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Strong limit theorems in noncommutative ${L}_{2}$-spaces. (English) Zbl 0743.46069
Lecture Notes in Mathematics. 1477. Berlin etc.: Springer-Verlag. x, 113 p. (1991).

The present book is a continuation of a previous volume [“Strong limit theorems in non-commutative probability”, Lect. Notes Math. 1110 (1985; Zbl 0554.46033)]. It is devoted to non-commutative versions of pointwise convergence theorems in ${L}^{2}$-spaces. The setting is as follows: let $M$ be a $\sigma$-finite von Neumann algebra with a faithful normal state $\phi$; let $H={L}^{2}\left(M,\phi \right)$ be the completion of $M$ under the norm $x↦{\left[\phi \left({x}^{*}x\right)\right]}^{1/2}$, $x\in M$. In $H$, a suitable notion of almost everywhere convergence is introduced; here, a number of definitions are possible; the author’s choice of definition has the advantage of giving clean theorems — besides being equivalent to the corresponding notion in the framework of the commutative ${L}^{2}$-spaces. The author has briefly discussed this point in Chapter 1.

He then goes on to prove the analogues of various ergodic theorems and their generalisations. These are related to the recent works of Gaposhkin, Goldstein, Lance and Yeadon. A non-commutative version of a theorem of Burkholder and Chow on the convergence of the iterates of two conditional expectations is given; this is proved via a non-commutative version of a theorem of E. Stein on the convergence of the iterates of a positive contraction in ${L}^{2}$. Non-commutative versions of the Rademacher-Menshov theorem for series of orthogonal functions and of the martingale convergence theorem are proven.

In a final short chapter, non-commutative versions of various other strong laws of large numbers are given. The monograph closes with six open problems. It has a helpful bibliography (116 items) containing a great deal of related recent publications.

The style of writing is such that only a modest knowledge of the vast theory of operator algebras would be sufficient for a reader familiar with basic probability theory and functional analysis. The monograph is a welcome addition to the growing literature on the subject.

##### MSC:
 46L51 Noncommutative measure and integration 46L53 Noncommutative probability and statistics 46L54 Free probability and free operator algebras 47A35 Ergodic theory of linear operators 46-02 Research monographs (functional analysis) 60F15 Strong limit theorems