Descriptions of conditional expectations induced by non-measure preserving transformations
HTML articles powered by AMS MathViewer
- by Alan Lambert and Barnet M. Weinstock PDF
- Proc. Amer. Math. Soc. 123 (1995), 897-903 Request permission
Abstract:
Given a measure-preserving transformation T acting on a $\sigma$-finite measure space $(X,\mathcal {A},m)$ and a $\sigma$-finite sigma algebra $\mathcal {B} \subset \mathcal {A}$, the conditional expectations $E( \cdot |\mathcal {B})$ acting on ${L^\infty }(\mathcal {A})$ and $E( \cdot |{T^{ - 1}}\mathcal {B})$ acting on ${L^\infty }({T^{ - 1}}\mathcal {A})$ are known to be related by the formula $[E(f|\mathcal {B})] \circ T = E(f \circ T|{T^{ - 1}}\mathcal {B})$ . In this note the conditional expectation $E( \cdot |{T^{ - 1}}\mathcal {B})$ is investigated in the non-measure-preserving case, and those transformations for which the above equation holds are characterized in terms of measurability conditions for $d(m \circ {T^{ - 1}})/dm$. It is precisely in the non-measure-preserving case that the measurability of $d(m \circ {T^{ - 1}})/dm$ plays an important role. Relatedly, it is shown that if composition by T intertwines $E( \cdot |\mathcal {B})$ and any mapping $\Lambda$, then $\Lambda$ is a conditional expectation induced by a measure equivalent to m. These results were motivated by a result concerning induced conditional expectation operators on ${C^ \ast }$-algebras, and the paper concludes with a brief description of this ${C^\ast }$-algebra setting.References
- James T. Campbell and James E. Jamison, On some classes of weighted composition operators, Glasgow Math. J. 32 (1990), no. 1, 87–94. MR 1045089, DOI 10.1017/S0017089500009095 J. Daughtry, A. Lambert, and B. Weinstock, A class of operator algebras induced by conditional expectations on ${C^\ast }$-algebras, Rocky Mountain Math. J. (to appear).
- Ulrich Krengel, Ergodic theorems, De Gruyter Studies in Mathematics, vol. 6, Walter de Gruyter & Co., Berlin, 1985. With a supplement by Antoine Brunel. MR 797411, DOI 10.1515/9783110844641
- Alan Lambert, Localising sets for sigma-algebras and related point transformations, Proc. Roy. Soc. Edinburgh Sect. A 118 (1991), no. 1-2, 111–118. MR 1113848, DOI 10.1017/S0308210500028948
- Alan Lambert and Barnet M. Weinstock, A class of operator algebras induced by probabilistic conditional expectations, Michigan Math. J. 40 (1993), no. 2, 359–376. MR 1226836, DOI 10.1307/mmj/1029004757
- William Parry, Topics in ergodic theory, Cambridge Tracts in Mathematics, vol. 75, Cambridge University Press, Cambridge-New York, 1981. MR 614142
- Şerban Strătilă, Modular theory in operator algebras, Editura Academiei Republicii Socialiste România, Bucharest; Abacus Press, Tunbridge Wells, 1981. Translated from the Romanian by the author. MR 696172
Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 123 (1995), 897-903
- MSC: Primary 28D99; Secondary 46N30, 60A10
- DOI: https://doi.org/10.1090/S0002-9939-1995-1231301-3
- MathSciNet review: 1231301