhoku Math. J. 29 (1977), 585–595. Google Scholar

[16] 16. Saito, K.-S., Invariant subspaces for finite maximal subdiagonal algebras, Pacific J. Math. 93 (1981), 431–434. Google Scholar

[17] 17. Saito, K.-S., Invariant subspaces and cocycles in nonselfadjoint crossed products, J. Funct. Anal. 45 (1982), 177–193. Google Scholar

[18] 18. Saito, K.-S., Nonself adjoint subalgebras associated with compact abelian group actions on finite von Neumann algebras, Tôhoku Math. J. 34 (1982), 485–494. Google Scholar

[19] 19. Saito, K.-S., Spectral resolutions of invariant subspaces by compact abelian group actions on von Neumann algebras, preprint. Google Scholar

[20] 20. Segal, I. E., A noncommutative extension of abstract integration, Ann. of Math. 57 (1953), 401–457. Google Scholar

[21] 21. Solel, B., Invariant subspaces for algebras of analytic operators associated with a periodic flow on a finite von Neumann algebra, preprint. Google Scholar | DOI

[22] 22. Solel, B., Algebras of analytic operators associated with a periodic flow on a von Neumann algebra, preprint. Google Scholar

[23] 23. Stratila, S., Modular theory in operator algebras, (Abacus Press, Tunbridge, England, 1981). Google Scholar

[24] 24. Takesaki, M., Duality for crossed products and the structure of von Neumann algebras of type III, Acta Math. 131 (1973), 249–308. Google Scholar

[25] 25. Takesaki, M., Theory of operator algebras I, (Springer-Verlag, Berlin-Heidelberg-New York, 1979). Google Scholar | DOI

[26] 26. Terp, M., Lp-spaces associated with von Neumann algebras, Rapport No. 3 (1981), The University of Odense. Google Scholar

[27] 27. Zsido, L., Spectral and ergodic properties of the analytic generators, J. Approximation Theory 20 (1977), 77–138. Google Scholar