IN L 2 OVER A VON NEUMANN ALGEBRA

VOL. LXIX 1995 FASC. 2

THE UNCONDITIONAL POINTWISE CONVERGENCE OF ORTHOGONAL SERIES

IN L 2 OVER A VON NEUMANN ALGEBRA

BY

EWA H E N S Z, RYSZARD J A J T E

AND

ADAM P A S Z K I E W I C Z ( L ´ OD´ Z)

1. Introduction. The paper is devoted to some problems concerning a convergence of pointwise type in the L 2 -space over a von Neumann algebra M with a faithful normal state Φ [3]. Here L 2 = L 2 (M, Φ) is the completion of M under the norm x → kxk ² = Φ(x ^∗ x) ^1/2 .

Intuitively, “reducing” L 2 to L ∞ , we can say that |f | is smaller than ε on a subset Z ∈ F , for f ∈ L 2 over a classical measure space (X, F , µ), if g n → f in L 2 and kg n 1 Z k _∞ < ε for some g n ∈ L _∞ (X, F , µ), n = 1, 2, . . .

This leads us to the following concept. Roughly speaking, ξ ∈ L 2 (M, Φ) has “modulus” less than ε on the subspace indicated by a projection p ∈ M if, for some x n ∈ M , kx _n − ξk → 0 and kx _n pk ∞ < ε, n = 1, 2, . . . , where kxk _∞ denotes the (operator) norm of x ∈ M .

We say that ξ n → 0 “almost surely” if, for any ε > 0, there exists a projection p ∈ M such that ξ n has “modulus” less than ε on the (subspace indicated by the) projection p for n large enough and, moreover, Φ(1−p) < ε.

Precise definitions are given in the next section.

It is worth noting that several limit theorems can be proved in the von Neumann algebras context by using the above concept of convergence [5; 6; 10; 11].

In comparison with other concepts of “almost sure” convergence for (un- bounded observables forming) L 2 (M, Φ) [4; 7; 9], our proposal seems to be intuitively clear. It gives a fairly natural extension of the almost uniform convergence in an algebra [2; 8; 12; 14; 15; 16; see also 9].

The main method used in the paper is the maximal inequality for a von Neumann algebra (Theorem 2.2) which can be proved quite elementarily and is very useful in the study of convergences on subspaces (cf. [7], compare also the maximal ergodic theorem of M. S. Goldstein [4]).

1991 Mathematics Subject Classification: 46L50, 60F15, 42C15.

Research supported by the KBN grant 211529101.

[167]

As an illustration, we prove the noncommutative extension of the rather advanced theorem of Tandori [17; 1] which gives the weakest condition im- plying the unconditional convergence of orthogonal series (Theorem 2.2).

Some discussion of the noncommutative Cauchy condition will also be nec- essary (cf. Section 3).

2. Notation and main results. Throughout the paper, M is a σ-finite von Neumann algebra with a faithful normal state Φ [3] (acting in a Hilbert space K), with the Hilbert space H = L 2 (M, Φ) being the com- pletion of the algebra M under the norm kxk = Φ(x ^∗ x) ^1/2 , given by the scalar product (x, y) = Φ(y ^∗ x). For x ∈ M , we put |x| ² = x ^∗ x. Proj M will denote the lattice of all self-adjoint projections in M . We write p ^⊥ = 1 − p for p ∈ Proj M . The operator norm in M will be denoted by k k ∞ .

For ξ ∈ H and p ∈ Proj M , we set S ξ,p = n

(x k ) ⊂ M :

∞

X

k=1

x k = ξ in H and

∞

X

k=1

x k p converges in norm in M o and

kξk _p = inf n

∞

X

k=1

x k p

∞ : (x k ) ∈ S ξ,p

o (with the usual convention inf ∅ = +∞).

We adopt the following definition.

2.1. Definition. A sequence (ξ ⁿ ) ⊂ H is said to be almost surely (a.s.) convergent to ξ ∈ H if, for each ε > 0, there exists a projection p in M such that Φ(p ^⊥ ) < ε and kξ n − ξk _p → 0 as n → ∞.

The following theorem gives a kind of maximal inequality and is crucial in our considerations.

2.2. Theorem. Let 0 < ε < 1/16, D n ∈ M ⁺ for n = 1, 2, . . . and (1)

∞

X

k=1

Φ(D k ) < ε.

Then there exists p ∈ Proj M such that

(2) Φ(p ^⊥ ) < ε ^1/4 ,

(3)

p

 X ⁿ

k=1

D k

 p

∞ < 4ε ^1/2 , n = 1, 2, . . .

As a main example of consequences of this maximal inequality we prove

the following extension to the noncommutative context of the classical Tan-

dori theorem on the unconditional almost sure convergence of orthogonal

series.

2.3. Theorem. Let (ξ n ) ^∞ _n=1 be a sequence of pairwise orthogonal ele- ments in H and

(4)

∞

X

k=0

 X

n∈I

kξ _n k ² log ² (n + 1)  1/2

< ∞,

where I k = {2 ²

+ 1, . . . , 2 ²

}. Then, for each permutation π of the set N of positive integers, the series P ∞

k=0 ξ π(k) is a.s. convergent.

The theorem below, an analogue of Orlicz’s theorem, can be deduced from the previous one.

2.4. Theorem. Let (ξ n ) ^∞ _n=1 be an orthogonal sequence in H. Let (w n ) be a nondecreasing sequence of positive numbers such that

∞

X

m=1

1/w ν

< ∞

for some increasing sequence (ν m ) of positive integers satisfying the inequal- ities

log ν m+1 ≤ c log ν m (c > 1, m = 1, 2, . . .).

If

∞

X

n=1

kξ _n k ² w n log ² (n + 1) < ∞, then, for each permutation π of N, the series P ∞

k=1 ξ _π(k) is a.s. convergent.

3. Noncommutative Cauchy condition. We start with some prop- erties of a.s. convergence, interesting in their own right.

3.1. Lemma. Let (η n ) ⊂ H and p ∈ Proj M . If P ∞

n=1 η n is convergent in H, then

(5)

∞

X

n=1

η n

p ≤

∞

X

n=1

kη n k p .

P r o o f. Without any loss of generality we may assume that (6)

∞

X

n=1

kη _n k _p < ∞.

Let ε > 0 and ε n = ε2 ⁻ⁿ , ε n,s = ε2 ^−n−s (n = 1, 2, . . .). Then there exist y n,k ∈ M (n, k = 1, 2, . . .) such that

(7)

η n −

s

X

k=1

y n,k

< ε n,s ,

(8) ky _n,s+1 pk ∞ < ε n,s ,

(9)

s

X

k=1

y n,k p

∞ < kη n k _p + ε n , n, s = 1, 2, . . . Put ξ = P ∞

k=1 η k and x n = P n

k,l=1 y k,l for n = 1, 2, . . . Then x n ∈ H.

First, we remark that kξ − x n k → 0. In fact, by (7), we have kξ − x _n k ≤

n

X

k=1

η k −

n

X

l=1

y k,l

+

∞

X

k=n+1

η k

≤

n

X

k=1

ε k,n + % n < ε n + % n , where % n = k P ∞

k=n+1 η k k → 0 as n → ∞. Now, we notice that (x _n p) ^∞ _n=1 is convergent in M . Indeed, by (8) and (9), we have

k(x _n+1 − x _n )pk ∞ ≤

n

X

k=1

ky _k,n+1 pk ∞ +

n+1

X

l=1

y n+1,l p _∞

<

n

X

k=1

ε k,n + kη n+1 k _p + ε n < 2ε n + kη n+1 k _p , and (6) yields the Cauchy condition for (x n p) ^∞ _n=1 . Finally, by (9), we have

kx _n pk ∞ ≤

n

X

k=1

n

X

l=1

y k,l p ∞ ≤

n

X

k=1

(kη k k _p + ε k ) <

n

X

k=1

kη _k k _p + ε.

Hence we get (5).

The next theorem gives a kind of noncommutative Cauchy condition for a.s. convergence.

3.2. Proposition. Let (σ n ) ⊂ H and kσ n − σk → 0 as n → ∞, where σ ∈ H. If , for each ε > 0, there exists some p ∈ Proj M with Φ(p ^⊥ ) < ε such that kσ n − σ _m k _p → 0 as n, m → ∞, then σ _n → σ a.s.

P r o o f. By the assumption, for ε > 0, there are p ∈ Proj M with Φ(p ^⊥ ) < ε and a sequence of indices m 0 < m 1 < . . . such that

(10) kσ _n − σ _m k _p < ε2 ^−k for n, m ≥ m k .

Let n ≥ m 0 . Fix k such that n < m k . Putting η 0 = σ m

− σ _n , η 1 = σ m

− σ m

, . . . , η j = σ m

− σ m

, . . . , we obtain σ − σ n = P ∞

j=0 η j , where the series is convergent in H. Moreover, by (10), we have kη ₀ k _p < ε, kη 1 k _p < ε2 ^−k , . . . , kη j k _p < ε2 ^−k−j+1 , . . . Thus P ∞

j=0 kη _j k _p < 2ε

and, by Lemma 3.1, we get kσ − σ n k _p < 2ε for n > m 0 . This completes the

proof.

4. Auxiliary consequences of the maximal inequality. For the sake of completeness we reproduce the proof given in [7, 3.10].

P r o o f o f T h e o r e m 2.2. For brevity, we define B n = P n k=1 D k , n = 1, 2, . . . Put

(11) p n = e B

([0, ε ^1/2 ]), n = 1, 2, . . . , where B n = R ∞

0 λ e B

(dλ) is the spectral representation. The sequence (p n ) ^∞ _n=1 of projections is conditionally weakly operator compact. Let a be a limit point, i.e.

(12) a = w.o.- lim

k→∞ p _n(k)

for some subsequence (n(k)). Obviously, a ∈ M , 0 ≤ a ≤ 1. Put (13) p = e a ([1 − ε ^1/4 , 1]),

where a = R 1

0 λe a (dλ).

By (1) and (11), we obtain

Φ(p ^⊥ _n ) = Φ(e B

((ε ^1/2 , ∞))) ≤ ε ^−1/2 Φ(B n ) < ε ^1/2 . Consequently, Φ(a) = lim k→∞ Φ(p _n(k) ) ≥ 1 − ε ^1/2 and, finally,

(14) Φ(1 − a) ≤ ε ^1/2 .

On the other hand, by (13), we can write p = e 1−a ([0, ε ^1/4 ]). Then, by (14), we obtain

Φ(p ^⊥ ) = Φ(e 1−a ((ε ^1/4 , 1])) ≤ e ^−1/4 Φ(1 − a) ≤ ε ^1/4 , which proves (2).

To show (3), we estimate (B n ξ, ξ) for all ξ ∈ pK with kξk = 1.

Obviously, the subspace pK is invariant for a. Moreover, by (13), the spectrum of the operator a p = a| pK is contained in the interval [1 − ε ^1/4 , 1].

Thus, a p is invertible, a ⁻¹ _p is defined on pK and ka ⁻¹ _p k _∞ ≤ (1 − ε ^1/4 ) ⁻¹ . Fix ξ ∈ pK, kξk = 1 and put ζ = a ⁻¹ _p ξ. Then ζ ∈ pK and

(15) kζk ≤ (1 − ε ^1/4 ) ⁻¹ .

Define η k = p n(k) ζ − ξ. By (12), η k converges weakly to 0 as k → ∞.

Therefore, by the positivity of B n , lim inf

k→∞ ((B n η k , η k ) + (B n η k , ξ) + (B n ξ, η k )) ≥ 0.

Hence, by (11) and (15), we get

(B n ξ, ξ) ≤ lim inf

k→∞ (B n (η k + ξ), η k + ξ) = lim inf

k→∞ (B n p _n(k) ζ, p _n(k) ζ)

≤ lim inf

k→∞ kp _n(k) B n p n(k) k _∞ kζk ²

≤ lim inf

k→∞ kp _n(k) B n(k) p n(k) k _∞ kζk ²

≤ ε ^1/2 (1 − ε ^1/4 ) ⁻² < 4ε ^1/2 , which gives (3). The proof is complete.

4.1. Lemma. Let 0 < ε < 1/16, D n ∈ M ⁺ , ζ n ∈ H for n = 1, 2, . . . and (16 )

∞

X

k=1

Φ(D k ) < ε,

(17 )

∞

X

k=1

kζ k k ^1/2 < ε.

Then there exists p ∈ Proj M such that

(18 ) Φ(p ^⊥ ) < 2ε ^1/4 ,

(19 )

p

 X ⁿ

k=1

D k

 p

∞ < 9ε ^1/2 , n = 1, 2, . . . , (20 )

∞

X

k=1

kζ _k k _p < 8ε ^5/4 .

Moreover , if condition (17) is replaced by (21)

∞

X

k=1

kζ _k k ² < ε, then (20) can be replaced by

(22) kζ _n k _p < 8ε ^1/4 , n = 1, 2, . . . P r o o f. Choose z k,l ∈ M such that ζ _k = P ∞

l=1 z k,l in H, and (23) kz _k,l k ≤ 2 ^−l+1 kζ _k k, k, l = 1, 2, . . .

Putting

D k,0 = D k , D k,l = 2 ^l kζ k k ⁻¹ |z k,l | ² , k, l = 1, 2, . . . ,

we obtain, by (23), Φ(D k,l ) ≤ 2 ^−l+2 kζ _k k for k, l = 1, 2, . . . Thus, by (16)

and (17),

∞

X

k=1

∞

X

l=0

Φ(D k,l ) ≤

∞

X

k=1

Φ(D k ) +

∞

X

k=1

∞

X

l=1

2 ^−l+2 kζ _k k

< ε + 4

∞

X

k=1

kζ _k k < 5ε.

Now, by Theorem 2.2, there exists p ∈ Proj M such that Φ(p ^⊥ ) < (5ε) ^1/4 <

2ε ^1/4 and p  X ⁿ

k=1 m

X

l=0

D k,l

 p

_∞ < 4(5ε) ^1/2 < 9ε ^1/2 , n = 1, 2, . . . , m = 0, 1, 2, . . . In particular, kp( P n

k=1 D k )pk ∞ < 9ε ^1/2 and kz k,l pk ² _∞ ≤ 2 ^−l kζ _k k 9ε ^1/2 for k, l = 1, 2, . . . Thus

kζ _k k _p ≤

∞

X

l=1

z k,l

∞

≤

∞

X

l=1

2 ^−l/2 3ε ^1/4 kζ _k k ^1/2 < 8ε ^1/4 kζ _k k ^1/2 , which, by (17), gives (20) easily.

The last part of the theorem can be proved in the same manner.

The following two propositions are simple consequences of Lemma 4.1.

4.2. Proposition. Let D n ∈ M ⁺ , ζ n ∈ H for n = 1, 2, . . . and

∞

X

k=1

Φ(D k ) < ∞,

∞

X

k=1

kζ _k k ^1/2 < ∞.

Then for each ε > 0 there exists p ∈ Proj M with Φ(p ^⊥ ) < ε such that the sequence (kp( P n

k=1 D k )pk ∞ ) ^∞ _n=1 is bounded and

∞

X

k=1

kζ _k k _p < ∞.

4.3. Proposition. Let D n ∈ M ⁺ , B n ∈ M ⁺ , ζ n ∈ H for n = 1, 2, . . . and

∞

X

k=1

Φ(D k ) < ∞,

∞

X

k=1

Φ(B k ) < ∞,

∞

X

k=1

kζ _k k ² < ∞.

Then for each ε > 0 there exists p ∈ Proj M with Φ(p ^⊥ ) < ε such that the sequence (kp( P n

k=1 D k )pk ∞ ) ^∞ _n=1 is bounded , kpB n pk ∞ → 0 and kζ _n k _p → 0 as n → ∞.

P r o o f. Consider (D k + B k ) ^∞ _k=1 and apply Lemma 4.1.

5. Other auxiliary results. The following simple lemma is in the spirit

of the classical Schwarz inequality and is very useful in many estimations.

5.1. Lemma. Let ε k > 0, x k ∈ M , E _k ∈ M ⁺ and |x k | ² ≤ ε _k E k for k = 1, . . . , n. Then

n

X

k=1

x k

∞ ≤

n

X

k=1

E k

1/2

∞

 X ⁿ

k=1

ε k

 1/2

. P r o o f. For ξ ∈ K with kξk = 1, we have

n

X

k=1

x k ξ

2

≤  X ⁿ

k=1

ε ^−1/2 _k kx _k ξkε ^1/2 _k

 2

≤  X ⁿ

k=1

ε ⁻¹ _k kx k ξk ²

 X ⁿ

k=1

ε k ≤  X ⁿ

k=1

E k ξ, ξ

 X ⁿ

k=1

ε k

≤

n

X

k=1

E k

∞

n

X

k=1

ε k .

The following lemma is, in fact, an easy modification of the Lemma of [8] (cf. also Proposition 4.2 of [9]).

5.2. Lemma. Let J ⊂ N have cardinality #J = µ. Let (y i ) ^∞ _i=1 be a sequence of operators in M such that y i = 0 for i 6∈ J . Then there exists an operator B ∈ M ⁺ such that

n

X

j=1

y j

2

≤ B, n = 1, 2, . . . , and

Φ(B) ≤ (1 + log µ) ² X

i,j∈J

|Φ(y _i ^∗ y j )|.

The next lemma is also a slight modification of Lemma 4.2 of [6] (cf.

also Lemma 5.2.2 of [11]).

5.3. Lemma. Let J ⊂ N with #J = µ. Let (η i ) ^∞ _i=1 be a sequence of pairwise orthogonal elements in H such that η i = 0 for i 6∈ J . Let (ε i ) be a sequence of positive numbers. Then there exist operators B ∈ M ⁺ and y i ∈ M (i ∈ N) with y i = 0 when η i = 0, such that

kη _i − y _i k < ε _i , i = 1, 2, . . . , 

n

X

i=1

y i

2

≤ B, n = 1, 2, . . . ,

Φ(B) ≤ 2(1 + log µ) ²

∞

X

i=1

kη _i k ² .

6. Proof of Tandori’s theorem

P r o o f o f T h e o r e m 2.3. First, we notice that, by assumption (4), the sequence (σ n ) = ( P n

k=1 ξ π(k) ) ^∞ _n=1 is convergent in H. By Proposition 3.2, it remains to prove the Cauchy condition for a.s. convergence. We put additionally I 0 = {1, . . . , 4}.

For brevity, we write

(24) α k =  X

n∈I

kξ _n k ² log ² (n + 1)  1/2

, k = 1, 2, . . . , and set

η k,i =

 α ⁻¹ _k ξ π(i) as π(i) ∈ I k ,

0 otherwise.

Fix k and, taking J = π ⁻¹ [I k ], apply Lemma 5.3 to the sequence (η k,i ) ^∞ _i=1 . Then there exist operators D k ∈ M ⁺ , y k,i ∈ M (i = 1, 2, . . .) such that

(26) kη _k,i − y _k,i k < 2 ⁻ⁱ if π(i) ∈ I k , y k,i = 0 if π(i) 6∈ I k , (27) |s _k,l | ² ≤ D _k for l = 1, 2, . . . , where s k,l = y k,1 + . . . + y k,l (l = 1, 2, . . .) and

Φ(D k ) ≤ 2(2 ^k+1 + 1) ²

∞

X

i=1

kη _k,i k ²

(the cardinality #I k is less than 2 ²

).

Thus, by (25), (24) and the definition of I k , we obtain Φ(D k ) ≤ 4 X

n∈I

(1 + log n) ² α ⁻¹ _k kξ _n k ²

≤ 16α ⁻¹ _k X

n∈I

log ² (n + 1)kξ n k ² = 16α k .

Therefore, by assumption (4) of our theorem, we get (28)

∞

X

k=1

Φ(D k ) < ∞.

Now, put

(29) ζ i = α ^1/2 _k (η k,i − y _k,i ), i = 1, 2, . . . ,

where k is uniquely determined by i via π(i) ∈ I k . From (26) and (4) we obtain

(30)

∞

X

i=1

kζ _i k ^1/2 < ∞.

From (28) and (30), by Proposition 4.2, for every ε > 0, there exists p ∈ Proj M with Φ(p ^⊥ ) < ε such that

p

 X ⁿ

k=1

D k

 p

∞ ≤ C, n = 1, 2, . . . , and

(31)

∞

X

k=1

kζ _k k _p < ∞.

Now, let us define two sequences of indices. Namely, for n = 1, 2, . . . , denote by k(n) the smallest k such that

{π(1), . . . , π(n)} ⊂ I ₁ ∪ . . . ∪ I _k , whereas j(n) is the greatest j satisfying

I 1 ∪ . . . ∪ I _j ⊂ {π(1), . . . , π(n)}.

Obviously, both (k(n)) and (j(n)) are nondecreasing and tend to infinity as n → ∞.

Then, for m < n, by (25), (27), (29), we have σ n − σ _m = ξ _π(m+1) + . . . + ξ _π(n) =

k(n)

X

k=j(m)+1 n

X

i=m+1

α ^1/2 _k η k,i

=

k(n)

X

k=j(m)+1

α ^1/2 _k (s k,n − s _k,m ) +

n

X

i=m+1

ζ i . Consequently,

(32) kσ _n − σ _m k _p ≤

k(n)

X

k=j(m)+1

α ^1/2 _k s k,n p _∞ +

k(n)

X

k=j(m)+1

α ^1/2 _k s k,m p _∞

+

n

X

i=m+1

kζ i k p .

By (27) and Lemma 5.1, we have

(33)

k(n)

X

k=j(m)+1

α ^1/2 _k s k,n p _∞ ≤

p 

k(n)

X

k=j(m)+1

D k

 p

1/2

∞



k(n)

X

k=j(m)+1

α k

 1/2

≤ C ^1/2 

k(n)

X

k=j(m)+1

α k

 1/2

. Analogously,

(34)

k(n)

X

k=j(m)+1

α ^1/2 _k s k,m

∞ ≤ C ^1/2 

k(n)

X

k=j(m)+1

α k

 1/2

. By (32), (33), (34), (1) and (31), the proof is complete.

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