Characterization of singular traces on the weak trace class ideal generated by exponentiation-invariant extended limits

vol. 55, no. 2 (2015), 127–145

Characterization of singular traces on the weak trace class ideal generated by exponentiation-invariant extended limits

Fedor Sukochev and Aleksandr Usachev

Summary. This paper studies the subset of singular traces generated by exponentiation-invariant extended limits. We describe relations between this subset and other important subsets of singular traces. We prove seve- ral conditions for measurability of operators from the weak trace class ideal with respect to the traces generated by exponentiation-invariant extended limits. We resolve an open question raised in [S. Lord, F. Su- kochev, Measure theory in noncommutative spaces, SIGMA Symmetry Integrability Geom. Methods Appl. 6 (2010), paper 072, 36] in the setting of the weak trace class ideal.

Keywords Dixmier traces;

measurable operators;

extended limits

MSC 2010 58B34; 46L52

Received: 2016-01-19, Accepted: 2016-03-13

We dedicate this paper to Professor Henryk Hudzik on the occasion of his 70th birthday.

1. Introduction

Let B(H) be the algebra of all bounded linear operators on a separable Hilbert space H.

Denote by µ(A) the generalised singular values function of an operator A ∈ B(H) [19].

For every A ∈ B(H) the function µ(A) is a step function and µ(n, A), n ⩾ 0, is the n-th singular value of the operator A. Let L _∞ = L _∞ (0, ∞) be the Banach space of all Lebesgue measurable bounded functions on (0, ∞) with the uniform norm.

Fedor Sukochev, School of Mathematics and Statistics, University of New South Wales, Kensington, 2052, NSW, Australia (e-mail: f.sukochev@unsw.edu.au)

Aleksandr Usachev, School of Mathematics and Statistics, University of New South Wales, Kensington, 2052, NSW, Australia (e-mail: a.usachev@unsw.edu.au)

DOI 10.14708/cm.v55i2.1102 © 2015 Polish Mathematical Society

The operator Marcinkiewicz space M _{1 ,∞} (also called Dixmier ideal and/or dual to Macaev ideal) is defined as follows:

M _{1 ,∞} ∶= {A ∈ B(H) is compact ∶ ∥A∥ M

∶= sup

t>0

1 log(1 + t) ∫

t

0

µ(s, A) d s < ∞}.

For an arbitrary translation and dilation-invariant extended limit ω on L _∞ , J. Di- xmier [9] proved that the weight

Tr _ω (A) ∶= ω(t ↦ 1 log(1 + t) ∫

t

0

µ(s, A) d s), 0 ⩽ A ∈ M _{1 ,∞} (1) extends by linearity to a trace (that is a linear positive unitarily invariant functional) on M _{1 ,∞} , which fails to be normal. In 1985 A. Connes employed Dixmier traces in the area of noncommutative geometry [8]. Since then, Dixmier traces play a crucial role in non- commutative geometry and analysis as they furnish a notion of an integral in the noncom- mutative aspect.

Various important formulae of noncommutative geometry were established assuming additional invariance of the extended limit ω defining Dixmier trace. In particular, the Connes’ formula for a representative of the Hochschild class of the Chern character for ( p, ∞)-summable spectral triples (see e.g. [4, Theorem 7] and [3, Theorem 6]) was esta- blished under the assumption that the extended limit ω is M-invariant, that is

ω = ω ○ M , where M∶ L _∞ → L ∞ is given by the formula

(M x )(t) = 1 log(t) ∫

t

1

x (s) d s

s

, t > 1.

Very recently these conditions have been relaxed in [6].

Also in [4, 5] the results on heat semigroup asymptotics and generalised ζ -function residues were established for Dixmier traces Tr ω , provided that ω was additionally chosen to be exponentiation invariant, that is, invariant under transformations P a , a > 0, given by the formula

(P a x )(t) = x (t ^a ), a > 0, x ∈ L _∞ .

These additional assumptions on ω single out various subclasses of Dixmier traces.

It has recently been proved in [30] that one can apply the Dixmier construction (1)

to any exponentiation-invariant extended limit ω instead of a dilation-invariant extended

limit and still obtain a non-normal trace. Moreover, these traces, denoted in [30] by D P ,

form a proper subset in the set of all Dixmier traces. However, the natural limitations of

the method used in [30] deny any reasonable description of the class D P (see beginning

of Section 2 below for more details).

More recently, Dixmier traces generated by exponentiation-invariant extended limits were employed to establish a non-commutative version of the classical Fubini Theorem [31].

Another result established in [30] asserts that there exists 0 ⩽ A ∈ M _{1 ,∞} which is D _P -measurable (that is, the values of all traces from D _P agree on A) and which is not Dixmier measurable (that is, not all values of Dixmier traces agree on A). However, due to the technical limitations mentioned above there is no way to calculate a common value of traces on D P -measurable operators, besides specifically tailored examples (see [30, The- orem 8]).

In [14] the construction of traces from D P originating in [30] has been extended to the setting of general operator Marcinkiewicz spaces. The authors proved formulae re- lating traces from D P with heat semigroup asymptotics and generalised ζ -function resi- dues. Using these results they established formulae that enable computation of traces of Hörmander–Weyl pseudo-differential operators in terms of their symbols. We state their result rigorously in the case of the Marcinkiewicz space M _{1 ,∞} . We need to introduce the functional Marcinkiewicz space M _{1 ,∞} which consists of all measurable functions f on (0, ∞) such that

sup

t>0

1 log(1 + t) ∫

t

0

f ^∗ (s) d s < ∞,

where f ^∗ is a non-increasing rearrangement of the function ∣ f ∣ (see [16]).

Let f be a function from the class of symbols corresponding to a Hörmander pair (g , m) (see [14,15] for details) and let OP W ( f ) be a Weyl pseudo-differential operator with symbol f . It was proved in [14] that for any symbol f ∈ S(m, g) the condition m ∈ M _{1 ,∞}

guarantees that OP _W ( f ) ∈ M _{1 ,∞} . Moreover, for every exponentiation-invariant extended limit ω on L _∞ the following formula holds:

Tr ω (OP W ( f )) = ω([r ↦ 1 log(r) ∫

R

f (x , ξ) ∶ ∣ f (x , ξ)∣ ^{1/ log(r)} d ⁿ x d ⁿ ξ]).

Note that lacking a description of D _P -measurability, there is no systematic way to compute Tr ω ( OP W ( f )) in the latter formula even in the case when this expression does not depend on ω.

In the present paper we restrict our study to the operator space which is a proper subspace of M _{1 ,∞} . This is the quasi-Banach operator space L _{1 ,∞} defined as follows:

L _{1 ,∞} ∶= {A ∈ B(H) is compact ∶ ∥A∥ L

∶= sup

n⩾0

(n + 1)µ(n, A) < ∞}.

We establish the criterion for D P -measurability and discuss relations between the class of all D P -measurable operators from L _{1 ,∞} and other classes of measurable operators.

In particular, we answer the question raised in [18, Question 7.4] by showing that the class

of D P -measurable operators and that of Tauberian operators do not coincide even on the

positive cone of L _{1 ,∞} (see Proposition 3.14).

We thank the referee for a very careful reading of the manuscript and suggesting a number of improvements.

2. Sucheston/Lorentz approach to Dixmier traces

The approach to Dixmier traces described in this section first appeared in [17] (see also [5, 7, 10, 11] and then was refined in [19, 28, 29]. It is based on the fundamental results of

G. G. Lorentz [20] and L. Sucheston [27].

First we introduce various classes of extended limits and describe their interconnec- tions. They are used to construct Dixmier traces. Note that in [28–30] the extended limits were termed “generalised limits”. Here we use the terminology which is in line with that of the recent book [19].

2.1. Definition. A linear functional ω on L _∞ (0, ∞) is called a dilation-invariant extended limit if

(i) ω(x ) ⩾ 0, whenever 0 ⩽ x ∈ L ∞ (0, ∞);

(ii) ω( χ _{(0 ,∞)} ) = 1 and ω( χ _{(0 , 1)} ) = 0, where χ S is a characteristic functions of a set S ⊆ R;

(iii) ω(σ

β

x ) = ω(x ) for every β > 0, x ∈ L ∞ (0, ∞), where operators σ

β

∶ L ∞ (0, ∞) → L _∞ (0, ∞) are given by (σ

β

x )(t) ∶= x (β t).

2.2. Definition. A linear functional ω on L _∞ is called an exponentiation-invariant exten- ded limit if

(i) ω(x ) ⩾ 0, whenever 0 ⩽ x ∈ L _∞ ; (ii) ω( χ _{(0 ,∞)} ) = 1 and ω( χ _{(0 , 1)} ) = 0;

(iii) ω(P _a x ) = ω(x ) for every x ∈ L _∞ and every a > 0.

The following remark links dilation- and exponentiation-invariant extended limits on L _∞ . This is based on the properties of dilation and exponentiation operators noticed in [5, Proposition 1.3] (see also [30]).

2.3. Remark. Let γ be a dilation-invariant extended limit on L _∞ . The functional ω ∶=

γ○exp on L _∞ defined by ω(x ) ∶= γ(x ○exp) = γ(t ↦ x (e ^t )) is an exponentiation-invariant extended limit on L _∞ . Conversely, if ω is an exponentiation-invariant extended limit on L _∞ , then the functional γ ∶= ω ○ log is a dilation-invariant extended limit on L _∞ .

The following Lorentz-type criterion (see [20]) was proved in [28].

2.4. Theorem. Let x ∈ L _∞ such that x ○ e x p is uniformly continuous. The equality ω(x ) = c holds for every dilation-invariant generalized limit ω on L _∞ if and only if

lim

t→+∞

1 log t ∫

t

1

x (αs) d s

s

= c

uniformly in α ⩾ 1.

Using Remark 2.3 and Theorem 2.4, we immediately infer the following Lorentz-type criteria for exponentiation-invariant extended limits.

2.5. Corollary. Let x ∈ L _∞ (0, ∞) such that x ○ exp ○ exp is uniformly continuous and c ∈ R. The equality ω(x) = c holds for every exponentiation-invariant extended limit ω on L _∞ (0, ∞) if and only if

lim

t→+∞

1 log log t ∫

t

1

x (s ^a ) d s s log s

= c

uniformly in a ⩾ 1.

Note that the condition on the function x ∈ L _∞ (0, ∞) in Corollary 2.5 is very strong.

It was proved in [30] that this condition makes the result of Corollary 2.5 inapplicable to the study of D _P -measurability of operators from M _{1 ,∞} . However, if we restrict our con- siderations to a smaller ideal L _{1 ,∞} , then we can obtain a description of D _P -measurability.

Before we state the result we explicitly define the notion of measurability and introduce some auxiliary notation.

2.6. Definition. Let A denote a subclass of Dixmier traces. An operator A ∈ L _{1 ,∞} is said to be A-measurable if Tr ω (A) takes the same value for all Tr ω ∈ A.

Recall that arbitrary A ∈ M _{1 ,∞} has a unique decomposition of the form A = A 1 − A 2 + i A 3 − i A 4 , 0 ⩽ A j ∈ M _{1 ,∞} , j = 1, 2, 3, 4.

We set

˜

µ(A) ∶= µ(A ₁ ) − µ(A 2 ) + i µ(A 3 ) − i µ(A 4 ), A ∈ B(H).

The theorem below is our first main result.

2.7. Theorem. An operator A ∈ L _{1 ,∞} is D P -measurable if and only if the limit

lim

t→+∞

1 log log t ∫

t

e

( 1 log(s ^a ) ∫

s

0

µ(z, A)d z) ˜ d s s log s

exists uniformly in a ⩾ 1; moreover, Tr _ω (A) is equal to the value of this limit for every trace

Tr _ω ∈ D P .

Proof. Fix A ∈ L _{1 ,∞} and let the operators A _j , j = 1, 2, 3, 4, come from the decomposition of A. We shall first check that the function t ↦ ¹

e

∫

e

t

0 µ(s, A)d s is uniformly continuous. ˜ We have

∣ d d t

(t ↦ 1 e ^t ∫

e

t

0

˜

µ(s, A)d s)∣ = ∣−

1 e ^t ∫

e

t

0

˜

µ(s, A) d s + e ^t ⋅ e ^e

t

⋅ ˜ µ(e ^e

t

, A) e ^t

∣

⩽ ∣ 1 e ^t ∫

e

t

0

˜

µ(s, A) d s∣ + ∣e ^e

t

⋅ ˜ µ(e ^e

t

, A)∣.

Since A ∈ L _{1 ,∞} , it follows that

∣∫

e

t

0

µ(s, A) d s∣ ⩽ c ∫ ˜

e

t

0

1 s + 1

d s = c log(1 + e ^e

t

),

where c = ∥A 1 ∥ L

+ ∥A 2 ∥ L

+ ∥A 3 ∥ L

+ ∥A 4 ∥ L

. Therefore,

∣ d d t

(t ↦ 1 e ^t ∫

e

t

0

µ(s, A)d s)∣ ⩽ c ⋅ ˜

log(1 + e ^e

t

) e ^t

) + c ⩽ 3c .

Hence, the function t ↦ ¹

e

∫

e

t

0 µ(s, A)d s is Lipschitz, so it is uniformly continuous. ˜ By definition, an operator A ∈ L _{1 ,∞} is D P -measurable if and only if

ω(t ↦ 1 log(t) ∫

t

0

˜

µ(s, A)d s) = c for every exponentiation-invariant extended limit ω and some c ∈ R.

Using Corollary 2.5, we conclude that an operator A ∈ L _{1 ,∞} is D P -measurable if and only if the limit

lim

t→+∞

1 log log t ∫

t

e

( 1 log(s ^a ) ∫

s

0

˜

µ(z, A)d z) d s s log s

= c exists uniformly in a ⩾ 1.

3. A new approach

We start this section by describing a new construction of singular traces on L _{1 ,∞} origina-

ting in [21–23] and modified and further developed in [26]. First, we recall the notion of

classical Banach limits.

By l _∞ we denote the Banach space of all complex bounded sequences x = (x ₀ , x ₁ , . . .) with the norm

∥x ∥ l

∶= sup

n⩾0

∣x n ∣.

3.1. Definition. A linear functional B on l ∞ is said to be a Banach limit if (i) B ⩾ 0, that is, Bx ⩾ 0 for x ⩾ 0,

(ii) B1 = 1, where 1 = (1, 1, . . .),

(iii) BT x = Bx for all x ∈ l _∞ , where T is a translation operator defined as follows T (x ₀ , x ₁ , . . .) = (x ₁ , x ₂ , . . .).

Denote by l _{1 ,∞} the linear space (frequently called the weak l ₁ -space) of all bounded sequences for which the quasi-norm

∥x ∥ _l

1 ,∞

= sup

n⩾0

(n + 1)x _n ^∗

is finite. By x ^∗ we denote a non-increasing rearrangement of ∣x ∣.

We will use a modified Pietsch operator D∶ l _∞ → l _{1 ,∞} defined as follows:

D(x ₀ , x 1 , x 2 , . . . ) = (x 0 , x 1

2 ¹ ,

x 1

2 ¹

´¹¹¹¹¸¹¹¹¹¶

2 ¹ times ,

x 2

2 ² , ⋯,

x 2

2 ²

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶

2 ² times , ⋯).

The following theorem is a combination of Corollary 4.2 and Theorem 6.3 from [26].

3.2. Theorem. There is a bijection between the set of all positive normalised traces on L _{1 ,∞}

and Banach limits given by the formulae

τ(A) = 1 log 2

B({

2

−2

∑

k =2

−1

µ(k , A)}

n⩾0

), 0 ⩽ A ∈ L _{1 ,∞} ,

B(x ) = log 2 ⋅ τ(diag(Dx )), x ∈ l _∞ . (2) Define the Cesàro operator C∶ l _∞ → l _∞ as follows

(C x ) n = 1 n + 1

n

∑

k =0

x _k , n ∈ N.

Define the Hardy (or integral Cesàro) operator H∶ L ∞ → L ∞ by the formula (H x )(t) ∶=

1 t ∫

t

0

x (s) d s, t > 0.

Let π be the isometric embedding π∶ l _∞ → L ∞ given by {x n } ^∞ _n=0

π

↦

∞

∑

n=0

x n χ _{[n , n+1)} .

Next we characterise the class D _P of Dixmier traces generated by exponentially inva- riant extended limits, in terms of Banach limits of a special type. We will use the following result which is a particular form of [13, Lemma 2.2].

3.3. Lemma. For every Lebesgue measurable function f on (0, ∞) such that ∣ f (s)∣ ⩽ ^α

s+1 , s > 0, for some α > 0 we have

∣∫

v

u

( f ^∗ − f )(s) d s∣ ⩽ 16α , ∀ 0 ⩽ u < v < ∞.

The following theorem is the second main result of this paper.

3.4. Theorem. A trace τ on L _{1 ,∞} belongs to the class D P if and only if the corresponding Banach limit B (given by formula (2)) is of the form B = θ ○ C, where θ = γ ○ π for some dilation-invariant extended limit γ on L ∞ .

Proof. Let τ ∈ D P and let B be its corresponding Banach limit given by Theorem 3.2. By Theorem 3.2 we have

B(x ) = log 2 ⋅ (τ ○ diag)(Dx ), x ∈ l _∞ . (3) By the definition of the set D _P , there exists an exponentiation-invariant extended limit ω on L _∞ (0, ∞) such that

τ(A) = ω(t ↦ 1 log(1 + t) ∫

t

0

µ(s, A) d s), 0 ⩽ A ∈ L _{1 ,∞} . (4) Note that µ(diag(Dx )) = π((Dx ) ^∗ ) for every x ∈ l _∞ . Combination of (3) and (4) yields

B(x ) = log 2 ⋅ ω(t ↦ 1 log(1 + t) ∫

t

0

π((Dx ) ^∗ )(s) d s), x ∈ l ∞ .

It follows from the definition of the modified Pietsch operator D that ∣(Dx ) k ∣ ⩽

∥D∥ l

→L

⋅ ∥x ∥ l

⋅ ¹

k +1 , k ⩾ 0. Lemma 3.3 implies that the function t ↦ ∫

t

0

(π(Dx ) − π((Dx ) ^∗ ))(s) d s is bounded on (0, ∞). So the function

t ↦ 1 log(1 + t) ∫

t

0

(π(Dx ) − π((Dx ) ^∗ ))(s) d s

vanishes at infinity. Since ω is an extended limit, it follows that ω(t ↦

1 log(1 + t) ∫

t

0

(π(Dx ) − π((Dx ) ^∗ ))(s) d s) = 0.

Therefore,

B(x ) = log 2 ⋅ ω(t ↦ 1 log(1 + t) ∫

t

0

π(Dx )(s) d s), x ∈ l _∞ .

For every x ∈ l ∞ and 2 ⁿ − 1 ⩽ t ⩽ 2 ⁿ⁺¹ − 2, we have

∫

t

0

π(Dx )(s) d s =

n−1

∑

i =0 2

−2

∑

k =2

−1

(Dx ) _k + ∫

t

2

−1

π(Dx )(s) d s

=

n−1

∑

i =0

x k + (t − 2 ⁿ + 1) x _n 2 ⁿ

= ∫

log

t

0

π(x )(s) d s − ∫

log

t

n

π(x )(s) d s + x _n

t − 2 ⁿ + 1 2 ⁿ

. Since the last two terms in the above expression are bounded functions of t, it follows that

∫

t

0

π(Dx )(s) d s = ∫

log

t

0

π(x )(s) d s + O(1), t → ∞.

Hence, for every x ∈ l _∞ the following chain of equalities holds B(x ) = ω(t ↦

log 2 log t ∫

log

t

0

π(x )(s) d s)

= ω(t ↦ 1 log t

1 log 2

∫

log t

1 log 2

0

π(x )(s) d s)

= (ω ○ P

log 2

○ log)(t ↦ 1 t ∫

t

0

π(x )(s) d s)

= (ω ○ log)(H π(x )), since ω is exponentiation invariant.

In was proved in [26, Equation 17] that the function H π(x ) − π(C x ) vanishes at infinity. Hence,

B(x ) = (ω ○ log)(π(C x )).

By Remark 2.3, an extended limit γ ∶= ω ○ log on L ∞ is dilation invariant. In other

words, B = γ ○ π ○ C, where γ is a dilation-invariant extended limit on L ∞ . This proves the

first assertion.

Suppose now that B = γ ○ π ○ C for some dilation-invariant extended limit γ on L _∞ . By Theorem 3.2, the functional

τ(A) = 1 log 2

⋅ B({

2

−2

∑

k =2

−1

µ(k , A)}

n⩾0

), 0 ⩽ A ∈ L _{1 ,∞}

extends by linearity to a trace on L _{1 ,∞} . For every 0 ⩽ A ∈ L _{1 ,∞} we obtain

τ(A) = 1 log 2

⋅ (γ ○ π ○ C)({

2

−2

∑

k =2

−1

µ(k , A)}

n⩾0

)

= 1 log 2

⋅ γ(π({

1 n

2

∑

k =0

µ(k , A)}

n⩾0

)).

A direct verification shows that for every 0 ⩽ A ∈ L _{1 ,∞} the bounded function t ↦ π({

1 n

2

∑

k =0

µ(k , A)}

n⩾0

)(t) − 1 t ∫

2

0

µ(s, A) d s vanishes at infinity.

Hence, for every positive A ∈ L _{1 ,∞} we obtain τ(A) =

1 log 2

⋅ γ(t ↦ 1 t ∫

2

0

µ(s, A) d s)

= 1 log 2

⋅ (γ ○ exp)(t ↦ 1 log t ∫

2

0

µ(s, A) d s)

= (γ ○ exp)(t ↦ 1 log t ^{log 2} ∫

t

0

µ(s, A) d s)

= (γ ○ exp ○P log 2 )(t ↦ 1 log t ∫

t

0

µ(s, A) d s).

Since γ is a dilation-invariant extended limit on L _∞ , it follows from Remark 2.3 that ω ∶= γ ○ exp is an exponentiation-invariant extended limit on L _∞ .

Therefore,

τ(A) = ω(t ↦ 1 log t ∫

t

0

µ(s, A) d s), 0 ⩽ A ∈ L _{1 ,∞} , where ω is an exponentiation-invariant extended limit, that is, τ ∈ D P .

3.5. Remark. As a direct corollary of Theorem 3.2 and Theorem 3.4, we obtain that an operator A ∈ L _{1 ,∞} is D P -measurable if and only if

(γ ○ π ○ C)({

2

−2

∑

k =2

−1

˜ µ(k , A)}

n⩾0

) = c

for every dilation-invariant γ on L _∞ and some constant c. This condition is equivalent to that given in Theorem 2.7.

Below we provide additional necessary and sufficient conditions for D P -measurability.

First we describe the relation between traces from D P and dilation-invariant Banach limits studied in [1,2]. For k ∈ N, k ⩾ 2 we denote by B(σ ^k ) the set of all Banach limits invariant under the dilation operator σ _k , where σ _k ∶ l ∞ → l ∞ is defined as follows:

σ k (x 0 , x 1 , . . . ) = (x 0 , . . . , x 0

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶

k times

, x 1 , . . . , x 1

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶

k times

, . . . ), x ∈ l ∞ .

3.6. Proposition. If τ ∈ D P , then the corresponding Banach limit B (given by Theorem 3.2) is of the form B = θ ○ C, where θ ∈ l _∞ ^∗ is such that θ = θ ○ σ _k , ∀ k ∈ N. In particular, B ∈ ⋂ ^∞ _{k =2} B(σ k ).

Proof. By Theorem 3.4, θ = γ ○ π for some γ on L _∞ such that γ = γ ○ σ _a , a > 0.

For k ∈ N, k ⩾ 2 we have

σ _k π(x ) = σ _k (t →

∞

∑

n=0

x _n χ _{(n , n+1]} (t))

= t →

∞

∑

n=0

x n χ _{(n , n+1]} (t/k)

= t →

∞

∑

n=0

x n χ (k n , k (n+1)] (t) = π(σ _k x ).

Hence,

θ ○ σ _k = γ ○ π ○ σ k = γ ○ σ k ○ π = γ ○ π = θ , that is, θ = θ ○ σ _k for every k ∈ N.

Since an operator C ○ σ _k − σ _k ○ C maps the space l _∞ to the space c 0 of vanishing sequences (see e.g. [25, Lemma q16]) and every Banach limit vanishes on c 0 , it follows that

B ○ σ k = θ ○ C ○ σ k = θ ○ σ k ○ C = θ ○ C = B, for every k ∈ N. Thus B ∈ ⋂ ^∞ k =2 B(σ k ).

The following result is a direct corollary of Theorem 3.2 and Proposition 3.6. It provi- des a necessary condition for D _P -measurability.

3.7. Corollary. If A ∈ L _{1 ,∞} is D _P -measurable, then there exists c ∈ C such that B({

2

−2

∑

k =2

−1

˜ µ(k , A)}

n⩾0

) = c

for every B ∈ ⋂ ^∞ _{k =2} B(σ k ).

With the aid of Proposition 3.6 the preceding result can be stated in a sharper form.

3.8. Proposition. If A ∈ L _{1 ,∞} is D P -measurable, then there exists c ∈ C such that

θ ({

1 n

2

∑

k =0

µ(k , A)} ˜

n⩾0

) = c

for every θ = θ ○ σ k , ∀ k ∈ N.

Similarly to the results of L. Sucheston [27] and G. G. Lorentz [20], it can be proved (see Appendix) that θ (x ) = c for every θ = θ ○ σ _k if and only if

lim

n→∞

1 n

n−1

∑

i =0

(σ _k ⁱ x ) _j = c (5)

uniformly in j ∈ N.

The following result is a necessary condition for D _P -measurability given in arithmetic form. Its proof is a combination of (5) and Proposition 3.8.

3.9. Theorem. If A ∈ L _{1 ,∞} is D P -measurable, then for every k ∈ N, k ⩾ 2 the limit

lim

m→∞

1 m

m−1

∑

i =0

(σ _k ⁱ { 1 n

2

∑

k =0

µ(k , A)} ˜

n⩾0

)

j

exists uniformly in j ∈ N.

Now we discuss the relation between the class D _P and the class D _M of all Dixmier tra- ces Tr _ω generated by M-invariant extended limit ω. The notion of M-invariant extended limits was given in Introduction.

3.10. Theorem. For every exponentiation-invariant extended limit ω there exists an M-in-

variant extended limit ω 0 such that

Tr _ω (A) = Tr _ω

0

(A)

for every A ∈ L _{1 ,∞} . That is, D P ⊆ D M . Proof. For A ∈ L _{1 ,∞} denote y n = ∑ ²

n+1

−2

k =2

−1 µ(k , A), n ⩾ 0. ˜

By Theorems 3.2 and 3.4, for every exponentiation-invariant extended limit ω on L _∞ we have

Tr _ω (A) = (γ ○ π ○ C)(

2

−2

∑

k =2

−1

˜

µ(k , A)) = γ(π(C y)),

where γ is a dilation-invariant extended limit.

In was proved in [26, Equation 17] that for every x ∈ l _∞ the function

(H π(x ) − π(C x ))(t) → 0 as t → ∞. (6)

So

Tr ω (A) = γ(H(π( y))).

For every x ∈ L _∞ (0, ∞) we have

∣ d d t

(t ↦ 1 e ^t ∫

e

0

x (s)d s)∣ = ∣ − 1 e ^t ∫

e

0

x (s)d s +

e ^t ⋅ x (e ^t ) e ^t

∣

⩽ ∣(H x )(e ^t )∣ + ∣x (e ^t )∣ ⩽ 2∥x ∥ l

.

Hence the function t ↦ (H x )(e ^t ) is Lipschitz and, consequently, uniformly conti- nuous. Therefore, by [28, Lemma 39], there exists a dilation and H-invariant extended limit γ 0 such that the equality γ(x ) = γ 0 (x ) holds for every x ∈ L ∞ (0, ∞) such that the function t ↦ x (e ^t ) is uniformly continuous. In particular,

Tr ω (A) = γ 0 (H(π( y))) = (γ 0 ○ π)(C y), where γ 0 is an H-invariant extended limit.

Using (6), we conclude that γ 0 ○ π is a Cesàro-invariant extended limit on l ∞ and so (γ 0 ○ π)(C y) = (γ 0 ○ π)( y). Thus,

Tr ω (A) = B({

2

−2

∑

k =2

−1

˜ µ(k , A)}

n⩾0

), (7)

where B ∶= γ 0 ○ π is a Cesàro-invariant extended limit on l ∞ . By [26, Theorem 5.15], the right-hand side of (7) coincides with

ω ₀ (t ↦ 1 log(t) ∫

t

0

˜

µ(s, A)d s),

where ω ₀ is an M-invariant extended limit on L _∞ (0, ∞). This proves the assertion.

3.11. Corollary. Every D _M -measurable operator from L _{1 ,∞} is D _P -measurable.

The following result is a sufficient condition for D P -measurability. It is based on re- sults about the Cesàro-invariant extended limits on l _∞ [25].

3.12. Proposition. An operator A ∈ L _{1 ,∞} is D _P -measurable if

lim

m→∞

lim inf

n→∞

C ^m {

2

−2

∑

k =2

−1

˜ µ(k , A)}

n⩾0

= lim

m→∞

lim sup

n→∞

C ^m {

2

−2

∑

k =2

−1

˜ µ(k , A)}

n⩾0

.

Proof. Let A ∈ L _{1 ,∞} . By [26, Proposition 7.8], if the condition above holds, then A is D M -measurable. Hence, A is D _P -measurable by Corollary 3.11.

We finish the paper by discussing the relation between the classes of D _P -measurable operators and Tauberian operators.

3.13. Definition. An operator A ∈ M _{1 ,∞} is said to be Tauberian if the function t ↦

1 log t ∫

t

0

˜

µ(z, A)d z is convergent as t → +∞.

It is straightforward from the definition of Dixmier traces that every Tauberian ope- rator A ∈ M _{1 ,∞} is Dixmier measurable (and, in particular, D _P -measurable). The converse fails to hold (see [24, Corollary 11]). However, it is known that every positive operator A ∈ M _{1 ,∞} is Tauberian if and only if it is Dixmier measurable [17] (see also [24, The- orem 5]).

In [18, Question 7.4] was raised the question whether the class of positive Tauberian operators from the Marcinkiewicz space M _{1 ,∞} coincides with the class of all D P -measu- rable operators. This question was answered in the negative in [30, Theorem 8], where it was proved that there is a positive operator from M _{1 ,∞} which is D P -measurable but not Dixmier measurable (and hence non-Tauberian (see [17])). The example below constructs such an operator from the smaller ideal L _{1 ,∞} .

3.14. Proposition. There exists a positive operator in L _{1 ,∞} which is D _P -measurable and not Dixmier measurable (and hence non-Tauberian).

Proof. Consider the sequence y =

∞

∑

n=1

χ ₍₂

_{, 2}

] + 1 ∈ l ∞

and a positive operator A = diag(D y) ∈ L _{1 ,∞} .

It was proved in [26, Theorem 7.9] that A is not Dixmier measurable and A is D _M -me- asurable. Applying Corollary 3.11, we conclude that the operator A is also D _P -measurable.

By [17], A is not Tauberian.

4. Appendix

In this section we prove analogues of the results of L. Sucheston [27] and G. G. Lorentz [20]

for extended limits on l _∞ invariant under the dilation operator σ _k , k ∈ N, k ⩾ 2.

For every k ∈ N, k ⩾ 2, consider the following functional on l ^∞ : p k (x ) = lim

n→∞

sup

j∈N

1 n

n−1

∑

i =0

(σ _k ⁱ x ) j , x ∈ l ∞ .

The following proposition show that p k is well defined on l ∞ for every k ∈ N, k ⩾ 2.

4.1. Proposition. For every k ∈ N, k ⩾ 2 the functional p ^k is well defined, convex and positively homogeneous on l ∞ .

Proof. Let k ∈ N, k ⩾ 2. Fix x ∈ l ^∞ and set a _n ∶= sup

j∈ N n−1

∑

i =0

(σ _k ⁱ x ) _j , n ⩾ 1.

For every n, m ⩾ 1 we have a _n+m = sup

j∈ N

(

n−1

∑

i =0

(σ _k ⁱ x ) _j +

m−1

∑

i =0

(σ _k ⁱ x ) _j+n ) ⩽ a n + a m , (8) that is, the sequence {a n } _n⩾1 is convex. Thus, by the classical result due to M. Fekete [12], the sequence {a n /n} _n⩾1 is convergent. In other words, the limit

lim

n→∞

sup

j∈N

1 n

n−1

∑

i =0

(σ _k ⁱ x ) j

exists for every x ∈ l ∞ .

The convexity of p k follows from (8) and, clearly, p k is positively homogeneous on l ∞ .

Now we prove some simple properties of the functionals p k , which we use below.

4.2. Lemma. For every k ∈ N, k ⩾ 2 the functional p ^k satisfies the following properties:

(i) p k (x ) ⩾ 0 whenever x ⩾ 0;

(ii) p k (1) = 1;

(iii) p k (σ k x − x ) = 0 for every x ∈ l ∞ .

Proof. The first two properties are straightforward. For the third one, we have

∣ p _k (σ k x − x )∣ = ∣ lim

n→∞

sup

j∈ N

1 n

n−1

∑

i =0

(σ _k ⁱ (σ k x − x )) _j ∣

= ∣ lim

n→∞

sup

j∈ N

1 n

n−1

∑

i =0

(σ _k ^{i +1} x − σ ⁱ

k x )) _j ∣

= ∣ lim

n→∞

sup

j∈ N

(σ ⁿ

k x − σ ⁰

k x )) j

n

∣ ⩽ lim

n→∞

2∥x ∥ l

n

= 0.

4.3. Remark. It follows immediately from the proof of Lemma 4.2 that the properties listed in Lemma 4.2 are also satisfied by the functional

− p k (−x ) = lim

n→∞

inf

j∈ N

1 n

n−1

∑

i =0

(σ _k ⁱ x ) j , x ∈ l ∞ .

The following theorem is an analogue of L. Sucheston’s result [27]. Note that in this theorem we consider real-valued sequences only. As usual, for complex-valued sequences extended limits are defined by the following formula:

θ (x ) ∶= θ (R(x)) + iθ(I(x)), x ∈ l _∞ , where (R(x)) n = R(x n ) and (I(x)) n = I(x n ), n ⩾ 0.

4.4. Theorem. Let k ∈ N, k ⩾ 2. For every x ∈ l ^∞ and a ∈ R the following assertions are equivalent:

(1) − p _k (−x ) ⩽ a ⩽ p _k (x ).

(2) There exists a σ _k -invariant extended limit θ on l _∞ such that θ (x ) = a.

Proof. (1)⇒(2) Let − p _k (−x ) ⩽ a ⩽ p k (x ). Define θ on R + xR by the formula θ (α ₁ + α 2 x ) = α ₁ + α 2 a.

We have that θ ⩽ p _k on R + xR. Indeed, since p ^k (1) = 1 and p _k is positively homoge- neous, it follows that

θ (α ₁ + α ₂ x ) = α ₁ + α ₂ a ⩽ α ₁ + ∣α ₂ ∣ p _k (sgn(α ₂ )x ) = p _k (α ₁ + α ₂ x ).

By the Hahn–Banach theorem, θ can be extended to l _∞ preserving the inequality θ ⩽ p _k . By the linearity of θ and the preceding inequality we obtain θ (x ) = −θ (−x ) ⩾ − p k (−x ).

By Lemma 4.2 and Remark 4.3, we have

(i) If x ∈ l ∞ is positive, then 0 ⩽ − p k (−x ) ⩽ θ (x );

(ii) 1 = − p k (−1) ⩽ θ (1) ⩽ p k (1) = 1;

(iii) 0 = − p k (−(σ k x − x )) ⩽ θ (σ _k x − x ) ⩽ p _k (σ k x − x ) = 0.

So θ is a σ k -invariant extended limit on l ∞ and θ (x ) = a.

(2)⇒(1) Suppose that there exists a σ _k -invariant extended limit θ on l _∞ such that θ (x ) = a.

We will show that − p _k (−x ) ⩽ θ (x ) ⩽ p _k (x ).

Fix n ∈ N and consider the sequence z _j ∶=

1 n

n−1

∑

i =0

(σ _k ⁱ x ) _j , j ∈ N.

Since the functional θ is σ k -invariant, it follows that θ (z) =

1 n

n−1

∑

i =0

θ (σ _k ⁱ x ) = 1 n

n−1

∑

i =0

θ (x ) = θ (x ).

Hence for every n ∈ N we have θ (x ) = θ (z) ⩽ sup

j∈N

z j = sup

j∈N

1 n

n−1

∑

i =0

(σ _k ⁱ x ) j .

Letting n → ∞ and using Lemma 4.1, we get θ (x ) ⩽ p _k (x ). By the linearity of θ and the preceding inequality, we obtain θ (x ) = −θ (−x ) ⩾ − p k (−x ).

The following theorem is an analogue of G. G. Lorentz’ result [20].

4.5. Theorem. Let x ∈ l _∞ , c ∈ C and k ∈ N, k ⩾ 2. The equality θ(x) = c holds for every extended limit θ on l _∞ such that θ = θ ○ σ _k if and only if

lim

n→∞

1 n

n−1

∑

i =0

(σ _k ⁱ x ) _j = c uniformly in j ∈ N.

Proof. Since

θ (x ) ∶= θ (R(x)) + iθ(I(x)), x ∈ l ∞ ,

it follows that the equality θ (x ) = c holds if and only if the following equalities hold:

θ (R(x)) = R(c) and θ (I(x)) = I(c).

Hence, without loss of generality, one can assume that the sequence x is real-valued and c ∈ R.

Let k ∈ N, k ⩾ 2. Suppose that θ(x) = c for every extended limit θ on l ^∞ such that θ = θ ○ σ k . Theorem 4.4 tells us that for every x ∈ l ∞ the set

{θ (x ) ∶ θ = θ ○ σ _k }

coincides with the closed interval [− p _k (−x ), p _k (x )]. Hence p _k (x ) = − p _k (−x ) = c. Thus, it follows from the definition of p _k that for every ε > 0 there exists N > 0 such that for all n > N we have

sup

j∈N

1 n

n−1

∑

i =0

(σ _k ⁱ x ) j < c + ε.

Hence

1 n

n−1

∑

i =0

(σ _k ⁱ x ) j < c + ε for every j ∈ N, n > N .

Applying similar arguments to the expression − p k (−x ), we obtain that there exists N > 0 such that for n > N we have

c − ε <

1 n

n−1

∑

i =0

(σ _k ⁱ x ) j < c + ε for every j ∈ N.

In other words,

lim

n→∞

1 n

n−1

∑

i =0

(σ _k ⁱ x ) _j = c uniformly in j ∈ N.

Now let

lim

n→∞

1 n

n−1

∑

i =0

(σ _k ⁱ x ) j = c

uniformly in j ∈ N. Then, for every ε > 0 there exists N > 0 such that for n > N we have

c − ε <

1 n

n−1

∑

i =0

(σ _k ⁱ x ) _j < c + ε for every j ∈ N.

Since the above inequalities hold for every j ∈ N, we infer that

c − ε ⩽ inf

j∈ N

1 n

n−1

∑

i =0

(σ _k ⁱ x ) j ⩽ sup

j∈ N

1 n

n−1

∑

i =0

(σ _k ⁱ x ) j ⩽ c + ε, ∀n > N .

Hence c − ε ⩽ − p _k (−x ) ⩽ p _k (x ) ⩽ c + ε. Since ε was arbitrary, we conclude that

− p _k (−x ) = p _k (x ) = c and so, by Theorem 4.4, that θ (x ) = c for every extended limit θ on l _∞ such that θ = θ ○ σ _k .

Acknowledgements. The work of both authors was partially supported by the ARC grant

DP 140100906.

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