On subdiagonal algebras associated with flows in operator algebras

Shinzô KAWAMURA; Jun TOMIYAMA

doi:10.2969/jmsj/02910073

Shinzô KAWAMURA, Jun TOMIYAMA

Author information

JOURNAL FREE ACCESS

1977 Volume 29 Issue 1 Pages 73-90

DOI https://doi.org/10.2969/jmsj/02910073

Details

Published: 1977 Received: January 07, 1976 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
Correction information
Date of correction: October 20, 2006 Reason for correction: - Correction: ABSTRACT Details: Wrong : The noncommutative Hardy spaces H^∞(α) and H¹(α) are introduced with respect to a σ-weakly continuous flow α={α_t} of *-automorphisms of a von Neumann algebra. In case that the algebra is α-finite the algebra H^∞(α) becomes a maximal subdiagonal algebra. The concept of C*-subdiagonal algebras will also be given for C*-algebras as a noncommutative counterpart of the algebras of generalized analytic functions. Examples of maximal C*-subdiagonal algebras and their structures are discussed.
Right : The noncommutative Hardy spaces H^∞(α) and H¹(α) are introduced with respect to a σ-weakly continuous flow α={α_t} of *-automorphisms of a von Neumann algebra. In case that the algebra is α-finite the algebra H^∞(α) becomes a maximal subdiagonal algebra. The concept of C^*-subdiagonal algebras will also be given for C^*-algebras as a noncommutative counterpart of the algebras of generalized analytic functions. Examples of maximal C^*-subdiagonal algebras and their structures are discussed.

Date of correction: October 20, 2006 Reason for correction: - Correction: AUTHOR Details: Wrong : Shinzo KAWAMURA¹⁾, Jun TOMIYAMA¹⁾
Right : Shinzô KAWAMURA¹⁾, Jun TOMIYAMA¹⁾

Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) R. Arens and I.M. Singer, Generalized analytic function, Trans. Amer. Soc., 81 (1956), 379-393.
2) W.B. Arveson, Analyticity in operator algebras, Amer. J. Math., 89 (1967), 578-642.
3) W.B. Arveson, Operator algebras and measure preserving automorphisms, Acta Math., 118 (1967), 95-109.
4) W.B. Arveson, On groups of automorphisms of operator algebras, J. Functional Analysis, 15 (1974), 217-243.
5) J. Dixmier, Les C^*-algebres et leurs représentation, Gauthier-Villars, Paris, 1969.
6) F. Forelli, Analytic and quasi-invariant measures, Acta Math., 118 (1967), 33-59.
7) F. Forelli, Measures orthogonal to polydisc algebra, J. Math. Mech., 17 (1968), 1073-1086.
8) F. Forelli, A maximal algebra, Math. Scand., 30 (1972), 152-158.
9) T. Gamelin, Uniform algebra, Printice-Hall, Englewood Cliffs. N.J. 1969.
10) T. Gamelin and G. Lumer, Theory of abstract Hardy spaces and the universal Hardy class, Advances in Math., 2 (1968), 118-174.
11) A. Guichardet and D. Kastler, Désintegration des etats quasi-invariants des C^*-algébres, J. Math. Pures Appl., 49 (1970), 349-380.
12) R.V. Kadison and I.M. Singer, Triangular operator algebras, Amer. J. Math., 82 (1960), 227-259.
13) N. Kamei, Simply invariant subspace theorems for antisymmetric finite subdiagonal algebras, Tohoku Math. J., 21 (1969), 467-473.
14) S. Kawamura, On the commutation relation and the spectral condition of weakly continuous representations of a locally compact abelian group on Banach spaces, Bull. Yamagata Univ. Natur. Sci., 8 (1975), 479-489.
15) I. Kovács and J. Szücs, Ergodic type theorems in von Neumann algebras, Acta Sci. Math., 27 (1966), 233-246.
16) R.I. Loebl and P.S. Muhly, Reductive algebras and automorphism groups of von Neumann algebras, Bull. Amer. Math. Soc., 81 (1975), 759-761.
17) P.S. Muhly, Function algebras and flows I, Acta Sci. Math., 35 (1973), 111-121.
18) P.S. Muhly, Function algebras and flows II, Ark. Math., 11 (1973), 203-213.
19) P.S. Muhly, Function algebras and flows III, Math. Z., 136 (1974), 253-260.
20) P.S. Muhly, Function algebras and flows VI, to appear.
21) D. Olesen, On spectral subspaces and their applications to automorphism groups, CNRS Notes, Marseille, 1974.
22) W. Rudin, Fourier analysis on groups, Interscience Publishers, Inc., New York, 1962.
23) K.H. Saito, The Hardy spaces associated with a periodic flow on a von Neumann algebra, to appear.
24) M. Takesaki, A note on the cross-norm of the direct product of operator algebras, Kodai Math. Sem. Rep., 10 (1958), 137-140.
25) M. Takesaki, Tomita's theory of modular Hilbert algebras and its applications, Lecture Notes in Math, 128, Springer-Verlag, 1970.
26) M. Takesaki, The structure of a von Neumann algebra with a homogeneous periodic state, Acta Math., 131 (1973), 79-121.
27) J. Tomiyama, Applications of Fubini type theorem to the tensor products of C^*-algebras, Tohoku Math. J., 19 (1967), 213-226.
28) J. Tomiyama, Application of Fubini mappings of tensor products of Banach algebras, Seminar (1971), University of Copenhagen, Copenhagen, Denmark.
29) J. Tomiyama, Function algebra and flow, Sûgaku, 28 (1976), 35-48 (Japanese).
30) G. Zeller-Meier, Produits croises d'une C^*-algèbre par un groupe d'automorphismes, J. Math. Pures Appl., 47 (1968), 101-239.

Right : [1] R. Arens and I. M. Singer, Generalized analytic function, Trans. Amer. Soc., 81 (1956), 379-393.
[2] W. B. Arveson, Analyticity in operator algebras, Amer. J. Math., 89 (1967), 578-642.
[3] W. B. Arveson, Operator algebras and measure preserving automorphisms, Acta Math., 118 (1967), 95-109.
[4] W. B. Arveson, On groups of automorphisms of operator algebras, J. Functional Analysis, 15 (1974), 217-243.
[5] J. Dixmier, Les C-algebres et leurs représentation, Gauthier-Villars, Paris, 1969.
[6] F. Forelli, Analytic and quasi-invariant measures, Acta Math., 118 (1967), 33-59.
[7] F. Forelli, Measures orthogonal to polydisc algebra, J. Math. Mech., 17 (1968), 1073-1086.
[8] F. Forelli, A maximal algebra, Math. Scand., 30 (1972), 152-158.
[9] T. Gamelin, Uniform algebra, Printice-Hall, Englewood Cliffs. N.J. 1969.
[10] T. Gamelin and G. Lumer, Theory of abstract Hardy spaces and the universal Hardy class, Advances in Math., 2 (1968), 118-174.
[11] A. Guichardet and D. Kastler, Désintegration des etats quasi-invariants des C-algébres, J. Math. Pures Appl., 49 (1970), 349-380.
[12] R. V. Kadison and I. M. Singer, Triangular operator algebras, Amer. J. Math., 82 (1960), 227-259.
[13] N. Kamei, Simply invariant subspace theorems for antisymmetric finite subdiagonal algebras, Tôhoku Math. J., 21 (1969), 467-473.
[14] S. Kawamura, On the commutation relation and the spectral condition of weakly continuous representations of a locally compact abelian group on Banach spaces, Bull. Yamagata Univ. Natur. Sci., 8 (1975), 479-489.
[15] I. Kovács and J. Szücs, Ergodic type theorems in von Neumann algebras, Acta Sci. Math., 27 (1966), 233-246.
[16] R. I. Loebl and P. S. Muhly, Reductive algebras and automorphism groups of von Neumann algebras, Bull. Amer. Math. Soc., 81 (1975), 759-761.
[17] P. S. Muhly, Function algebras and flows I, Acta Sci. Math., 35 (1973), 111-121.
[18] P. S. Muhly, Function algebras and flows II, Ark. Math., 11 (1973), 203-213.
[19] P. S. Muhly, Eunction algebras and flows III, Math. Z., 136 (1974), 253-260.
[20] P. S. Muhly, Function algebras and flows VI, to appear.
[21] D. Olesen, On spectral subspaces and their applications to automorphism groups, CNRS Notes, Marseille, 1974.
[22] W. Rudin, Fourier analysis on groups, Interscience Publishers, Inc., New York, 1962.
[23] K. H. Saito, The Hardy spaces associated with a periodic flow on a von Neumann algebra, to appear.
[24] M. Takesaki, A note on the cross-norm of the direct product of operator algebras, Kodai Math. Sem. Rep., 10 (1958), 137-140.
[25] M. Takesaki, Tomita's theory of modular Hilbert algebras and its applications, Lecture Notes in Math, 128, Springer-Verlag, 1970.
[26] M. Takesaki, The structure of a von Neumann algebra with a homogeneous periodic state, Acta Math., 131 (1973), 79-121.
[27] J. Tomiyama, Applications of Fubini type theorem to the tensor products of C-algebras, Tôhoku Math. J., 19 (1967), 213-226.
[28] J. Tomiyama, Application of Fubini mappings of tensor products of Banach algebras, Seminar (1971), University of Copenhagen, Copenhagen, Denmark.
[29] J. Tomiyama, Function algebra and flow, Sûgaku, 28 (1976), 35-48 (Japanese).
[30] G. Zeller-Meier, Produits croises d'une C-algèbre par un groupe d'automorphismes, J. Math. Pures Appl., 47 (1968), 101-239.

Date of correction: October 20, 2006 Reason for correction: - Correction: PDF FILE Details: -

Abstract

The noncommutative Hardy spaces H^∞(α) and H¹(α) are introduced with respect to a σ-weakly continuous flow α={α_t} of *-automorphisms of a von Neumann algebra. In case that the algebra is α-finite the algebra H^∞(α) becomes a maximal subdiagonal algebra. The concept of C^*-subdiagonal algebras will also be given for C^*-algebras as a noncommutative counterpart of the algebras of generalized analytic functions. Examples of maximal C^*-subdiagonal algebras and their structures are discussed.

Corresponding author

Correction information

Register with J-STAGE for free!