By Ryszard Jajte

The noncommutative models of primary classical effects at the nearly convinced convergence in L2-spaces are mentioned: person ergodic theorems, powerful legislation of huge numbers, theorems on convergence of orthogonal sequence, of martingales of powers of contractions and so on. The proofs introduce new concepts in von Neumann algebras. The reader is thought to grasp the basics of useful research and chance. The e-book is written quite often for mathematicians and physicists conversant in chance thought and attracted to purposes of operator algebras to quantum statistical mechanics.

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**Strong Limit Theorems in Noncommutative L2-Spaces**

The noncommutative models of primary classical effects at the nearly yes convergence in L2-spaces are mentioned: person ergodic theorems, powerful legislation of huge numbers, theorems on convergence of orthogonal sequence, of martingales of powers of contractions and so forth. The proofs introduce new recommendations in von Neumann algebras.

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**Extra info for Strong Limit Theorems in Noncommutative L2-Spaces**

**Example text**

Tinuous i n t on We assume that the state lows that all st t h e ~unation [0,~), ~or each is u-invariant. ~ are positive x ~ M, system of c o n t r a c t i o n s t ~ 0). continuous. O En > 0 (St(x~J,y~) the asymptotic behaviour of the averages T ~ 0 (local ergodic theorem). of Goldstein's maximal the ergodic theorem (for one kernel). be a quantum dynamical semigroup (n = 1,2 ..... ). ~ 2 Z Then th6~e ~ I ~ in M. 1 Let at(x)~, u. S. ~ . 4 . 1 . L~MHA. s. c o n v e r g e n c e as For it fol- H In the sequel, we discuss T 3" o (et)tZ0 (M(et)t~0,~) and c o n t r a c t i v i t y T -I x ~ M.

2q; M. in Namely, (12) q = 1 ..... n). ,tn; < t # v). ). From estimations ting n n tZ= Rn,q,jq(Zt)~t" 1 (15), Yn,m = ~n,m + ~n,m' we have 2n-I (18) Z n=l ~ m=l 2 llYn,mll < ~. ,2n-i), Dn, where (20) Dn = n ~ q=l 2 q2 2q X j=l Id 12 n,q,j Indeed, we have lYn,m 12 n Iq=l ~ = ~qdn'q'Jq 12 _< n (X q=l n q -2 )( X q21d n I2) q=l "q'Jq n 2 Z q=l q21d n . 2, ¢(I - p) < 4e IIPDsP J{~ (23) + there m=iZ llYs,mll2 exists a + llg2sl12) < projection ~. ). iig2s - gnilp < Ce~/2, Consequently, p ~ Proj M (25) for an arbitrary E > 0, ~(I - p) < 4E with we can such that, find for a projection 2 s ~ n < 2 s+l, llSn(~) - E(z E ~ : Ii - z I < 2-s)~lip llg2s - gnlip ~ C£I/2s ~ 0 which means that formula (26) n=l 2s < n Z q=l (i) holds.

W i t h the y o n N e u m a n n algebra L (T,m,M) bounded the n o r m llfll = s u p ess t6T B d o ~ not depend on h, t E T. the w e a k * - t e n s o r sentially on h ~ ultraweakly m-measurable Jlf(t)II . functions The tensor product of f : T ~ M state eswith ~ = ¢ ® m is g i v e n b y the f o r m u l a ~(f) = f ¢(f(t))m(dt), f ~ B. T W e s p l i t the p r o o f Let Q of o u r t h e o r e m be a m e a s u r e preserving i n t o a f e w steps. transformation of T and let 32 : T ~ F(M,#) for e a c h be a m a p s u c h t h a t h E H.