Lectures on Ergodic Theory of Group Actions (a Von Neumann Algebra Approach)
نویسنده
چکیده
1.1. Probability spaces as von Neumann algebras. The “classical” measure theoretical approach to the study of actions of groups on the probability space is equivalent to a “non-classical” operator algebra approach due to a well known observation of von Neumann, showing that measure preserving isomorphisms between standard probability spaces (X,μ) are in natural correspondence with ∗-algebra isomorphisms between their function algebras L∞X = L∞(X,μ) preserving the functional given by the integral, τμ = ∫ ·dμ. More precisely: 1.1.1. Theorem. 1◦. Let T : (X,μ) → (Y, ν) be a measurable map with ν ◦ T = μ. Then ρT : L ∞Y → L∞X defined by ρT (x)(s) = x(Ts), s ∈ X, is an injective ∗-algebra morphism satisfying τμ◦ρT = τν . Conversely, if (X,μ), (Y, ν) are probability spaces and ρ : L∞Y → L∞X is an injective ∗-algebra morphism such that τμ ◦ j = τν , then there exists a measurable map T : X → Y , such that ρ = ρT . Moreover, T is unique and onto, modulo a set of measure 0, and the correspondence T 7→ ρT is “contravariant” functorial, i.e. ρS◦T = ρT ◦ ρS. Also, T is a.e. 1 to 1 if and only if ρ is onto and if this is the case then T−1 is also measurable and measure preserving. 2◦. If (X,μ) is a non-atomic probability space then (X,μ) ' (T, λ) and (L∞X, τμ) ' (L∞T, τλ). Proof. The fact that ρT is a ∗-algebra isomorphism preserving the integral is trivial by the definition. Also, T 7→ ρT is clearly functorial. If (X,μ) has no atoms then one can easily construct recursively finite “diadic” partitions Pn = {pk | 1 ≤ k ≤ mn} with projections in L∞X such that τμ(pk ) = 2−mn ,∀k, Pn ⊂ Pn+1,∀n, and ∪nΣkCpk = L∞X, thus giving an isomorphism ρ of (L∞X, τμ)
منابع مشابه
Reiter’s Properties for the Actions of Locally Compact Quantum Goups on von Neumann Algebras
متن کامل
Set Theory and von Neumann algebras
The aim of the lectures is to give a brief introduction to the area of von Neumann algebras to a typical set theorist. The ideal intended reader is a person in the field of (descriptive) set theory, who works with group actions and equivalence relations, and who is familiar with the rudiments of ergodic theory, and perhaps also orbit equivalence. This should not intimidate readers with a differ...
متن کاملVon Neumann Algebras with Unique Cartan Decomposition
These are the lectures notes from a minicourse given at the Masterclass on “Ergodic theory and von Neumann algebras” at the University of Copenhagen in October 2013. In the Lectures 2, 3 and 4, we will follow S. Vaes’s exposition for his mini-course at the “ Intensive Month on Operator Algebra and Harmonic Analysis” at ICMAT in Madrid in May 2013. The exposition in itself is very close to the o...
متن کاملErgodic Actions of Universal Quantum Groups on Operator Algebras
We construct ergodic actions of compact quantum groups on C∗-algebras and von Neumann algebras, and exhibit phenomena of such actions that are of different nature from ergodic actions of compact groups. In particular, we construct: (1). an ergodic action of the compact quantum Au(Q) on the type IIIλ Powers factor Rλ for an appropriate positive Q ∈ GL(2, R); (2). an ergodic action of the compact...
متن کاملThe Fundamental Group of the Von Neumann Algebra of a Free Group with Infinitely
In this paper we show that the fundamental group !T of the von Neumann algebra 2(Foo) of a free (noncommutative) group with infinitely many generators is lR+ \ {O}. This extends the result of Voiculescu who previously proved [26,27] that Q+ \ {O} is contained in !T(2(Foo)). This solves a classical problem in the harmonic analysis of the free group F 00. In particular, it follows that there exis...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2011