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Владикавказский математический журнал январь-март, 2007, Том 9, Выпуск 1
UDC 517.98
ON NON-COMMUTATIVE ERGODIC TYPE THEOREMS FOR FREE FINITELY GENERATED SEMIGROUPS
Grabarnik G. Ya., Katz A. A., Shwartz L. A.1
In the paper the authors generalize Bufetov’s Ergodic Type Theorems to the case of the actions of free
finitely generated semigroups on von Neumann algebras.
Mathematics Subject Classification (2000): 46L51, 46L53, 46L54.
Key words and phrases: von Neumann algebras, non-commutative ergodic type theorems, actions of
free semigroups.
1. Introduction
First Ergodic Theorems for actions of arbitrary countable groups were obtained by Oseledets [26], who followed an idea of Kakutani [17]. For actions of free groups Guivarc’h [14] considered uniform averages over spheres of increasing radii in a group and proved the related mean ergodic theorem. Grigorchuk [12] announced the Pointwise Ergodic Theorem for Cesaro averages of the spherical averages. Nevo [24] and Nevo and Stein [25] published a proof of the Pointwise Ergodic Theorem. In [13] Grigorchuk announced an Ergodic Theorem for Actions of Free Semigroups. In [3] Bufetov generalized classical and recent Ergodic Theorems of Kakutani, Oseledets, Guivarc’h, Grigorchuk, Nevo and Nevo and Stein for measure-preserving actions of free semigroups and groups.
The first results in the field of non-commutative Ergodic Theorems were obtained by Sinai and Anshelevich [29] and Lance [22]. Developments of the subject are reflected in the monographs of Jajte [15] and Krengel [21].
Majorant ergodic theorem for the operators affiliated to tracial von Neumann algebras was proved in [6].
The aim of the present paper is to generalize Bufetov’s results from [3] to the non-commutative case to obtain non-commutative Ergodic Theorems for the actions of free finitely generated semigroups on von Neumann algebras.
RemarkI. The paper extends results presented by the authors in [7] and [8].
2. Non-commutative Operator Ergodic Theorems
Let the pair (M, t) be a non-commutative probability space, where M is a von Neumann algebra with a faithful, normal tracial state t.
© 2007 Grabarnik G. Ya., Katz A. A., Shwartz L. A.
1The third author is thankful to her Mentor, Dr. Louis E. Labuschagne (UNISA, South Africa) for constant support and careful reading.
Let ai,a2,---,am : M ^ M be 'positive kernels or linear maps satisfying following
conditions: (a,(M+) C M+; a,1 ^ 1; toa, ^ t).
All the {a,}’s could be extended to operators Li(M,t) ^ Li(M,t), which we will also call {a,} without loss of generality.
Let Qm = {w = W1W2 ... wn... : u, = 1,...,m} be the space of all one-sided infinite sequences in the symbols 1,..., m.
We denote by am the shift on Qm, defined by the formula (amu)i = u,+i.
Consider the set Wm = {w = W1W2 ... wn : w, = 1,..., m} of all finite words in the symbols 1, . . . , m.
Denote by |w| the length of the word w. For each w G Wm, let C(w) C Qm be the set of all sequences starting with the word w. For an arbitrary Borel measure ^ on Qm, set ^(w) = ^(C (w)).
Measure ^ on Qm invariant with respect to shift am we call am-invariant measure.
For each w G Wm, introduce the operator
aw --- awn awn-1 . . . aWl . (2.1)
Let ^ be a Borel am-invariant probability measure on Qm. Consider the words w with |w| = l, and the sum of the corresponding operators aw with the weights ^(w),
si(a) = ^2 Kw)aw.
\w\=i
Average si (a) over l = 0,..., n — 1,
n—1
ci(a) = nEsl (a).
1=0
Definition 2.1. A sequence {Xn} C L1(M, t) is said to converge to Xo G L1(M, t) doubleside almost everywhere if for every e ^ 0 and 5 ^ 0 there exists N G N and projection E G M such that t(I — E) <5 and E(Xn — Xo)E G M and ||E(Xn — Xo)E||^ ^ e for n ^ N.
Suppose ^ is a am-invariant Markov measure on ^m. We will show that the averages cl(a)p converge both doubleside almost everywhere and in L1(M, t) for any operator p G L1 (A, t).
Definition 2.2. A matrix Q with non-negative entries is said to be irreducible if, for some n > 0, all entries of the matrix Q + Q2 + ... + Qn are positive (if Q is stochastic, then this is equivalent to saying that in the corresponding Markov chain any state is attainable from any other state).
Definition 2.3. A matrix P with non-negative entries is said to be strictly irreducible if P and PPT are irreducible (here PT stands for the transpose of the matrix P).
Definition 2.4. A Markov chain is said to be strictly irreducible if the corresponding transition matrix is strictly irreducible.
Let (M, t) be a non-commutative probability space, a1,..., am : M ^ M positive kernels, and a1,..., am : L1(M, t) ^ L1(M, t) their corresponding extensions. Let ^ be a am-invariant Markov measure on ^m. Then, for any element p G L1(M, t), there exists an element p G L1(M, t), such that cn(a)p ^ p in L1(M, t) norm as n ^ to.
Theorem 2.1. The following equality holds whenever a1,..., am preserve the state t: t(p) = t(p). If the measure ^ is strictly irreducible, then ajp = p, for j = 1,... ,m. If
p Є Lp(M, т), p ^ І, then cn(a)p ^ p both doubleside almost everywhere and in Lp(M, т) norm as well.
Remark 2. Theorem 2.1 generalizes Ergodic Theorems of Grigorchuk [13], Nevo [24], Nevo and Stein [25], and Bufetov [3] to the non-commutative case.
Now we discuss an operator version of Theorem 2.1.
Let (M, т) be a non-commutative probability space and al,..., am : Li (M, т) ^
Ll(M, т) be linear operators. The operators aw, sf(a), and cn(a) are introduced as above.
Recall the standard terminology. A linear operator on a Banach space is called a contraction, if its norm is not greater than one.
Definition 2.5. A linear operator a : L1(M, т) ^ L1(M, т) is said to be positive, if the
image of each non-negative element is a non-negative element.
Definition 2.6. A linear operator a : Ll(M, т) ^ Ll(M, т) is called an Ll-L^-con-traction, if HaH^ ^ І and HaH^ ^ І.
Definition 2.7. A linear operator a : Ll(M, т) ^ Ll(M, т) is said to be т-preserving, if т(p) = т(a(p)) for any p Є L1(M, т).
The following is a non-commutative Operator Ergodic Theorem:
Theorem 2.2. Let ^ be a am-invariant Markov measure on Qm, let (M, т) be a non-commutative probability space, and let al,...,am be positive Ll-L^-contractions. Then for each p Є Ll(M, т), there exists p Є Ll(M, т), such that cn(a)p ^ p, as n ^ то both doubleside almost everywhere and in Ll(M, т). If the measure ^ is strictly irreducible, then a*p = p for all i = І,..., m. If the operators al,..., am preserve the state т, then т(p) = т(p). If p ^ І, then cn(a)p ^ p, in Lp(M,т) norm as well (modulo the definition of the actions in Lp(M, т)).
It is easy to see that Theorem 2.1 is a consequence of Theorem 2.2.
The following is a generalized version of the Mean Ergodic Theorem for operators on Hilbert spaces.
Theorem 2.3. Let ^ be a am-invariant Markov measure on Qm, let H = L2(M, т), be the Hilbert space constructed using non-commutative probability space (M, т), and let the linear operators al,..., am : L2(M, т) ^ L2(M, т), be contractions. Then for each operator h Є L2(M, т), there exists an operator h Є L2(M, т), such that cn(a)h ^ h, in L2(M, т) as n ^ то. If the measure ^ is strictly irreducible, then a*h = h for all i = І,..., m.
The following is a non-commutative version of the Ergodic Theorem for operators on Lp(M, т).
Theorem 2.4. Let ^ be a am-invariant Markov measure on Qm. Let (M, т) be a non-commutative probability space and let p > І. Suppose that all operators al, . . . , am : Lp(M, т) ^ Lp(M, т), are positive contractions. Then for each p Є Lp(M, т), there exists p Є Lp(M, т), such that cn(a)p ^ p, as n ^ то both doubleside almost everywhere and in Lp(A, т). If the measure ^ is strictly irreducible, then a*p = p for all i = І,..., m.
3. Convergence of Multiparametric Cesaro Averages
In this section we discuss the convergence of time averages in Theorems 2.1-2.4. The main idea here is to use the operator a^ introduced later in this section.
Let L be a Real or Complex linear space, let al,..., am : L ^ L, be linear operators, and let ^ be a am-invariant Markov measure on ^m with initial distribution (pl,..., pm) and transition probability matrix P = (p*j). We always assume in what follows that p* > О for any i = І , . . . , m.
Consider the weighted sum of operators aw over all words of length l with given last symbol,
sf’*(a) = ^2 • (3.1)
{w:|w|=l,wi=i}
For the sake of convenience, we set aw with |w| =0 equal to an identical operator on M. Now we average sj’^a) over l = 1,..., n — 1,
1 n—1
cj,i(a) = nE sHa)-1=0
The following lemma describes relation between sj’^a) and sj+j(a).
Lemma 3.1. For any positive integer l and any j G {1,..., m}, we have
m
sj++ji(a) = Epij a sf,i(a).
i=i
< The proof of the lemma follows directly from definition (3.1) of sj’*(a) and aw. >
We can rewrite expression from the above lemma as follows:
sj+i (a) ^PiPj a. (fHa) ^
pj j v pi /.
Now we consider the space Lm, i. e., the m-th Cartesian power of L. We introduce operators
j
a^ : Lm ^ Lm defined by the formula
aM(vi, . . . , Vm) = (t>1, . . . ,!)m), (3.2)
where
y^pipij
pj
vj = 7 aj vi.
Lemma 3.2. For any v G L and n G N or (n ^ 1), we have
sn,1(a)v sn,m (a)v
<(v,---,v) =
p1 pm
< Follows by induction from the formulae 3.2 above. >
Corollary 3.3. For any v G L and n G N,
1 n^ it (cn,1(a)v cn,m(a)v^
-> aj(v,..., v) = ---------,...,---------- .
n 1=0 j V p1 Pm /
< Follows from the previous lemma. >
Applying the classical non-commutative Individual Ergodic Theorem of Goldstein [5], Theorem 1.1 to the operator a^ and using Corollary 3.3, we obtain statements on the convergence of the averages cm (a).
Lemma 3.4. Let ^ be a am-invariant Markov measure on Qm, let H = L2(M, t) be the Hilbert space constructed using non-commutative probability space (M, t), and let linear operators a1,..., am : L2(M, t) ^ L2(M, t) be contractions. Then for any operator h G L2(M, t) and i G {1,..., m}, the sequence cjj,i(a)h ^ hi in H as n ^ to, where
hi G L2(M,t), and aM(hi,..., hm) = (hi,..., hm).
< If a1,..., am are contractions on L2(M, t), then a^ is a contraction on (L2(A, t))m. Corollary 1 and the Mean Ergodic Theorem for a^ complete the proof. >
The relation cj(a) = cj’1 (a) + ... + cj,m(a) yields the following assertion.
Corollary 3.5. Under the assumptions of previous Lemma, for any h G L2(M, t), the sequence cn(a)h converges in L2(M, t) norm as n ^ to.
Corollary 3.5 proves the convergence of time averages in Theorem 2.3.
Similarly, the following results are valid.
Theorem 3.6. Let ^ be a am-invariant Markov measure on Qm. Let (M, t) be a non-commutative probability space and let p > 1. Suppose that all operators a1,...,am : Lp(M, t ) ^ Lp(M, t ) are contractions. Then, for any v G Lp(M, t ) and i G {1,...,m},
the sequence ^p-j cn,i(a)v ^ v in Lp(M, T) as n ^ to, where the operator v G Lp(M, T),
and aM(v1,.. ..«m) = («1, ..., vm).
< If a1,..., am are contractions on Lp(M, t), then a^ is a contraction on (Lp(M, t))m. The result follows from Corollary 3.3 and Lorch’s Ergodic Theorem applied to the contraction a^ (see [3] or [21, p. 73, Theorem 1.2]). >
Corollary 3.7. Under assumptions of the previous Theorem, for any v G Lp(M, T), the sequence cn(a)v converges in Lp(M, t) as n ^ to.
Now let (M,t) be a non-commutative probability space as above, and a1,...,am : L1(M, t) ^ L1 (M, t), be linear operators.
Now we specialize the construction of a^ from condition of Corollary 3.2 to the case of L1(M,t ).
Let ^ be a am-invariant Markov measure on ^m with initial distribution p = (p1,... ,pm), and transition probability matrix P = (pj), and let a^ : (L1(M, t))m ^ (L1(M, t))m, be the operator defined as before.
The space (L1 (M, t))m, can be identified with the space L1(M x {1,... ,m},T xp), where T x p is the product of the state T on the algebra A and the probability distribution p = (p1,... ,pm) on {1,..., m}. Now the operator a^ becomes an operator on the space L1(M x {1,..., m}, T x p). It is clear that, if a1,..., am are positive, then so is aj if a1,..., am are L1(A, T)-contractions, then so is aj if a1,..., am are Lro(M, t)-contractions then so is aj if a1,..., am preserve the state t , then a^ preserves the measure T x p.
Lemma 3.8. Let (M,t) be a non-commutative probability space and let a1,...,am be positive L1-L^-contractions. Then, for any operator p G L1 (M,T), and i = 1,...,m, sequence cn,i(a)p converges as n ^ to both doubleside almost everywhere and in L1 (M, T).
If p = lim (-1 ) cn’i(a)p, then . . . ,pm) = (pL . . . ,Pm).
n^-ro \p- /
To prove the lemma, we use the following standard fact [5]:
Theorem 3.9. If a is a positive L1-Lro-contraction on the non-commutative probability space (M, T), then, for any p G L1 (M, T), there exists an operator p G L1(M, T) such that n (p + ap + ... + an—1p) ^ p as n ^ to both doubleside almost everywhere and in L1 (M, t). The operator p satisfies the relation ap = p.
< (of lemma 3.8) Applying Theorem 3.9 to the operator а^, and using Corollary 3.7, we obtain statement of the Lemma. >
The Lemma 3.8 proves the convergence of time averages in Theorem 2.2.
The doubleside almost everywhere convergence in the Theorem above also holds for spaces of infinite measure; therefore, we have the following Lemma.
Lemma З.10. Let (M, т) be a von Neumann algebra with faithful normal semifinite trace т and let а-,..., ат, be positive L--L^-contractions. Then for any operator p Є L-(M, т), and i = 1,..., m, the sequence сП’^а^ converges doubleside almost everywhere as n ^ то.
Now consider contractions on Lp(M, т), for p > 1.
Lemma З.11. Let (M, т) be a non-commutative probability space, let p > 1, and let аl,...,аm be positive Lp(M,т)-contractions. For any operator p Є Lp(M,т), and i = 1,..., m, the sequence (— ^П’^а^, converges as n ^ то both doubleside almost everywhere and in Lp(M, т) to an operator pi Є Lp(M, т). We have аД^-,... , pm) = (p—,... , pm).
< If а-,..., аm are contractions, then so is а^. Applying Theorem 2.2 from [33] (see also [3] or [21, p. 73]) to the operator а^ and using Corollary 3.7, we obtain the result. >
Corollary З.12. Under the assumptions of the previous Lemma, for any operator p Є Lp(M, т), the sequence c^^p converges both doubleside almost everywhere and in Lp(M, т) norm.
Corollary 3.12 completes the proof of the convergence of time averages in Theorem 2.4.
4. Invariance of the Limit
In this section we establish the invariance of the limit in Theorems 2.1-2.4 and complete the proof of these theorems.
The following theorem allows to conclude invariance of the limit in Theorem 2.2 from the Lemma 3.8 and as consequence invariance of the limit in the Theorem 4.1.
Theorem 4.1. Let (M, т) be a non-commutative probability space and let а-,..., аm be positive L--L^-contractions on L-(M, т). Let ^ be a strictly am-invariant Markov measure on Qm. Suppose that the operators p-,..., pm Є L- (M, т), satisfy the condition
ам (pi,..., pm) = (pi,..., pm). (4.1)
Then p- = ... = pm = p and аіp = p for all i = 1,..., m.
In order to prove Theorem 4.1 we first establish a similar result for contractions on the Hilbert space H = L2(M, т).
Theorem 4.2. Let (M, т) be a non-commutative probability space, and let the linear operators а-,..., аm : L2 (M, т) ^ L2 (M, т), be contractions. Let ^ be a am-invariant Markov measure on Qm, and let h-,..., hm Є L2(M, т), be such that ам(^,..., hm) = (h-,..., hm). If measure ^ is strictly irreducible, then h- = ... = hm = h, and аіh = h, for each i = 1,..., m.
The main idea of the proof is just this: if v-, v2, and v3 are operators from L2(M,т) such that ||vi|| = ||v2|| = ||v3||, and v- = (v2 ++v3), then v- = v2 = v3. Y. Guivarc’h used this observation in [14] to prove the invariance of the limit function in his ergodic theorem.
< Let (pi,...,pm) be initial distribution of the measure ^, and let P = (pj) be the transition probability matrix of ^. For any i, j Є {1,..., m} and n Є N, denote by pj the n-step transition probability from i to j (in other words, pj = (Pn)ij).
We partition proof of the theorem 4.2 into series of steps.
Proposition 4.3. Let (M, t) be a non-commutative probability space, and let linear operators ai,...,am : L2(M, t ) ^ L2(M, t ) be contractions. Let ^ be a am-invariant Markov measure on ^m, such that the corresponding Markov chain is irreducible. Suppose that operators hi,, hm G L2(M, t) satisfy the relation
aM(hi,..., hm) = (hi,..., hm). (4.2)
Then there is an r G R, such that ||hi^ = ... = ||hm|| = r and, if pj > 0, then ||ajhj|| = r.
< Assume that ||hi | ^ ||hi||, for any i = 1,... ,m. Since equality 4.11 implies
hi = jr () aA,
V Pi /
and invariancy of initial vector (pi,..., pm) with respect to transition matrix PT implies
m
1 = ^2 EiEii. it follows from the triangle inequality that ||hi|| = 11aihi| = ||hi||, if Pii > 0.
i=i
(2)
Similarly, ||hi|| = 11hi|, for any i such that > 0, and so on. The Markov chain
corresponding to the measure ^ is irreducible; hence, ||hi|| = ... = ||hm||, and ||hj|| = ||ajhi||
if pij > 0. >
Proposition 4.4. Suppose that hi,...,hn, h G L2(M, t ) satisfy the condition ||hi | = ||h2|| = ... = ||hn|| = ||h|. Let h = cihi + ... + cnhn for some ci > 0,..., cn > 0 such that
ci + ... + cn = 1. Then hi = h2 = ... = hn = h.
< This immediately follows from equality condition for the Cauchy-Bunyakowsky-Schwartz inequality in the Hilbert space. >
Proposition 4.5. Let (M, t) be a non-commutative probability space, a : L2(M, t) ^ L2(M, t), be a contraction, and let operators hi,h2 G L2(M, t), satisfy the relations ||hi|| = ||h21 = |ahi| = ||ah2||. Then ahi = ah2, implies hi = h2.
< Indeed, if hi = h2, then (hl++h2) < || hi | by Proposition 4.4. Since a( (hl++h2)) = 11 hi ||
and a is a contraction, we arrive at a contradiction. >
In what follows, (PPT)ij stands for the (i, j)-entry of the matrix PPT.
Proposition 4.6. Let (M, t) be a non-commutative probability space, and let linear operators ai,..., am : L2(M, t) ^ L2(M, t), be contractions. Let ^ be a am-invariant Markov measure on Qm, and let hi,..., hm G L2(M, t), be such that aM(hi,..., hm) = (hi,..., hm). Let transition matrix P of ^ be irreducible. Then (PPT)j > 0 implies hi = hj.
< By Proposition 4.3, if P is irreducible, then ||hi|| = ... = ||hm||. Note that (PPT)j > 0
m / \
if and only, if there is a k for which Pik > 0, and pjk > 0. Since hk = ^ I I akh;, it
i=i V /
follows from Proposition 4.4 and 2.4 that hk = akhi = akhj, and ||hk|| = 11hi| = ||hj||, by Proposition 4.3. By Proposition 4.5 this yields hi = hj, which completes the proof. >
Combination of statements of the propositions 4.3-4.6 finishes the proof of the
theorem 4.2. >
Let us return to the proof of Theorem 4.1.
Suppose that there exist pi and pj G Li(M, t) with i, j G {1,... ,m} and ||pi — pj |l1 ^ e > 0, and satisfying equality 4.2. Since L2(M, t) is dense in the Li(M, t) in Li(M, t) norm, we can find hj G L2(M, t) satisfying ||hj — pj|l1 < e/3, for each j G {1,... , m}. Let
_ _ 1 N-i
(h^ . . . , hm) = lim — ^ . . . , hm). (4.3)
n=0
The limit in equation 4.3 exists in Li and L2 norm. Hence, h; G L2(M, t). Since a^ is contraction in Li(M, t), then ||h; — p; |l1 ^ e/3. In addition, the following equality holds a^(hi,..., hm) = (hi,..., hm). Hence, from Theorem 4.2 the following equality holds: hi = h2 = ... = hm. The latter equality implies that e ^ |pi — pj |l1 ^ |pi — hi|L1 + ||hj — pj |l1 ^
2e/3. We came to contradiction with the suggestion that e ^ ||pi — pj|l1 . Theorem 4.1 is established.
Theorem 4.7. Let (M, t) be a non-commutative probability space, let p > 1 and let ai,..., am : Lp(M, t) ^ Lp(M, t), be contractions. Let ^ be a am-invariant Markov measure on Qm, and let operators pi,..., pm G Lp(M, t), be such that a^(pi,..., pm) = (pi,..., pm). If the measure ^ is strictly irreducible, then pi = ... = pm = p, and aip = p, for any i = 1,..., m.
< The proof of the latter Theorem reproduces that of Theorem 4.2 above. The key observation is that Proposition 4.4 holds for the space Lp(M, t) since Lp(M, t) is a strictly convex space (see for example [27]). >
Theorems 4.7 and 4.1 imply Theorem 2.4.
5. Ergodic type theorem for the action of finitely generated locally free semigroups
Definition 5.1. A locally free semigroup (see [31] and references there) LFSm+i with m generators is defined as a semigroup determined by generators satisfying the following relations: LFSm+i = {gi,... ,gm : gigj = gjgi; i, j G {1,... ,m}, |i — j| > 1}.
Semigroup LFSm+i is associated with a topological Markov chain with states {1, . . . , m} and transition matrix
/ s /1, if |i — j I < 1 or i ^ j;
m = (mi,j), mi,j = < .
0, otherwise.
The set of admissible words in the chain corresponds to the Wm, the set of admissible onesided sequences corresponds to ^m and left shift am corresponds to shift on ^m. Each word Wi ... corresponds to gw = gwi ... gWn.
The correspondence w ^ gw defines a bijection between Wm and LFSm+i, and from (4.1) it follows that system (^m, am) mixes topologically, hence ergodic measure has a positive measure on cylinders corresponding to the words Wm.
Now we assume that semigroup LFSm+i acts as a semigroup with generators gi mapped
to the kernels ai acting on a tracial von Neumann Algebra (M, t). Applying Theorem 2.1, we
obtain an ergodic theorem for the action of LFSm+i.
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Received March 17, 2005.
Genady Ya. Grabarnik
IBM T. J. Watson Research Center,
19 Skyline Dr., Hawthorne, NY 10510, USA E-mail: [email protected]
Alexander A. Katz
Department of Mathematics and Computer Science,
St. John’s University,
300 Howard Ave., Staten Island, NY 10301, USA E-mail: [email protected]
Larisa A. Shwartz Department of Mathematics,
Applied Mathematics and Astronomy,
University of South Africa,
Pretoria 0003, Republic of South Africa E-mail: [email protected]
