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 • ay) (A G L^M, т), у G M\

Definition 2. A sequence {A_n} of measurable operators is said to converge stochastically to operator Xq, if for every e > 0,

т({|А_п — A_o > 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, x_p, 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 {A_n}_nev of measurable operators affiliated with M is said to converge stochastically to operator A_o, 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 0 and sup (A*/z — Д)(ж)/т(ж) —> 0 when the multiindex n —> oo.

iEEM+E

Let Pi be a map:

v —> lim A^v.