Noncommutative Ergodic Theorems

Question

I am looking for books presenting the noncommutative version of ergodic theorems.

The only book I have found is Krengel, Ergodic Theorems, 1985.

Are there other references other than Krengel's book ?

Answer 1

This is a sufficiently advanced topic so I do not think there is much in book form yet. You can look up

• Jajte, Ryszard, Strong limit theorems in non-commutative probability, Lecture Notes in Mathematics. 1110. Berlin etc.: Springer-Verlag. VI, 152 p. DM 26,50 (1985). ZBL0554.46033.

• Jajte, Ryszard, Strong limit theorems in noncommutative $L^2$-spaces, Lecture Notes in Mathematics. 1477. Berlin etc.: Springer-Verlag. x, 113 p. (1991). ZBL0743.46069.

Although I suspect that both are a little bit outdated. You can also look at

• Defant, Andreas, Classical summation in commutative and noncommutative $L_p$-spaces, Lecture Notes in Mathematics 2021. Berlin: Springer (ISBN 978-3-642-20437-1/pbk; 978-3-642-20438-8/ebook). x, 171 p. (2011). ZBL1267.46002.

which is quite good and has a chapter on noncommutative theory, but it is focused on strong limit theorems, not in Ergodic ones. Nevertheless the language and technicalities are shared.

Apart from that, you would have to look at articles. The usual references for the weak $(1,1)$-inequality of the ergodic maximal are:

• Yeadon, F. J., Ergodic theorems for semifinite von Neumann algebras I, J. Lond. Math. Soc., II. Ser. 16, 326-332 (1977). ZBL0369.46061.
• Yeadon, F. J., Ergodic theorems for semifinite von Neumann algebras. II, Math. Proc. Camb. Philos. Soc. 88, 135-147 (1980). ZBL0466.46056.

The $L_p$ case as well as the real interpolation of maximal inequalities is covered in

• Junge, Marius; Xu, Quanhua, Maximal ergodic theorems in non-commutative $L_p$-spaces, C. R., Math., Acad. Sci. Paris 334, No. 9, 773-778 (2002). ZBL1030.46082.
• Junge, Marius; Xu, Quanhua, Noncommutative maximal ergodic theorems, J. Am. Math. Soc. 20, No. 2, 385-439 (2007). ZBL1116.46053.