On the distinction between the classes of Dixmier and Connes-Dixmier traces

In the present paper we prove that the classes of Dixmier and Connes-Dixmier traces differ even on the Dixmier ideal $\mathcal M_{1,\infty}$. We construct a Marcinkiewicz space $\mathcal M_\psi$ and a positive operator $T\in \mathcal M_\psi$ which is Connes-Dixmier measurable but which is not Dixmier measurable.


Introduction and preliminares
In [5] J. Dixmier proved that there exists a non-normal trace (a Dixmier trace) on the non-commutative Marcinkiewicz spaces M ψ for every ψ such that In [4] A. Connes introduced a subclass of Dixmier traces, later termed in [9] Connes-Dixmier traces. In this paper, we investigate the relationship between these two classes and show that they differ even on the classical Dixmier ideal M 1,∞ . Furthermore, we prove that there is a Marcinkiewicz ideal M ψ , with ψ satisfying (1.1) such that these two classes of traces generate distinct sets of measurable elements (see [ A normalized positive linear functional on l ∞ which equals the ordinary limit on convergent sequences is called a generalized limit. For every n ∈ N we define a dilation operator σ n : l ∞ → l ∞ as follows If a generalized limit ω on l ∞ satisfies the condition ω(σ n x) = ω(x) for every x ∈ l ∞ and any n ∈ N, then ω is called a dilation invariant generalized limit.
Let L ∞ = L ∞ (0, ∞) be the space of all real-valued bounded Lebesgue measurable functions on (0, ∞) equipped with the norm A normalized positive linear functional on L ∞ which equals the ordinary limit on convergent (at infinity) sequences is called a generalized limit. For every x ∈ L ∞ and for any generalized limit γ on L ∞ the following inequalities hold By Hahn-Banach extension theorem, for every x ∈ L ∞ there exist generalized limits γ 1 and γ 2 such that We define a dilation operator σ s : L ∞ → L ∞ as follows for every x ∈ L ∞ and any s > 0. Let π be the isometric embedding π : l ∞ → L ∞ given by x n χ (n,n+1] .
The following natural way to generate dilation invariant generalized limits was suggested in [4, Section IV, 2β]. A. Connes observed that for any generalised limit γ on L ∞ a functional ω := γ • M • π is a dilation invariant generalized limit on l ∞ . Here, the bounded operator M : L ∞ → L ∞ is given by the formula Throughout the paper we denote by log t the natural logarithm and by log 2 t the logarithm with base 2. For a compact operator T , it can be proven that µ(k, T ) is the k-th largest eigenvalue of an operator |T |, k ≥ 0.
Since B(H) is an atomic von Neumann algebra and traces of all atoms equal to 1, it follows that µ(T ) is a step function and µ(T ) = π(µ(k, T )) for every T ∈ B(H).
Let ψ ∈ Ω. Consider the Banach ideal (M ψ , · M ψ ) of compact operators in B(H) given by (see e.g. [2,8,9]) where f * denotes the decreasing rearrangement of the function |f | that is We define the Marcinkiewicz function space M ψ of real-valued measurable functions f on (0, ∞) by setting The subclass C ⊂ D of all Dixmier traces Tr ω defined by ω = γ •M •π was termed Connes-Dixmier traces in [9]. A priori, C ⊆ D and the question about precise relationship between these two classes arises naturally. Recently the distinction between C and D was studied by A. Pietsch in terms of density characters (see [11]- [13]). For the discussion of various classes of singular traces we refer to [1,2,10].
The first main result of the present paper (Theorem 2.2 below) shows that the inclusion C ⊂ D is proper. Our approach is completely different from that of A. Pietsch and the proof provided here is much shorter.
It has become traditional to reduce various problems about Dixmier traces to its commutative analogues.
For every dilation invariant generalized limit ω on L ∞ one can define a commutative analogue of Dixmier trace (a Dixmier functional on M 1,∞ ) as follows and extend it to M 1,∞ by linearity. It was shown in [7,8] that, for a general Marcinkiewicz space M ψ , the following conditions are equivalent (i) The space M ψ admits non-trivial Dixmier traces.
(ii) The function ψ ∈ Ω satisfies the following condition

FEDOR SUKOCHEV, ALEXANDR USACHEV, AND DMITRIY ZANIN
(iii) There exists a dilation invariant limit ω on l ∞ such that It was also proven in [8, Proposition 9, Theorem 11] that for ψ ∈ Ω satisfying (1.4), the weight extends to a Dixmier trace on M ψ if and only if a dilation invariant generalized limit ω on l ∞ satisfies (1.5).
Similarly to (1.3), we define Dixmier and Connes-Dixmier functionals τ ω for every dilation invariant generalized limit ω on L ∞ satisfying The converse implication also holds.
Lemma 1.2. Let ψ ∈ Ω satisfy (1.4) and let ω be a dilation invariant generalized limit on L ∞ satisfying (1.6). For every f ∈ M ψ , we have Proof. Since ω is a dilation invariant generalized limit, Since ω satisfies (1.6), it follows from [8,Proposition 4] that Hence, This result naturally raises the question, whether for an arbitrary function ψ ∈ Ω satisfying (1.4) the Connes-Dixmier measurability is equivalent to Dixmier measurability on the cone of all positive elements from M ψ . Our second main result (Theorem 3.4 below) shows that the answer is (surpisingly) negative.

FEDOR SUKOCHEV, ALEXANDR USACHEV, AND DMITRIY ZANIN
In view of the difference between (2.1) and (2.2), the following question arises naturally: "Is the constant c in (2.2) necessarily strictly greater than 1?" The following theorem shows that the inclusion C ⊂ D is proper and answers this question in the affirmative. Proof. Let T 0 be such that We set f 0 = µ(T 0 ). By Lemma 2.1, we have Clearly, f 0 = f * 0 . For every 2 2 n ≤ t < 2 2 n+1 we have a(t, f 0 ) = 1 log(1 + t) It is easy to check that f 0 ∈ M 1,∞ and, hence, T 0 ∈ M 1,∞ .
= 0 for every generalized limit γ on L ∞ and, appealing to (1.2), we conclude Define the function x ∈ L ∞ by setting Hence, we obtain from ( For 2 2 n ≤ t < 2 2 n+1 , we have it follows that The function has extrema at We have g(t n ) = 2 e log 2 for every n ∈ N. Since g(2 2 n ) = 1 for every n ∈ N and since g is continuous on (1, ∞), it follows that lim sup t→∞ g(t)

The classes of Dixmier and Connes-Dixmier measurable elements are distinct
The following Lemma is taken from [15] (see Theorem 18 or [14, Theorem 6.1.3]).  Proof. The mapping t → a(e t , f ) is uniformly continuous since The following Theorem strengthens the result from [9,Corollary 3.9] in the case when ψ ∈ Ω satisfies (1.1). Proof. Suppose that T ∈ M ψ is Dixmier measurable positive operator, that is Tr ω (T ) = A for every Dixmier trace Tr ω on M ψ . According to Remark 1.1, we have τ ω (µ(T )) = A for every Dixmier functional τ ω on M ψ . Denote, for brevity, f := µ(T ). By Corollary 3.2 we have So, for any N > 0 one can find such t 0 = t 0 (N ) that for every t > t 0 we have By the definition of a limit superior, there exists α > t 0 such that a(t, f ).
A direct computation shows that ψ(t) = 2 √ log 2 (1+t) − 1 does not satisfy (1.8). The following Theorem provides an example of a positive operator T 0 ∈ M ψ which is Connes-Dixmier measurable, however it is not Dixmier measurable.
There exists a positive Connes-Dixmier measurable operator T 0 ∈ M ψ such that the limit in (1.9) does not exist.
However, direct computation shows that We conclude that a(·, f 0 ) has no limit at infinity and, so, a limit in (1.9) does not exist.