Transformations induced in the state space of a -algebra and related ergodic theorems

Author:
A. Łuczak

Journal:
Proc. Amer. Math. Soc. **96** (1986), 617-625

MSC:
Primary 46L50

DOI:
https://doi.org/10.1090/S0002-9939-1986-0826491-2

MathSciNet review:
826491

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a norm-separable -algebra with unit **1**, -weakly dense in a -algebra , and let be a positive linear mapping of into itself leaving **1** invariant. We show that induces a transformation defined "almost everywhere" on the state space of with values in . If is a -automorphism of , then there exists

**[1]**O. Bratelli and D. W. Robinson,*Operator algebras and quantum statistical mechanics*. I, Springer-Verlag, Berlin and New York, 1979.**[2]**M. S. Goldstein,*Theorems on almost everywhere convergence in von Neumann algebras*, J. Operator Theory**6**(1981), 233-311. (Russian) MR**643693 (84g:46096)****[3]**E. C. Lance,*Ergodic theorems for convex sets and operator algebras*, Invent. Math.**37**(1976), 201-214. MR**0428060 (55:1089)****[4]**A. Paszkiewicz,*Convergences in**-algebras*, J. Funct. Anal. (to appear). MR**865218 (87m:46130)****[5]**D. Petz,*Quasi-uniform ergodic theorems in von Neumann algebras*, Bull. London Math. Soc.**16**(1984), 151-156. MR**737243 (85c:46066)****[6]**C. Radin,*Automorphism of von Neumann algebras as point transformations*, Proc. Amer. Math. Soc.**39**(1973), 343-346. MR**0313829 (47:2383)****[7]**-,*Pointwise ergodic theory on operator algebras*, J. Math. Phys.**19**(1978), 1983-1985. MR**0495989 (58:14621)****[8]**M. Takesaki,*Theory of operator algebras*. I, Springer-Verlag, Berlin and New York, 1979. MR**548728 (81e:46038)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
46L50

Retrieve articles in all journals with MSC: 46L50

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1986-0826491-2

Keywords:
Convergence in von Neumann algebras,
state space of a -algebra,
ergodic theorems

Article copyright:
© Copyright 1986
American Mathematical Society