ERGODIC THEOREMS IN SYMMETRIC SPACES OF τ

arXiv:1410.1451v1 [math.OA] 6 Oct 2014

ERGODIC THEOREMS IN SYMMETRIC SPACES OF τ −MEASURABLE OPERATORS VLADIMIR CHILIN AND SEMYON LITVINOV Abstract. In [11], a maximal ergodic theorem in noncommutative Lp −spaces was established and, among other things, corresponding individual ergodic theorems were obtained. In this article we derive maximal ergodic inequalities in noncommutative Lp −spaces and apply them to prove individual and Besicovitch weighted ergodic theorems in noncommutative Lp −spaces. Then we extend these results to noncommutative fully symmetric Banach spaces with Fatou property and non-trivial Boyd indices, in particular, to noncommutative Lorentz spaces Lp,q . Norm convergence of ergodic averages in noncommutative fully symmetric Banach spaces is also studied.

1. Introduction and preliminaries Let H be a Hilbert space over C, B(H) the algebra of all bounded linear operators in H, k · k∞ the uniform norm in B(H), I the identity in B(H). If M ⊂ B(H) is a von Neumann algebra, denote by P(M) = {e ∈ M : e = e2 = e∗ } the complete lattice of all projections in M. For every e ∈ P(M) we write e⊥ = IV− e. If T ei . ei (H) is denoted by {ei }i∈I ⊂ P(M), the projection on the subspace i∈I

i∈I

A linear operator x : Dx → H, where the domain Dx of x is a linear subspace of H, is said to be affiliated with the algebra M if yx ⊆ xy for every y from the commutant of M. Assume now that M is a semifinite von Neumann algebra equipped with a faithful normal semifinite trace τ . A densely-defined closed linear operator x affiliated with M is called τ -measurable if for each ǫ > 0 there exists such e ∈ P(M) with τ (e⊥ ) ≤ ǫ that e(H) ⊂ Dx . Let us denote by L0 (M, τ ) the set of all τ -measurable operators. It is well-known [21] that if x, y ∈ L0 (M, τ ), then the operators x + y and xy are densely-defined and preclosed. Moreover, the closures x + y (the strong sum) and xy (the strong product) and x∗ are also τ -measurable and, equipped with these operations, L0 (M, τ ) is a unital ∗-algebra over C. For every subset If X ⊂ L0 (M, τ ), the set of all self-adjoint operators in X is denoted by X h , whereas the set of all positive operators in X is denoted by X + . The partial order ≤ in Lh0 (M, τ ) is defined by the cone L+ 0 (M, τ ). The topology defined in L0 (M, τ ) by the family V (ǫ, δ) = {x ∈ L0 (M, τ ) : kxek∞ ≤ δ for some e ∈ P(M) with τ (e⊥ ) ≤ ǫ} Date: October 3, 2014. 2010 Mathematics Subject Classification. 47A35(primary), 46L52(secondary). Key words and phrases. Semifinite von Neumann algebra, maximal ergodic inequality, noncommutative ergodic theorem, bounded Besicovitch sequence, noncommutative fully symmetric space, Boyd indices. 1

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VLADIMIR CHILIN AND SEMYON LITVINOV


W (ǫ, δ) = {x ∈ L0 (M, τ ) : kexek∞ ≤ δ for some e ∈ P(M) with τ (e⊥ ) ≤ ǫ} ,

ǫ > 0, δ > 0, of (closed) neighborhoods of zero is called the measure topology (resp., the bilaterally measure topology). It is said that a sequence {xn } ⊂ L0 (M, τ ) converges to x ∈ L0 (M, τ ) in measure (bilaterally in measure) if this sequence converges to x in measure topology (resp., in bilaterally measure topology). It is known [3, Theorem 2.2] that xn → x in measure if and only if xn → x bilaterally in measure. For basic properties of the measure topology in L0 (M, τ ), see [19]. A sequence {xn } ⊂ L0 (M, τ ) is said to converge to x ∈ L0 (M, τ ) almost uniformly (a.u.) (bilaterally almost uniformly (b.a.u.)) if for every ǫ > 0 there exists such e ∈ P(M) that τ (e⊥ ) ≤ ǫ and k(x − xn )ek∞ → 0 (resp., ke(x − xn )ek∞ → 0). It is clear that every a.u. convergent (b.a.u. convergent) to x sequence in L0 (M, τ ) converges to x in measure (resp., bilaterally R ∞in measure, hence in measure). For a positive self-adjoint operator x = 0 λdeλ affiliated with M one can define  Z ∞ Z n λdτ (eλ ). τ (x) = sup τ λdeλ = n

0

0

If 1 ≤ p < ∞, then the noncommutative Lp −space associated with (M, τ ) is defined as Lp = (Lp (M, τ ), k · kp ) = {x ∈ L0 (M, τ ) : kxkp = (τ (|x|p ))1/p < ∞},

where |x| = (x∗ x)1/2 , the absolute value of x (see [24]). Naturally, L∞ = (M, k·k∞ ). If xn , x ∈ Lp and kx − xn kp → 0, then xn → x in measure [12, Theorem 3.7]. Besides, utilizing the spectral decomposition of x ∈ L+ p , it is possible to find a ∩ M such that 0 ≤ x ≤ x for each n and xn ↑ x; in sequence {xn } ⊂ L+ n p particular, kxn kp ≤ kxkp for all n and kx − xn kp → 0. Let T : L1 ∩ M → L1 ∩ M be a positive linear map that satisfies the condition (Y ) T (e) ≤ I and τ (T (e)) ≤ τ (e) ∀ e ∈ P(M) with τ (e) < ∞. It is known [25, Proposition 1] that such T admits unique positive continuous linear extensions T : Lp → Lp , 1 ≤ p < ∞, and a unique positive ultraweakly continuous linear extension T : M → M. In addition, kT (x)kp ≤ kxkp whenever x ∈ Lhp , 1 ≤ p < ∞, and kT (x)k∞ ≤ kxk∞ if x ∈ L1 ∩ Mh . In fact, T is norm-contimuous on all of M: Proposition 1.1. kT (x)k∞ ≤ kxk∞ for every x ∈ Mh . Proof. Assume first that x =

n P

k=1

λk ek , where λk ∈ R, 0 6= ek ∈ P(M), 1 ≤ k ≤ n,

and ei ej = 0 for all 1 ≤ i, j ≤ n with i 6= j. Since the trace τ is semifinite, for any (k) given k there exists a net {pα }α∈Λ ⊂ P(M), where Λ is a base of neighborhoods of (k) (k) zero of ultraweak topology ordered by inclusion, such that pα ≤ ek , 0 < τ (pα ) < (k) ∞ for all k and α, and pα → ek ultraweakly for every k. n P (k) λk pα , then {xα }α∈Λ ⊂ L1 ∩Mh and xα → x ultraweakly, If we denote xα = k=1

thus T (xα ) → T (x) ultraweakly. Since kT (xα )k∞ ≤ kxα k∞ = kxk∞ and the unit ball of M is closed in ultraweak topology, we conclude that kT (x)k∞ ≤ kxk∞ . If x ∈ Mh , due to spectral theorem, x is the uniform limit of elements of the n P λk ek , which implies that kT (x)k∞ ≤ kxk∞ .  form k=1

ERGODIC THEOREMS IN SYMMETRIC SPACES OF τ −MEASURABLE OPERATORS

3

If x ∈ Lp , 1 ≤ p ≤ ∞, is not necessarily self-adjoint, then x = Re(x) + i · Im(x), where Re(x) = 2−1 (x + x∗ ) ∈ Lhp and Im(x) = (2i)−1 (x − x∗ ) ∈ Lhp , and the inequality kT (x)kp ≤ 2kxkp follows.

We will write T ∈ AC + = AC + (M, τ ), where AC stands for absolute contraction, to indicate that T : L1 + M → L1 + M is the unique positive linear exstension of a positive linear map T : L1 ∩M → L1 ∩M satisfying condition (Y). As we know, T is continuous on each Lp , 1 ≤ p ≤ ∞. In the commutative case, similar transformations were considered in [16] where they were called L1 − L∞ − contractions. Namely, it was assumed that a T : L1 → L1 is a contaction that also contracts the L∞ −norm in L∞ ∩ L1 . The class of positive linear maps T : M → M that was considered in [11] is given by the condition (JX) kT (x)k∞ ≤ kxk∞ ∀ x ∈ M and τ (T (x)) ≤ τ (x) ∀ x ∈ L1 ∩ M+ .

It is clear that (JX) =⇒ (Y). Besides, by Lemma 1.1 in [11], a positive linear map T : M → M that satisfies (JX) uniquely extends to a positive linear contraction T in Lp , 1 ≤ p < ∞.

We shall write T ∈ DS + = DS + (M, τ ) to indicate that T : L1 +M → L1 +M is the unique positive linear extension of a positive linear map T : M → M satisfying condition (JX). Such T is often called positive Dunford-Schwartz transformation (see, for example, [26]). Clearly, 2−1 T ∈ DS + whenever T ∈ AC + . Assume that T ∈ AC + and form its ergodic averages: n 1 X k (1) Mn = Mn (T ) = T , n = 1, 2, ... n+1 k=0

The following theorem provides a maximal ergodic inequality in L1 for the ergodic averages given by (1). Theorem 1.1. [25] If T ∈ AC + , then for every x ∈ L+ 1 and ǫ > 0, there is such e ∈ P(M) that kxk1 and sup keMn (x)ek∞ ≤ ǫ. τ (e⊥ ) ≤ ǫ n Here is a corollary of Theorem 1.1, a noncommutative individual ergodic theorem of Yeadon: Theorem 1.2. [25] If T ∈ AC + , then for every x ∈ L1 the averages Mn (x) converge b.a.u. to some x b ∈ L1 .

The following noncommutative individual ergodic theorem was established in [11] as an application of their maximal ergodic theorem [11, Theorem 4.1]. Theorem 1.3. Let 1 < p < ∞. If T ∈ DS + , then (i) for every x ∈ Lp the averages Mn (x) converge b.a.u. to some x b ∈ Lp ; (ii) if p ≥ 2, these averages converge also a.u.

The next result was obtained in [18] by utilizing the notion of uniform equicontinuity at zero of a family of additive maps into L0 (M, τ ). Theorem 1.4. Let 1 < p < ∞. If T ∈ AC + , then (i) for every x ∈ Lp the averages Mn (x) converge b.a.u. to some x b ∈ Lp ; (ii) if p ≥ 2, these averages converge also a.u.

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VLADIMIR CHILIN AND SEMYON LITVINOV

In what follows, assuming that T ∈ AC + (M, τ ), we derive maximal ergodic inequalities for the averages (1) in Lp with 1 < p < ∞ (see Theorem 2.1 below). As an application, a simplified proof of Theorem 1.4 is given. Next, we use these maximal inequalities to prove Besicovitch weighted noncommutative ergodic theorem in Lp , 1 < p < ∞, (see Theorem 3.3 below). This theorem is an extension of the corresponding result of [3] for L1 . Having available an individual ergodic theorem for noncommutative Lp −spaces with 1 ≤ p < ∞, allows us to establish its validity for noncommutative fully symmetric spaces with Fatou property and non-trivial Boyd indices. As a consequence, we obtain an individual ergodic theorem in noncommutative Lorentz spaces Lp,q . The last section of the article is devoted to a study of the mean ergodic ergodic theorem in noncommutative fully symmetric spaces in the case where T ∈ DS + (M, τ ). 2. Maximal ergodic inequalities in noncommutative Lp −spaces Here is an extension of Theorem 1.1 to Lp , 1 < p < ∞: Theorem 2.1. Let 1 ≤ p < ∞, and let T ∈ AC + . Then for every x ∈ Lp and ǫ > 0 there is e ∈ P(M) such that p  kxkp and sup keMn (x)ek∞ ≤ 8ǫ. (2) τ (e⊥ ) ≤ 4 ǫ n R∞ Proof. Fix ǫ > 0. Assume that x ∈ L+ p , and let x = 0 λdeλ be its spectral decomposition. Since λ ≥ ǫ implies λ ≤ ǫ1−p λp , we have Z ∞ Z ∞ λdeλ ≤ ǫ1−p λp deλ ≤ ǫ1−p xp . ǫ

Then we can write (3)

x=

ǫ

Z

ǫ

λdeλ +

0

Z

ǫ

∞

λdeλ ≤ xǫ + ǫ1−p xp ,

Rǫ where xǫ = 0 λdeλ . As xp ∈ L1 , Theorem 1.1 entails that there exists e ∈ P(M) satisfying p  kxkp kxp k1 and sup keMn (xp )ek∞ ≤ ǫp . = τ (e⊥ ) ≤ ǫp ǫ n It follows from (3) that 0 ≤ Mn (x) ≤ Mn (xǫ ) + ǫ1−p Mn (xp ) and 0 ≤ eMn (x)e ≤ eMn (xǫ )e + ǫ1−p eMn (xp )e

for every n. Since xǫ ∈ M+ , by Proposition 1.1, the inequality (4)

kT (xǫ )k∞ ≤ kxǫ k∞ ≤ ǫ

holds, and we conclude that (5)

sup keMn (x)ek∞ ≤ ǫ + ǫ = 2ǫ. n

ERGODIC THEOREMS IN SYMMETRIC SPACES OF τ −MEASURABLE OPERATORS

5

If x ∈ Lp , then x = (x1 − x2 ) + i(x3 − x4 ), where xj ∈ L+ p and kxj kp ≤ kxkp for every j = 1, . . . , 4. As we have shown, there exist ej ∈ P(M) such that  p  p kxj kp kxkp τ (e⊥ ) ≤ ≤ and sup kej Mn (xj )ej k∞ ≤ 2ǫ, j ǫ ǫ n j = 1, . . . , 4. Taking into consideration Theorem 1.1 for p = 1, we see that the 4 V ej satisfies (2) for every 1 ≤ p < ∞.  projector e = j=1

Remark 2.1. In the terminology of [11, Section 3], Theorem 2.1 asserts that the sequence {Mn } is of weak type (p, p). To refine Theorem 2.1 when p ≥ 2 we turn to the following fundamental result of Kadison [13]. Theorem 2.2 (Kadison’s inequality). Let S : M → M be a positive linear map such that S(I) ≤ I. Then S(x)2 ≤ S(x2 ) for every x ∈ Mh . We will need the following lemma (for a proof, see the proof of [3, Theorem 2.7] or [18, Theorem 3.1]). Lemma 2.1. Let {amn }∞ m,n=1 ⊂ L0 (M, τ ) be such that for any n the sequence ∞ {amn }∞ m=1 converges in measure to some an ∈ L0 (M, τ ). Then there exists {amk n }k,n=1 such that for any n we have amk n → an a.u. as k → ∞. Proposition 2.1 (cf. [11], proof of Remark 6.5). Assume that 2 ≤ p < ∞ and T ∈ AC + . Then for every x ∈ Lhp and ǫ > 0, there exists e ∈ P(M) such that τ (e⊥ ) ≤ ǫ and keMn (x)2 ek∞ ≤ keMn (x2 )ek∞ , n = 1, 2, . . . R∞ Proof. Let x = −∞ λdeλ be the spectral decomposition of x ∈ Lhp , and let Rm xm = −m λdeλ . Then, since x ∈ Lp , we clearly have kx − xm kp → 0. Besides kx2 − x2m kp/2 → 0, so kMn (x2 ) − Mn (x2m )kp/2 → 0 for every n, which implies that Mn (x2m ) → Mn (x2 ) in measure, n = 1, 2, . . . Also kMn (x) − Mn (xm )kp → 0 for every n, hence Mn (xm ) → Mn (x) in measure and Mn (xm )2 → Mn (x)2 in measure, n = 1, 2, . . . In view of Lemma 2.1, it is possible to find a subsequence {xmk } ⊂ {xm } such that Mn (x2mk ) → Mn (x2 ) and Mn (xmk )2 → Mn (x)2 a.u., n = 1, 2, . . .

Then one can construct such e ∈ P(M) that τ (e⊥ ) ≤ ǫ and

keMn(x2mk )ek∞ → keMn (x2 )ek∞ and keMn (xmk )2 ek∞ → keMn (x)2 ek∞

for every n. Since, by Kadison’s inequality, keMn (xmk )2 ek∞ ≤ keMn (x2mk )ek∞ , k, n = 1, 2, . . . , the result follows.



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VLADIMIR CHILIN AND SEMYON LITVINOV

Theorem 2.3. Let 2 ≤ p < ∞ and T ∈ AC + . Then for every x ∈ Lp and ǫ > 0 there is such e ∈ P(M) that p  √ kxkp and sup kMn (x)ek∞ ≤ 2 2ǫ. (6) τ (e⊥ ) ≤ 4 ǫ n Proof. Assume first that x ∈ Lhp . Since x2 ∈ L+ p/2 , by the proof of Theorem 2.1, there exists e1 ∈ P(M) such that  2 p p/2  kx kp/2 kxkp τ (e⊥ ) ≤ and sup ke1 Mn (x2 )e1 k∞ ≤ 2ǫ2 . = 1 ǫ2 ǫ n By Proposition 2.1, there is e2 ∈ P(M) such that p  kxkp ⊥ and sup ke2 Mn (x)2 e2 k∞ ≤ sup ke2 Mn (x2 )e2 k∞ . τ (e2 ) ≤ ǫ n n p  kxkp and Then, letting e = e1 ∧ e2 , we obtain τ (e⊥ ) ≤ 2 ǫ sup kMn (x)ek∞ n

=

 1/2 2 = sup kMn (x)ek∞ = n

1/2  1/2  √ sup keMn (x)2 ek∞ ≤ sup keMn (x2 )ek∞ ≤ 2ǫ. n

n

If x ∈ Lp , then x = x1 + ix2 , where xj ∈ Lhp and kxj kp ≤ kxkp , j = 1, 2, and we arrive at (6).  Let us now derive Theorem 1.4 using maximal ergodic inequalities given in Theorems 2.1 and 2.3. Lemma 2.2 (see [2], Lemma 1.6). Let X be a linear space, and let Sn : X → L be a sequence of additive maps. Assume that x ∈ X is such that for every ǫ > 0 there exists a sequence {xk } ⊂ X and a projection e ∈ P(M) satisfying the following conditions: (i) the sequence {Sn (x + xk )} converges a.u. (b.a.u.) as n → ∞ for each k; (ii) τ (e⊥ ) ≤ ǫ; (iii) supn kSn (xk )ek∞ → 0 (resp., supn kean (xk )ek∞ → 0) as k → ∞. Then the sequence {Sn (x)} also converges a.u. (resp., b.a.u.) Corollary 2.1. Let 1 ≤ p < ∞ (2 ≤ p < ∞) and T ∈ AC + . Then the set {x ∈ Lp : {Mn (x)} converges b.a.u.} is closed in Lp .

(resp., {x ∈ Lp : {Mn (x)} converges a.u.})

Proof. Denote C = {x ∈ Lp : {an (x)} converges b.a.u.}. Fix ǫ > 0. Theorem 2.1 implies that for every given k ∈ N there is such γk > 0 that for every x ∈ Lp with kxkp < γk it is possible to find ek,x ∈ P(M) for which τ (e⊥ k,x ) ≤

1 ǫ and sup kek,x an (x)ek,x k∞ ≤ . 2k k n

ERGODIC THEOREMS IN SYMMETRIC SPACES OF τ −MEASURABLE OPERATORS

7

Pick x ∈ C, the closure of C in Lp . Given k, let yk ∈ C be such that kyk − xkp < γk . Denoting yk − x = xk , choose a sequence {ek } ⊂ P(M) to be such that ǫ 1 τ (e⊥ , k = 1, 2, ... . k ) ≤ k and sup kek an (xk )ek k∞ ≤ 2 k n V ek , we have Then we have x + xk = yk ∈ C for every k. Also, letting e = k≥1

τ (e⊥ ) ≤ ǫ and sup keMn (xk )ek∞ n

1 ≤ . k

Consequently, Lemma 2.2 yields x ∈ C. Analogously, applying Theorem 2.3 instead of Theorem 2.1, we obtain the remaining part of the statement.  Now we can prove Theorem 1.4. Proof. (ii) Since the map T generates a contraction in the real Hilbert space (Lh2 , (·, ·)τ ) [25, Proposition 1], where (x, y)τ = τ (xy), x, y ∈ Lh2 , it is easy to verify that the set H0 = {x ∈ Lh2 : T (x) = x} + {x − T (x) : x ∈ L2h }

is dense in (Lh2 , k · k2 ) (see, for example [10, Ch.VIII, §5]). Therefore, because the set Lh2 ∩ M is dense in Lhp and T contracts Lhp , we conclude that the set H1 = {x ∈ Lh2 : T (x) = x} + {x − T (x) : x ∈ L2h ∩ M}

is also dense in (Lh2 , k · k2). Besides, if y = x − T (x), x ∈ L2h ∩ M, then the sequence Mn (y) = (n + 1)−1 (x − T n+1 (x)) converges to zero with respect to the norm k · k∞ , hence a.u. Therefore H1 + iH1 is a dense in L2 subset on which the averages Mn converge a.u. This, by Corollary 2.1, implies that {Mn (x)} converges a.u. for all x ∈ L2 . Further, since the set Lp ∩ L2 is dense in Lp , Corollary 2.1 implies that the sequence {Mn (x)} converges a.u. for each x ∈ Lp (to some x b ∈ L0 (M, τ )). Then {Mn (x)} converges to x b in measure. Since Mn (x) ∈ Lp and kMn (x)kp ≤ 2, n = 1, 2, . . . , by Theorem 1.2 in [3], x b ∈ Lp . (i) By part (ii), we have b.a.u. convergence of the sequence {Mn (x)} for all x ∈ L2 . But Lp ∩ L2 is dense in Lp , and Corollary 2.1 entails b.a.u. convergence of the averages Mn (x) for all x ∈ Lp . Remembering that b.a.u. convergence implies convergence in measure (see Section 1), we conclude, as in part (ii), that the b.a.u. limit of the sequence {Mn (x)} belongs to Lp .  3. Besicovitch weighted ergodic theorem in noncommutative Lp −spaces

Everywhere in this section T ∈ AC + . Assume that a sequence of complex numbers {βk }∞ k=0 is such that |βk | ≤ C for every k. Let us denote n

(7)

Mβ,n =

1 X βk T k . n+1 k=0

Theorem 3.1. If 1 ≤ p < ∞, then for every x ∈ Lp and ǫ > 0 there is e ∈ P(M) such that p  kxkp ⊥ and sup keMβ,n(x)ek∞ ≤ 192Cǫ. (8) τ (e ) ≤ 16 ǫ n

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VLADIMIR CHILIN AND SEMYON LITVINOV

Proof. We have x = (x1 − x2 ) + i(x3 − x4 ), where xj ∈ L+ p and kxj kp ≤ kxkp for each j = 1, 2, 3, 4. For a given j, by Theorem 2.1, there is ej ∈ P(M) such that p  kxj kp ⊥ and sup kej Mn (xj )ej k∞ ≤ 8ǫ. τ (ej ) ≤ 4 ǫ n Since 0 ≤ Reβk + C ≤ 2C, 0 ≤ Imβk + C ≤ 2C, and T k (xj ) ≥ 0 for every k and j, it follows from the decomposition n n i X 1 X k (Reβk + C)T + (Imβk + C)T k − C(1 + i)Mn (9) Mβ,n = n+1 n+1 k=0

k=0

that sup kej Mβ,n (xj )ej k∞ ≤ 6C sup kej Mn (xj )ej k∞ ≤ 48Cǫ, j = 1, 2, 3, 4. n

n

4 V

Now, letting e =

ej , we obtain (8).



j=1

Theorem 3.2. If 2 ≤ p < ∞, then for every x ∈ Lp and ǫ > 0 there is e ∈ P(M) such that  p √ √ √ kxkp ⊥ (10) τ (e ) ≤ 8 and sup kMβ,n(x)ek∞ ≤ 4 2C( 2 + C)ǫ. ǫ n Proof. Referring to decomposition (9), let us denote n n 1 X 1 X (R) (I) Mβ,n = (Reβk + C)T k , Mβ,n = (Imβk + C)T k . n+1 n=1 k=0

k=0

Lhp

We have x = x1 + ix2 , where xj ∈ and kxj kp ≤ kxkp , j = 1, 2. Since x21 ∈ L+ , it follows from the proof of Theorem 2.1 that there is f1 ∈ P(M) such p/2 that p p/2   2 kx1 kp/2 kx1 kp and sup kf1 Mn (x21 )f1 k∞ ≤ 2ǫ2 . = τ (f1⊥ ) ≤ ǫ2 ǫ n Therefore we have (R)

sup kf1 Mβ,n (x21 )f1 k∞ ≤ 4Cǫ2

(I)

sup kf1 Mβ,n (x21 )f1 k∞ ≤ 4Cǫ2 .

and

n

(R)

n

(I)

Since Mβ,n : M → M and Mβ,n : M → M are positive linear maps that are (R)

(I)

continuous in (Lp , k · kp ) and such that (2C)−1 Mβ,n (I) ≤ I and (2C)−1 Mβ,n (I) ≤ I for each n, we can find, as in Proposition 2.1, such g11 , g12 ∈ P(M) ⊥ with τ (g1j ) ≤ (kx1 kp /ǫ)p , j = 1, 2, that (R)

(R)

sup kg11 Mβ,n (x1 )2 g11 k∞ ≤ sup kg11 Mβ,n (x21 )g11 k∞ n

n

and

(I)

(I)

sup kg12 Mβ,n (x1 )2 g12 k∞ ≤ sup kg12 Mβ,n (x21 )g12 k∞ . n

n

Furthemore, Proposition 2.1 implies that there exists h1 ∈ P (M) with p τ (h⊥ 1 ) ≤ (kx1 kp /ǫ) such that sup kh1 Mn (x1 )2 h1 k∞ ≤ sup kh1 Mn (x21 )h1 k∞ . n

n

ERGODIC THEOREMS IN SYMMETRIC SPACES OF τ −MEASURABLE OPERATORS

Setting e1 = f1 ∧ g11 ∧ g12 ∧ h1 , we see that

p τ (e⊥ 1 ) ≤ 4(kx1 kp /ǫ) , sup kMn (x1 )e1 k∞ ≤

9

√ 2ǫ,

n

(R) sup kMβ,n (x1 )e1 k∞ n

√ √ (I) ≤ 2 Cǫ, and sup kMβ,n (x1 )e1 k∞ ≤ 2 Cǫ. n

Now it follows from the decomposition (9) that √ √ √ sup kMβ,n (x1 )e1 k∞ ≤ 2 2C( 2 + C)ǫ. n

p Similarly, we can find e2 ∈ P(M) with τ (e⊥ 2 ) ≤ 4(kx2 kp /ǫ) such that √ √ √ sup kMβ,n (x2 )e2 k∞ ≤ 2 2C( 2 + C)ǫ. n

Finally, e = e1 ∧ e2 ∈ P(M) satisfies (10).



Using maximal ergodic inequalities given in Theorems 3.1 and 3.2 as in the proof of Corollary 2.1, we obtain the following. Corollary 3.1. Let 1 ≤ p < ∞ (2 ≤ p < ∞). Then the set

{x ∈ Lp : {Mβ,n (x)} converges b.a.u.}

is closed in Lp .

(resp., {x ∈ Lp : {Mβ,n (x)} converges a.u.})

Let C1 = {z ∈ C : |z| = 1} be the unit circle inPC. A function P : Z → C s is said to be a trigonometric polynomial if P (k) = j=1 zj λkj , k ∈ Z, for some s s ∞ s ∈ N, {zj }1 ⊂ C, and {λj }1 ⊂ C1 . A sequence {βk }k=0 ⊂ C is called a bounded Besicovitch sequence if (i) |βk | ≤ C < ∞ for all k; (ii) for every ǫ > 0 there exists a trigonometric polynomial P such that n 1 X lim sup |βk − P (k)| < ǫ. n+1 n k=0

Assume now that M has a separable predual. The reason for this assumption is that our argument essentially relies on Theorem 1.22.13 in [20]. Since L1 ∩ M ⊂ L2 , using Theorem 1.4 for p = 2 (or [5, Theorem 3.1]) and repeating steps of the proof of [3, Lemma 4.2], we arrive at the following. Proposition 3.1. For any trigonometric polynomial P and x ∈ L1 ∩ M, the averages n 1 X P (k)T k (x) n+1 k=0 converge a.u. Then it is easy to verify the following (see the proof of [3, Theorem 4.4]). Proposition 3.2. If {βk } is a bounded Besicovitch sequence, then the averages (7) converge a.u. for every x ∈ L1 ∩ M. Here is an extension in [3, Theorem 4.6] to Lp −spaces. Theorem 3.3. Assume that M has a separable predual. Let 1 < p < ∞, and let {βk } be a bounded Besicovitch sequence. If T ∈ AC + , then for every x ∈ Lp the averages (7) converge b.a.u. to some x b ∈ Lp . If p ≥ 2, these averages converge a.u.

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VLADIMIR CHILIN AND SEMYON LITVINOV

Proof. In view of Proposition 3.2 and Corollary 3.1, we only need to recall that the set L1 ∩ M is dense in Lp . The inclusion x b ∈ Lp follows as in the proof of Theorem 1.4.  4. Individual ergodic theorems in noncommutative fully symmetric spaces Let x ∈ L0 (M, τ ), and let {eλ }λ≥0 be the spectral family of projections for the absolute value |x| of x. If t > 0, then the t-th generalized singular number of x (see [12]) is defined as µt (x) = inf{λ > 0 : τ (e⊥ λ ) ≤ t}. A Banach space (E, k · kE ) ⊂ L0 (M, τ ) is called fully symmetric if the conditions x ∈ E, y ∈ L0 (M, τ ),

Zs 0

µt (y)dt ≤

Zs

µt (x)dt for all s > 0

0

imply that y ∈ E and kykE ≤ kxkE . It is known [6] that if (E, k · kE ) is a fully symmetric space, xn , x ∈ E, and kx − xn kE → 0, then xn → x in measure. A fully symmetric space (E, k · kE ) is said to possess Fatou property if the conditions xα ∈ E + , xα ≤ xβ for α ≤ β, and sup kxα kE < ∞ α

imply that there exists x = sup xα ∈ E and kxkE = sup kxα kE . The space (E, k·kE ) α

α

is said to have order continuous norm if kxα kE ↓ 0 whenever xα ∈ E and xα ↓ 0. Let L0 (0, ∞) be the linear space of all (equivalence classes of) almost everywhere finite complex-valued Lebesgue measurable functions on the interval (0, ∞). We identify L∞ (0, ∞) with the commutative von Neumann algebra acting on the Hilbert space L2 (0, ∞) via multiplication by the elements from L∞ (0, ∞) with the trace given by the integration with respect to Lebesgue measure. A Banach space E ⊂ L0 (0, ∞) is called fully symmetric Banach space on (0, ∞) if the condition above holds with respect to the von Neumann algebra L∞ (0, ∞). Let E = (E(0, ∞), k · kE ) be a fully symmetric function space. For each s > 0 let Ds : E(0, ∞) → E(0, ∞) be the bounded linear operator given by Ds (f )(t) = f (t/s), t > 0. The Boyd indices pE and qE are defined as pE = lim

s→∞

log s log s , qE = lim . s→+0 log kDs kE log kDs kE

It is known [17, Ch.2, Proposition 2.b.2, Theorem 2.b.3] that 1 ≤ pE ≤ qE ≤ ∞ and if 1 ≤ p < pE ≤ qE < q ≤ ∞, then (11)

Lp (0, ∞) ∩ Lq (0, ∞) ⊂ E(0, ∞) ⊂ Lp (0, ∞) + Lq (0, ∞),

with the inclusion maps being continuous relative to kf kLp∩Lq = max{kf kp , kf kq } and kf kLp+Lq = inf{kgkp + khkq : f = g + h, g ∈ Lp (0, ∞), h ∈ Lq (0, ∞)}. A fully symmetric function space is said to have non-trivial Boyd indices if 1 < pE ≤ qE < ∞. Note that the space Lp (0, ∞), 1 < p < ∞, has non-trivial Boyd indices: pLp (0,∞) = qLp (0,∞) = p [1, Ch.4, §4, Theorem 4.3].

ERGODIC THEOREMS IN SYMMETRIC SPACES OF τ −MEASURABLE OPERATORS

11

If E(0, ∞) is a fully symmetric function space, define E(M) = E(M, τ ) = {x ∈ L0 (M, τ ) : µt (x) ∈ E} and set kxkE(M) = kµt (x)kE , x ∈ E(M).

It is shown in [6] that (E(M), k · kE(M) ) is a fully symmetric space. If 1 ≤ p < ∞ and E = Lp (0, ∞), the space (E(M), k ·kE(M) ) coincides with the noncommutative Lp −space (Lp (M, τ ), k · kp ) because ∞ 1/p Z kxkp =  µpt (x)dt = kxkE(M) 0

[24, Proposition 2.4]. It was shown in [4, Proposition 2.2] that if M is non-atomic, then every noncommutative fully symmetric (E, k · kE ) ⊂ L0 (M, τ ) is of the form (E(M), k · kE(M) ) for a suitable fully symmetric function space E(0, ∞). Let Lp,q (0, ∞), 1 ≤ p, q < ∞, be the classical function Lorentz space, that is, the space of all such functions f ∈ L0 (0, ∞) that 1/q ∞ Z dt < ∞. kf kp,q =  (t1/p µt (f ))q  t 0

It is known that for q ≤ p the space (Lp,q (0, ∞), k·kp,q ) is a fully symmetric function space with Fatou property and order continuous norm. In addition, Lp,p = Lp . In the case 1 < p < q, the function k · kp,q is a quasi-norm on Lp,q (0, ∞), but there exists a norm || · ||(p,q) on Lp,q (0, ∞) that is equivalent to the norm k · kp,q and such that (Lp,q (0, ∞), || · ||(p,q) ) is a fully symmetric function space with Fatou property and order continuous norm [1, Ch.4, §4]. In addition, if 1 ≤ q ≤ p < ∞ (1 < p < ∞, 1 ≤ q < ∞), then p(Lp,q (0,∞),k·kp,q ) = q(Lp,q (0,∞),k·kp,q ) = p [1, Ch.4, §4, Theorem 4.3] (resp., p(Lp,q (0,∞),k·k(p,q) ) = q(Lp,q (0,∞),k·k(p,q) ) = p [1, Ch.4, §4, Theorem 4.5]).

Using function Lorentz space (Lp,q (0, ∞), k · kp,q ) ((Lp,q (0, ∞), k · k(p,q) )), one can define noncommutative Lorentz space   1/q ∞   Z   1/p q dt   p > 1). In addition, the norm k · kp,q (resp., || · ||(p,q) ) is order continuous [7, Proposition 3.6] and satisfies Fatou property [8, Theorem 4.1]. These spaces were first introduced in the paper [14]. Following [15], a Banach couple (X, Y ) is a pair of Banach spaces, (X, k · kX ) and (Y, k · kY ), which are algebraically and topologically embedded in a Housdorff

12

VLADIMIR CHILIN AND SEMYON LITVINOV

topological space. With any Banach couple (X, Y ) the following Banach spaces are associated: (i) the space X ∩ Y equipped with the norm kxkX∩Y = max{kxkX , kxkY }, x ∈ X ∩ Y ; (ii) the space X + Y equipped with the norm kxkX+Y = inf{kykX + kzkY : x = y + z, y ∈ X, z ∈ Y }, x ∈ X + Y. Let (X, Y ) be a Banach couple. A linear map T : X + Y → X + Y is called a bounded operator for the couple (X, Y ) if both T : X → X and T : Y → Y are bounded. Denote by B(X, Y ) the linear space of all bounded linear operators for the couple (X, Y ). Equipped with the norm kT kB(X,Y ) = max{kT kX→X , kT kY →Y }, this space is a Banach space. A Banach space Z is said to be intermediate for a Banach couple (X, Y ) if X ∩ Y ⊂ Z ⊂ X + Y with continuous inclusions. If Z is intermediate for a Banach couple (X, Y ), then it is called an interpolation space for (X, Y ) if every bounded linear operator for the couple (X, Y ) acts boundedly from Z to Z. If Z is an interpolation space for a Banach couple (X, Y ), then there exists a constant C > 0 such that kT kZ→Z ≤ CkT kB(X,Y ) for all T ∈ B(X, Y ). An interpolation space Z for a Banach couple (X, Y ) is called an exact interpolation space if kT kZ→Z ≤ kT kB(X,Y ) for all T ∈ B(X, Y ). Every fully symmetric function space E = E(0, ∞) is an exact interpolation space for the Banach couple (L1 (0, ∞), L∞ (0, ∞)) [15, Ch.II, §4, Theorem 4.3]. We need the following noncommutative interpolation result for the spaces E(M).

Theorem 4.1. [6, Theorem 3.4] Let E, E1 , E2 be fully symmetric function spaces on (0, ∞). Let M be a von Neumann algebra with a faithful semifinite normal trace. If (E1 , E2 ) is a Banach couple and E is an exact interpolation space for (E1 , E2 ), then E(M) is an exact interpolation space for the Banach couple (E1 (M), E2 (M)). It follows now from [15, Ch.II, Theorem 4.3] and Theorem 4.1 that every noncommutative fully symmetric space E(M), where E = E(0, ∞) is a fully symmetric function space, is an exact interpolation space for the Banach couple (L1 (M), M). Let T ∈ AC + (M, τ ). Let E(0, ∞) be a fully symmetric function space. Since the noncommutative fully symmetric space E(M) is an exact interpolation space for the Banach couple (L1 (M, τ ), M), we conclude that T (E(M)) ⊂ E(M) and T is a positive continuous linear map on (E(M), k · kE(M) ). Thus n

Mn (x) =

1 X k T (x) ∈ E(M) n+1 k=0

for each x ∈ E(M) and all n. Besides, the inequalities

kT (x)k1 ≤ kxk1 , x ∈ Lh1 , kT (x)k∞ ≤ kxk∞ , x ∈ Mh

imply that sup kMn kLh1 →Lh1 ≤ 1 and sup kMn kMh →Mh ≤ 1. n≥1

n≥1

ERGODIC THEOREMS IN SYMMETRIC SPACES OF τ −MEASURABLE OPERATORS

13

Therefore, due to the decomposition x = Re(x) + i · Im(x), we have sup kMn kL1 →L1 ≤ 2 and sup kMn kM→M ≤ 2. n≥1

n≥1

Since the noncommutative fully symmetric space E(M) is an exact interpolation space for the Banach couple (L1 (M, τ ), M), we have (12)

sup kMn kE(M)→E(M) < 2.

n≥1

If T ∈ DS + (M, τ ), then (13)

sup kMn kE(M)→E(M) ≤ 1.

n≥1

The following theorem is a version of Theorem 1.4 for noncommutative fully symmetric Banach spaces with non-trivial Boyd indices. Theorem 4.2. Let E(0, ∞) be a fully symmetric function space with Fatou property and non-trivial Boyd indices. If T ∈ AC + (M, τ ), then for any given x ∈ E(M, τ ), (i) the averages Mn (x) converge b.a.u. to some x b ∈ E(M, τ ); (ii) if pE(0,∞) > 2, these averages converge a.u.

Proof. (i) According to [17, Theorem 2.b.3] (see (11)) and Theorem 4.1, there exist such 1 < p, q < ∞ that E(M, τ ) ⊂ Lp (M, τ ) + Lq (M, τ ).

Then x = x1 + x2 , where x1 ∈ Lp (M, τ ), x2 ∈ Lq (M, τ ), and, by Theorem 1.4(i), there exist such x b1 ∈ Lp (M, τ ) and x b1 ∈ Lq (M, τ ) that Mn (xj ) converge b.a.u. to x bj , j = 1, 2. Therefore Mn (x) → x b=x b1 + x b2 ∈ Lp (M, τ ) + Lq (M, τ ) ⊂ L0 (M, τ )

b.a.u., hence Mn (x) → x b in measure. Since E(M) satisfies Fatou property, the unit ball of E(M) is closed in the measure topology [8, Theorem 4.1], and (12) implies that x b ∈ E(M). (ii) If pE(0,∞) > 2, then the numbers p and q in part (i) can be chosen such that 2 < p, q < ∞. Utilizing Theorem 1.4(ii) and repeating the argument above, we conclude that the averages Mn (x) converge to x b a.u. 

Since any function Lorentz space E = Lp,q (0, ∞) with 1 < p < ∞ and 1 ≤ q < ∞ has non-trivial Boyd indices pE = qE = p, we have the following corollary of Theorem 4.2. Theorem 4.3. Let 1 < p < ∞ and 1 ≤ q < ∞. Then, given x ∈ Lp,q (M, τ ), (i) the averages Mn (x) converge b.a.u. to some x b ∈ Lp,q (M, τ ); (ii) if p > 2, these averages converge a.u.

Remark 4.1. If 1 ≤ q ≤ p, then Lp,q (M, τ ) ⊂ Lp,p (M, τ ) = Lp (M, τ ) (see [14] and [23, Lemma 1.6]). Then it follows directly from Theorem 1.4 along with the ending of the proof of part (i) of Theorem 3.1 that for every x ∈ Lp,q (M, τ ) the averages Mn (x) converge to some x b ∈ Lp,q (M, τ ) b.a.u. (a.u. for p ≥ 2).

Utilizing Theorem 3.3 and following the proof of Theorem 3.1, we arrive at the following version of Theorem 3.3 for noncommutative fully symmetric Banach spaces with non-trivial Boyd indices.

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VLADIMIR CHILIN AND SEMYON LITVINOV

Theorem 4.4. Let E = E(0, ∞) be a fully symmetric function space with Fatou property and non-trivial Boyd indices. Then for every x ∈ E(M) the averages (7) converge b.a.u. to some x b ∈ E(M). In addition, if pE(0,∞) > 2, then averages (7) converge a.u.

As a corollary of Theorem 4.4, we obtain Besicovitch weighted ergodic theorem in noncommutative Lorentz spaces. Theorem 4.5. Let 1 < p < ∞ and 1 ≤ q < ∞. Then for any x ∈ Lp,q (M, τ ) the averages (7) converge b.a.u. to some x b ∈ Lp,q (M, τ ). If p > 2, these averages converge a.u.

Remark 4.2. If 1 ≤ q ≤ p, then Lp,q (M, τ ) ⊂ Lp (M, τ ), and it follows directly from Theorem 3.3 along with the ending of the proof of part (i) of Theorem 3.1 that for every x ∈ Lp,q (M, τ ) the averages (7) converge to some x b ∈ Lp,q (M, τ ) b.a.u. (a.u. for p ≥ 2). 5. Mean ergodic theorems in noncommutative fully symmetric spaces Let M be a von Neumann algebra with a faithful normal semifinite trace τ . In [24] the following mean ergodic theorem for noncommutative fully symmetric spaces was proven. Theorem 5.1. Let E(M) be a noncommutative fully symmetric space such that (i) L1 ∩ M is dense in E(M); (ii) ken kE(M) → 0 for any sequence of projections {en } ⊂ L1 ∩ M with en ↓ 0; (iii) ken kE(M) /τ (en ) → 0 for any increasing sequence of projections {en } ⊂ L1 ∩ M with τ (en ) → ∞. Then, given x ∈ E(M) and T ∈ DS + (M, τ ), there exists x b ∈ E(M) such that kb x − Mn (x)kE(M) → 0.

It is clear that any noncommutative fully symmetric space (E(M), k·kE(M) ) with order continuous norm satisfies conditions (i) and (ii) of Theorem 5.1. Besides, in the case of noncommutative Lorentz space Lp,q (M, τ ), the inequality p > 1 together with  1/q p τ (e)1/p , e ∈ L1 ∩ P(M) kekp,q = q imply that condition (iii) is also satisfied. Therefore Theorem 5.1 entails the following. Corollary 5.1. Let 1 < p < ∞, 1 ≤ q < ∞, T ∈ DS + , and x ∈ Lp,q (M, τ ). Then there exists x b ∈ Lp,q (M, τ ) such that kb x − Mn (x)kp,q → 0.

The next theorem asserts convergence in the norm k · kE(M) of the averages Mn (x) for any noncommutative fully symmetric space (E(M), k·kE(M) ) with order continuous norm, under the assumption that τ (I) < ∞. Theorem 5.2. Let τ be finite, and let E(M, τ ) be a noncommutative fully symmetric space with order continuous norm. Then for any x ∈ E(M) and T ∈ DS + there exists x b ∈ E(M) such that kb x − Mn (x)kE(M) → 0.

Proof. Since the trace τ is finite, we have M ⊂ E(M, τ ). As the norm k · kE(M) is order continuous, applying spectral theorem for selfadjoint operators in E(M, τ ),

ERGODIC THEOREMS IN SYMMETRIC SPACES OF τ −MEASURABLE OPERATORS

15

we conclude that M is dense in (E(M, τ ), k · kE(M) ). Therefore M+ is a fundamental subset of (E(M, τ ), k · kE(M) ), that is, the linear span of M+ is dense in (E(M, τ ), k · kE(M) ). Show that the sequence {Mn (x)} is weakly sequentially compact for every x ∈ M+ . Without loss of generality, assume that 0 ≤ x ≤ I. Since T ∈ DS + , we have 0 ≤ Mn (x) ≤ Mn (I) ≤ I for any n. By [9, Proposition 4.3], given y ∈ E + (M, τ ), the set {a ∈ E(M, τ ) : 0 ≤ a ≤ y} is weakly compact in (E(M, τ ), k · kE(M) ), which implies that the sequence {Mn (x)} is weakly sequentially compact in (E(M, τ ), k · kE(M) ). Since sup kMn kE(M)→E(M) ≤ 1 (see (13)) and n≥1

n

T (x)

0≤ n

E(M)

≤

kxkE(M) →0 n

whenever x ∈ M+ , the result follows by Corollary 3 in [10, Ch.VIII, §5].



Remark 5.1. In the commutative case, Theorem 5.2 was established in [22]. It was also shown that if M = L∞ (0, 1), then for every fully symmetric Banach function space E(0, 1) with the norm that is not order continuous there exists such T ∈ DS + and x ∈ E(M) that the averages Mn (x) do not converge in (E(M), k · kE(M) ). The following proposition is a version of Theorem 5.1 for noncommutative fully symmetric space with order continuous norm with condition (iii) being replaced by non-triviality of the Boyd indices of E(0, ∞). Note that we do not require T to be positive. Proposition 5.1. Let E(0, ∞) be a fully symmetric function space with non-trivial Boyd indices and order continuous norm. Then for any x ∈ E(M, τ ) and T ∈ DS(M, τ ) there exists such x b ∈ E(M, τ ) that kb x − Mn (x)kE(M) → 0.

Proof. By [17, Theorem 2.b.3], it is possible to find such 1 < p, q < ∞ that Lp (0, ∞) ∩ Lq (0, ∞) ⊂ E(0, ∞) ⊂ Lp (0, ∞) + Lq (0, ∞)

with continuous inclusion maps. Therefore Theorem 4.1 implies that the space L = Lp (M, τ ) ∩ Lq (M, τ ) is continuously embedded in E(M, τ ). Besides, it follows as in Theorem 5.2 that L+ is a fundamental subset of (E(M, τ ), k · kE(M) ). Show that for everty x ∈ L+ the sequence {Mn (x)} is weakly sequentially compact in (E(M, τ ), k · kE(M) ). Since p, q > 1, the spaces Lp (M, τ ) and Lq (M, τ ) are reflexive. Taking into account that T ∈ DS and x ∈ Lp (M, τ ) ∩ Lq (M, τ ), we conclude that the averages {Mn (x)} converge in (Lp (M, τ ), k · kp ) and in (Lq (M, τ ), k·kq ) to x b1 ∈ Lp (M, τ ) and to x b2 ∈ Lq (M, τ ), respectively [10, Ch.VIII, §5, Corollary 4]. This implies that the sequence {Mn (x)} converges to x b1 and to x b2 in measure, hence x b1 = x b2 := x b. Since L is continuously embedded in E(M, τ ), the sequence {Mn (x)} converges to x b with respect to the norm k · kE(M) , thus, it is weakly sequentially compact in (E(M, τ ), k · kE(M) ). Now we can proceed as in the ending of the proof of Theorem 5.2. 

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VLADIMIR CHILIN AND SEMYON LITVINOV

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