On neveu decomposition and ergodic type theorems for semi-finite von Neumann algebras
Автор: Grabarnik Genady Ya., Katz Alexander A.
Журнал: Владикавказский математический журнал @vmj-ru
Статья в выпуске: 2 т.5, 2003 года.
Бесплатный доступ
Some ergodic type theorems for automorphisms of semi-finite von Neumann algebras are considered. Neveu decomposition is employed in order to prove stochastical convergence. This work is a generalization of authors results from [5] to the case of semi-finite von Neumann algebras.
Короткий адрес: https://sciup.org/14318083
IDR: 14318083
Текст научной статьи On neveu decomposition and ergodic type theorems for semi-finite von Neumann algebras
ON NEVEU DECOMPOSITION AND ERGODIC TYPE THEOREMS FOR SEMI-FINITE VON NEUMANN ALGEBRAS
G. Ya. Grabarnik, A. A. Katz
Some ergodic type theorems for automorphisms of semi-finite von Neumann algebras are considered. Neveu decomposition is employed in order to prove stochastical convergence. This work is a generalization of authors results from [5] to the case of semi-finite von Neumann algebras.
1. Introduction and Notations
This work is devoted to some results concerning ergodic type theorems for semi-finite von Neumann algebras. The first results in this field were obtained by Sinai and Anshelevich [17] and Lance [14]. Developments of the subject are reflected in the monographs of Jajte [7] and Krengel [13].
The notion of a. weakly wandering set (in commutative context) was introduced by Hajian and Kakutani [9] in order to establish conditions which are equivalent to the existence of finite invariant measures. The non-commutative case was first considered by Jajte [7], and later, for the case of finite von Neumann algebras, by Grabarnik and Katz [5] and Katz [2].
In section 2 we consider Neveu decomposition which gives a. characterization of the existence of the invariant measures in terms of a. weakly wandering operator.
Section 3 is devoted to a. presentation of the Krengel’s Stochastic Ergodic Theorem for the actions of an automorphism on semi-finite von Neumann algebra. [4].
In section 4 we consider a. multiparametric version of the Stochastic Ergodic Theorem [5, 2].
Remark 1. The Multiparametric Superadditive Stochastic Ergodic Theorem will be separately presented in the forthcoming paper [6].
We use the following notations: everywhere below M is assumed to be a. о-finite von Neumann algebra with semi-finite faithful normal trace т (semi-finite algebra), M, is a predual of M, and M* is the Banach dual space to M.
1 denotes the unit of M. For p E M*, the support of p will be denoted by S(p).
2. Neven Decomposition and the Weakly Wandering Operator
Let a be an automorphism of algebra. M, and let a* be an operator acting in M*, to which a is conjugated.
By An (A") we denote the Cesaro average of a (a*).
DEFINITION 1. An operator h E M^. is said to be a weakly wandering operator, if
||An/t|| —> 0 when n —> oo.
The following theorem is valid:
Theorem 1. Let M, a and т be as defined above. The following conditions are equivalent:
-
(i) There exists an «„-invariant normal state p on M with support S(p) = E, t(E) < oo, such that the support of every «„-invariant normal state p is less then or equal to E; in symbols
S^ < E.
-
(ii) E is the maximal projection such that for every projection P < E, P E M,
Ыт(а"Р) > 0.
-
(iii) There exists a weakly wandering operator ho E M+ with support
S^ho) = 1 — E such that the support of every weakly wandering operator is less then or equal to 1 — E.
It follows immediately from the theorem, that:
Corollary 1 (Neven Decomposition). Let « be an automorphism of von Neumann algebra M with «-invariant semi-finite normal trace т. Then there exist projections Ey and E2,
Si + P2 = l (1)
such that:
(i) There exists an «„-invariant normal state p with support S(p) = Ei,
(ii) There exists a weakly wandering operator h E M with Sljij = E2.
3. Stochastic Ergodic Theorem
The space M„ of normal functionals on von Neumann algebra M with «-invariant semifinite normal trace т is naturally identified with the space Li(M, t) of locally measurable operators, each affiliated to M and integrable with modulus. Action «' is defined as an operator conjugated to « with respect to duality:
т<а'Х
• у) =
t
Definition 2. A sequence {An} of measurable operators is said to converge stochastically to operator Xq, if for every e > 0,
т({|Ап — Ao > e}) —> 0 when n —> 00.
Theorem 2 (Stochastic Ergodic Theorem). Let a be an automorphism of von Neumann algebra M with «-invariant semi-finite normal trace t. Then for X E L^M,t\ the Cesaro averages A'nX converge stochastically to X E Li^M,^. The limit X is «'-invariant and
E2XE2 = 0 (2)
(where E^ is a projection from Neven decomposition (1)).
To prove the Theorem (2), we need the following variant of non-commutative Individual Ergodic Theorem:
Theorem 3 (Individual Ergodic Theorem). Let M be a von Neumann algebra with a-invariant semi-finite normal trace t, t(1) = 1. Let a be an automorphism of M, p be a normal faithful state on M, p о а = p.
Then for every p E M, there exists an «„-invariant normal functional p such that for every e > 0 there exists a projection E E M with т(1 — E) < e and sup \(А"p — "р( (яр/т(ар\ ^ 0 when n ^ oo. iXENgE
Let (Tip, xp, 9Л) be a representation of algebra M constructed by a faithful normal state p. Then Ш1 is a von Neumann algebra isomorphic to M. Let a be an image of automorphism ct and ct' be an associated transformation on Ш1':
(dX • EQ, Q) = (X • a'EQ, Q), X E ЭЛ, Y E TH, where Q is a bicyclic vector with (XQ,Q) = p(X), X E Ж
The following theorem is a variant of the Maximal Hopf Lemma.
Theorem 4 (Maximal Hopf Lemma). Let p ЕЖ be a Hermitian functional and e > 0 be such that ||/z||-s—1 < 1. Then, for afixedN there exists a projection E E Ж, p(E±) < ||/z||-s—1 such that sup \(An (а*, p( (ж) / p(x) \ < E,n = 1,2,... , N.
хСЕ9Л+Е
4. Multiparametric Stochastic Ergodic Theorem (the case of d-commuting automorphisms)
Now we will consider the case of (/-commuting automorphisms. Let d > 1 be a natural number and V = {0,1, 2,... }d be an additive semigroup of d-dimentional vectors with natural coordinates. For и = (mJ, у = (yj E V, relation и ^ у (и > у) means гц ^ vt (гц > yj for г = 1,... , d. By [u, y[ we denote the set {w E V : и < го < у}. For the finite set В let card(B) or \B\ means the number of elements of B. For n = (ni,..., n^ E V let
d
irH = П ^y = [0,n[| .
For n G V and operators Pi, 32т ■ ■ т Pd^
Pn = PVP^ ...p^; s„= ptt; A^ttH^S^
uG[o,n[ expression n —» oo means that ny tends to infinity independently for v = l,2,...,d. Let «i, «2,... , ad be automorphisms of algebra M.
DEFINITION 3. An operator h E M^ is called a weakly wandering if
HA"/^^ —> 0 when n —> oo.
Definition 4. A multisequence {An}nev of measurable operators affiliated with M is said to converge stochastically to operator Ao, if for every e > 0,
v({ A„-Ao >0}) —> 0
holds when the multiindex n —> oo.
The following theorem is valid:
Theorem 5. Let «i, a^,... , ад be commuting automorphisms on von Neumann algebra M with faithful normal semi-finite trace т. The following conditions are equivalent:
-
(i) There exists an «„^-invariant normal state p on M with support E such that the support of every normal state does not exceed E (i = 1,2,... ,d).
-
(ii) There exists a weakly wandering operator ho E M+ with support 1 — E such that the support of every weakly wandering operator does not exceed 1 — E.
Moreover, d d
E= /\Ер 1-E = \/ (1-SJ, i=l i=l where Ei is the «maximal» support of the invariant normal states of the automorphism oq, i = 1, 2,... , d. The following Stochastic Multiparametric Ergodic Theorem is valid:
Theorem 6 (Stochastic Multiparametric Ergodic Theorem). Let oq be automorphisms of semi-finite von Neumann algebra M with semi-finite weight т, г = l,2,...,d. Then for X E Li^M,^, the averages A*nA converge stochastically to X E Li^M,^, where n = (ni, П2,..., n^. The limit X is «„-invariant and
EXE = 0, where d
E= V(l"^):
i=l and Ei are projections that were constructed by Theorem 5.
The proof of the above theorem is based on the following:
Theorem 7. Let M be a semi-finite von Neumann algebra, «i be automorphisms of algebra M, i = 1,2,... , d; т be a normal semi-finite «i-invariant trace and p be a faithful normal «i-invariant (i = 1,2,...,^ state on M. Then for every p E M, there exists an cq-invariant functional p such that for every e > 0 there exists a projection
E EM, t
iEEM+E
Let Pi be a map:
v —> lim A^v.
Список литературы On neveu decomposition and ergodic type theorems for semi-finite von Neumann algebras
- Akcoglu M., Sucheston L. A stochastic ergodic theorem for superadditive precesses//Ergodic Theory and Dynamical Systems.-1983.-V. 3.-P. 335-344.
- Cunze J. P., Dang-Nqoc N. Ergodic theorems for noncommutative dynamical systems//Inventiones Mathematicae.-1978.-V. 46.-P. 1-15.
- Goldstein M. S., Grabarnik G. Ya. Almost sure convergence theorems in von Neumann algebras//Israel J. Math.-1991.-V. 76.-P. 161-182.
- Dixmier J. Les algebres d'operateurs dans l'espace hilbertien (algebres de von Neumann).-Paris: Gauthier-Villar, 1969.-367 p.
- Grabarnik G. Ya., Katz A. A. Ergodic type theorems for finite von Neumann algebras//Israel J. Math.-1995.-V. 90.-P. 403-422.
- Grabarnik G. Ya., Katz A. A. On multiparametric superadditive stochastic ergodic theorem for semi-finite von Neumann algebras/to appear.
- Jajte R. Strong limit theorem in noncommutative probability//Lecture Notes in Math.-V. 1110.-Berlin: Spring-Verlag, 1985.-162 p.
- Jajte R. On the existence of invariant states in W*-algebras//Bull. Polish Acad. Sci.-1986.-V. 34.-P. 617-624.
- Hajian A., Kakutani S. Weakly wandering sets and invariant measures//Trans. Amer. Math. Soc.-1964.-V. 110.-P. 131-151.
- Katz A. A. Ergodic type theorems in von Neumann algebras.-Ph. D. Thesis.-Pretoria: University of South Africa, 2001.-84 p.
- Kingman J. F. C. Subadditive ergodic theory//Annals of Probability.-1973.-V. 1.-P. 883-909.
- Kovacs I., Szucs J. Ergodic type theorem in von Neumann algebras//Acta Scientiarum Mathematicarum (Szeged).-1966.-V. 27.-P. 233-246.
- Krengel U. Ergodic Theorems de Greuter.-Berlin, 1985.
- Lance E. C. Ergodic theorems for convex sets and operator algebras//Inventiones Mathematicae.-1976.-V. 37.-P. 201-214.
- Petz D. Ergodic theorems in von Neumann algebras//Acta Scientiarum Mathematicarum (Szeged).-1983.-V. 46.-P. 329-343.
- Segal I. E. A noncommutative extension of abstract integration//Archiv der Math.-1953.-V. 57.-P. 401-457.
- Синай Я. Г., Аншелевич В. В. Некоторые проблемы некоммутативной эргодической теории//Успехи мат. наук.-1976.-Т. 32.-С. 157-174.
- Takesaki M. Theory of Operator Algebras. I.-Berlin: Springer-Verlag, 1979.-vii+415 p.
- Yeadon F. J. Convergence of measurable operators//Math. Proc. Cambridge Philos. Soc.-1973.-V. 74.-P. 257-269.
- Yeadon F. J. Ergodic theorems for semi-finite von Neumann algebras, I//J. London Math. Soc.-1977.-V. 16.-P. 326-332.
- Yeadon F. J. Ergodic theorems for semi-finite von Neumann algebras, II//Math. Proc. Cambridge Philos. Soc.-1980.-V. 88.-P. 135-147.