zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Strong limit theorems in noncommutative $L\sb 2$-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 $L\sp 2$-spaces. The setting is as follows: let $M$ be a $\sigma$-finite von Neumann algebra with a faithful normal state $\phi$; let $H=L\sp 2(M,\phi)$ be the completion of $M$ under the norm $x\mapsto[\phi(x\sp*x)]\sp{1/2}$, $x\in M$. 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 $L\sp 2$-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 $L\sp 2$. 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.

46L51Noncommutative measure and integration
46L53Noncommutative probability and statistics
46L54Free probability and free operator algebras
47A35Ergodic theory of linear operators
46-02Research monographs (functional analysis)
60F15Strong limit theorems
Full Text: DOI