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Владикавказский математический журнал Апрель-июнь, 2003, Том 5, Выпуск 2
УДК 517.98
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
f5, 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 a-finite von Neumann algebra with semi-finite faithful normal trace r (semi-finite algebra), A/„, is a predual of M, and M* is the Banach dual space to M.
1 denotes the unit of M. For p £ A/,„. the support of p will be denoted by S(p).
Let a be an automorphism of algebra M, and let a* be an operator acting in A/,„. to which a is conjugated.
By An (AJ) we denote the Cesaro average of a (a*).
© 2003 Grabarnik G. Ya., Katz A. A.
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G. Ya.. Gra.ba.rnik, A. A. Katz
2. Neveu Decomposition and the Weakly Wandering Operator
Definition 1. An operator h G M| is said to be a weakly wandering operator, if
\\Anh\\ 0 when n oo. The following theorem is valid:
Theorem 1. Let A/, a and r be as defined above. The following conditions are equivalent:
(i) There exists an ck*-invariant normal state p on M with support S(p) = E, t(E) < oo, such that the support of every ck*-invariant normal state p is less then or equal to E; in symbols
S(p) <
(ii) is the maximal projection such that for every projection P ^ E, P G M,
inf r(a"P) > 0.
n
(iii) There exists a weakly wandering operator ho G 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 (Neveu Decomposition). Let a be an automorphism of von Neumann algebra M with a-invariant semi-finite normal trace r. Then there exist projections Ej and E->.
El+E2 = l (1)
such that:
(i) There exists an ck*-invariant normal state p with support S(p) = E\,
(ii) There exists a weakly wandering operator h G M with S(h) = E->.
3. Stochastic Ergodic Theorem
The space A/„, of normal functionals on von Neumann algebra M with a-invariant semi-finite normal trace r is naturally identified with the space L\ (M, r) of locally measurable operators, each affiliated to M and integrable with modulus. Action a' is defined as an operator conjugated to a with respect to duality:
t(o'X ■ y) = t(X ■ ay) (X G ¿i(M,r), y G M).
Definition 2. A sequence {Xn} of measurable operators is said to converge stochastically to operator Xq, if for every e > 0,
r({\Xn — Xo| > e}) —0 when n —oo.
Theorem 2 (Stochastic Ergodic Theorem). Let a be an automorphism of von Neumann algebra M with a-invariant semi-finite normal trace r. Then for X G L\ {M,t), the Cesa.ro averages .4'" A" converge stochastically to X G L\(M, r). The limit X is a'-invariant and
l>2 X l>2 = 0
(2)
(where /-A. is a projection from Neveu 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 ainvariant semi-finite normal trace t, t(1) = 1. Let a be an automorphism of M, p be a normal faithful state on M,
p o a = p.
Then for every p £ A/„, there exists an a^-invariant normal functional JJ such that for every e > 0 there exists a projection E £ M with r(l — E) < e and
sup \(A™p -JI) (x)/t(x) | —0 when n —oo. xeEM+E
x^O
Let (Hp,irp,iM) be a representation of algebra M constructed by a faithful normal state p. Then 9Jt is a von Neumann algebra isomorphic to M. Let a be an image of automorphism a and a' be an associated transformation on 9Jt':
1.1V ■ Yil. fi) = (X ■ a'YQ, <>). X em, Y £ M,
where Q is a bicyclic vector with (Xil, Q) = p(X), X £ 9Jt.
The following theorem is a variant of the Maximal Hopf Lemma.
Theorem 4 (Maximal Hopf Lemma). Let p. £ QJt be a Hermitian functional and e > 0 be such that ||/i|| < 1. Then, for a fixed N there exists a projection E £ 9Jt, p(E-1) < ||/i|| such that
sup \(An(a*, p) (x)/p(x)| < ii 1.2.....N.
xeEM+E
x^O
4. Multiparametric Stochastic Ergodic Theorem (the case of d-commuting automorphisms)
Now we will consider the case of d-commuting automorphisms. Let d ^ 1 be a natural number and ¥ = {0,1, 2,... }d be an additive semigroup of <i-dimentional vectors with natural coordinates. For u = («¿), v = (vi) £ ¥, relation u ^ v (u > v) means U{ ^ V{ (U{ > Vi) for % = 1,..., d. By [u, v[ we denote the set {w £ ¥ : u ^ w < v\. For the finite set B let card(B) or \B\ means the number of elements of B. For n = (ni,..., n^) £ ¥ let
d
n(n) = = |[0,n[| .
v=l
For n £ ¥ and operators /3\, ■ ■ ■, /3d,
Pn = fti1P22 ■ ■ ■ d' = /3«; An = 7r(n) 1 Sn;
u£[o,n[
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G. Ya. Gra.ba.rnik, A. A. Katz
expression n —oo means that tends to infinity independently for v = 1,2,...,d. Let «1, «2, ■ ■ ■ 5 «d be automorphisms of algebra M.
Definition 3. An operator /, e A/1 is called a weakly wandering if
II^IL 0 when rw oo.
Definition 4. A multisequence {In}„£v of measurable operators affiliated with M is said to converge stochastically to operator X$, if for every e > 0,
Ti{\Xn — -X"o| > 0}) —0
holds when the multiindex n —oo. The following theorem is valid:
Theorem 5. Let ai, ■ ■ ■, ay be commuting automorphisms on von Neumann algebra M with faithful normal semi-finite trace r. The following conditions are equivalent:
(i) There exists an ck*¿-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 € M+ with support 1 — E such that the support of every weakly wandering operator does not exceed 1 — E.
Moreover,
d d E = /\Ef, 1 — E = \J (1 — Ej).
i=1 i=1
where E{ is the «maximal» support of the invariant normal states of the automorphism ai, i = 1,2,... ,d. The following Stochastic Multiparametric Ergodic Theorem is valid:
Theorem 6 (Stochastic Multiparametric Ergodic Theorem). Let oii be automorphisms of semi-finite von Neumann algebra M with semi-finite weight t, % = 1,2 ,...,d. Then for X € Li(M, t), the averages .4„,„ A" converge stochastically to X € ¿i(M, r), where n = {ni,n,2;... ,n<i). The limit X is ««-invariant and
EX E = 0,
where
d
E=\/(l^Et), i=1
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, ai be automorphisms of algebra M, i = 1, 2,..., d; r be a normal semi-finite on-invariant trace and p be a faithful normal on-invariant (i = 1,2,...,d) state on M. Then for every p £ A/,„ there exists an ai-invariant functional ~p such that for every e > 0 there exists a projection
E (EM, t{El) < e-
moreover, \\A™ ^p\h —0 and
sup — p)(x)/t(x)\ —0 when the multiindex n oo.
xeEM+E
x^O
Let Pj, be а. map:
v lim AkHu.
k—>00
The map Pi is a projection on the set of a^-stationary points and
-jj = pd ■ pd_x.....
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Статья поступила 11 апреля, 2003 Dr. Genady Ya. Grabarnik,
IBM T.J. Watson Research Center, 19 Skyline Dr., Hawthorne, NY 10510, USA. E-mail: [email protected]
Alexander A. Katz, Ph.D.
Department of Mathematics & CS, St. John's University, 300 Howard Ave., Staten Island, NY 10301, USA. E-mail: [email protected]
