Advances in Analysis, Probability and Mathematical Physics”

Proceedings of the Conference “Advances in Analysis, Probability and Mathematical Physics”

Blaubeuren, 1992

Mathematics and its Applications - Kluwer Academic Publishers

1 2

Singular traces and non standard analysis

S. Albeverio1,4,5, D. Guido2, A. Ponosov1,6, S.Scarlatti3

1: FakultätfürMathematik, Ruhr-UniversitätBochum, Germany 2: Dipartimento di Matematica, Universitádi Roma Tor Vergata, Italy 3: Dipartimento di Matematica, Universitádi L’Aquila, Italy 4: SFB-237, Essen-Bochum-Düsseldorf 5: CERFIM, Locarno 6: Supported by the Deutsche Forschungsgemeinschaft

Abstract: We discuss non trivial singular traces on the compact operators, ex- tending some results by Dixmier and Varga. We also give an explicit description of these traces and associated ergodic states using tools of non standard analysis.

Section 1. Introduction

Let H be a complex separable Hilbert space and let B(H) be the Banach algebra of all bounded linear operators on H. It is well-known that every non trivial normal (i.e. continuous in some sense; see Section 2 for precise definitions) trace on B(H) is proportional to the usual one (see e.g. [D1]). On the other hand Dixmier proved in 1966 [D2] that there exist non trivial traces on B(H) which are not normal (moreover they are ”singular” in the sense of definition 2.2 below). The basic idea of Dixmier for constructing singular traces was to consider the compact operators for which the usual trace diverges at a given suitable rate and to associate to any such compact operator a suitable element of the Banach space `∞(N) of all bounded sequences. The singular trace is then obtained evaluating on such elements a state on `∞(N) which is invariant under ”2-dilations”. The importance of singular traces is well-known now due to their applications in non commutative geometry and quantum field theory (see e.g. [C]). The general question which operator ideals in B(H) support traces has been recently studied by Varga [V] who used different kinds of states on `∞(N). In our two preceding papers [AGPS1], [AGPS2] we gave explicit formulas for Dixmier-type traces and introduced a new class of singular traces. The present paper is organized as follows. In the first part of Section 2 we deal with the problem of describing which kind of sequences can be “summed” by a non trivial singular trace. We shall give a complete answer to this problem, generalizing some results by Varga [V]. In the second part of Section 2 we generalize the construction of non-normal traces given by Dixmier in [D2]. 3

In Section 3 we describe ergodic states giving rise to both kinds of singular traces introduced in Section 2. The basic technique we use is related to non standard analysis (NSA). The Section 3 also involves the representation of Banach-Mazur limits by NSA. Such representation have been discussed before - e.g. in [KM], [L]. In Section 4 we work out explicitly the computation of the Dixmier traces of an operator again using the NSA framework in an essential way.

Section 2. Singular traces and generalized eccentric operators

The content of this section is purely standard. So we will omit all the proofs which can be found in [AGPS3]. We start recalling some basic deﬁnitions and results. Let R a von Neumann algebra and R+ the cone of its positive elements. A weight on R is a linear map

φ : R+ → [0, +∞]

Any weight can be extended by linearity on the natural domain given by the linear span of {T ∈ R+|φ(T ) < +∞} A weight τ which has the property:

τ(T ∗T ) = τ(TT ∗) ∀T ∈ R is called a trace on R. The natural domain of a trace is a two–sided ideal denoted by Iτ . For instance the natural domains of the trivial traces on R given by τ ≡ 0 and τ ≡ +∞ are respectively the ideals R and {0} while the usual trace on B(H), the bounded linear operators on a complex, separable Hilbert space H, is associated with the ideal L1(H) of the trace class operators. A weight φ on R is called normal if for every monotonically increasing generalized + sequence {Tα, α ∈ I} of elements of R such that T = supα Tα one has

φ(T ) = limφ(Tα) α

From now on the von Neumann algebra R will be ﬁxed to be B(H). A classical result [D1] concerning normal traces on B(H) is the following:

2.1 Theorem (Dixmier). Every non trivial normal trace on B(H) is proportional to the usual trace. By a theorem of Calkin (see [GK]), each proper two-sided ideal in B(H) contains the ﬁnite rank operators and is contained in the ideal K(H) of the compact linear operators on H. Therefore all traces on B(H) live on the compact operators, and the following deﬁnition makes sense:

2.2 Deﬁnition. A trace τ on B(H) will be call singular if it vanishes on the set F (H) of ﬁnite rank operators.

2.3 Proposition. Any trace τ on K(H) can be uniquely decomposed as τ = τ1 + τ2, where τ1 is a normal trace and τ2 is a singular trace. 4

In view of this proposition, in the rest of the paper we shall restrict our attention to the singular traces. ∞ For T a compact operator on H, {µn(T )}n=1 will denote the non increasing sequence of the eigenvalues of |T | with multiplicity. n P We shall also set σn(T ) ≡ µk(T ). k=1

2.4 Deﬁnition. Let T be a compact operator. We call integral sequence of T the ∞ sequence {Sn(T )}n=0 which is an indeﬁnite integral (w.r.t. the counting measure) ∞ of {µn(T )}n=1, i.e. Sn(T ) − Sn−1(T ) = µn(T ), n ≥ 1, and such that 0 T/∈ L1(H) S0(T ) ≡ −tr(T ) T ∈ L1(H)

1 1 Notice that if T/∈ L (H),Sn(T ) = σn(T ), n ≥ 1, while if T ∈ L (H), then Sn(T ) = σn(T ) − tr(T ) → 0. We also give the following

2.5 Remark. If T does not belong to L1(H) and τ is a trace which is ﬁnite and non-zero on |T | then τ is necessarily singular, that is, the existence of traces which are non trivial on T is equivalent to the existence of non trivial singular traces on T. Since for T ∈ L1(H) the existence of a non trivial trace is obvious, it follows that the relevant question is not the mere “traceability” of a compact operator T, but the existence of a singular trace which is non trivial on |T |. Let us also notice that a trace τ is ﬁnite on |T | if and only if the principal ideal I(T ), i.e. the (two–sided) ideal generated by T in B(H), is contained in Iτ .

2.6 Lemma. Let T be a compact operator. The following are equivalent ∞ n S2n(T ) o (i) 1 is a limit point of the sequence S (T ) n n=0

Skpk (T ) (ii) There exists an increasing sequence of natural numbers {pk} such that lim S (T ) = k→∞ pk 1

2.7 Deﬁnition. A compact operator T which satisﬁes one of the equivalent properties of Lemma 2.6 will be called generalized eccentric.

2.8 Remark. The class of generalized eccentric operators which are not in L1(H) coincides with the class of eccentric operator considered in [V]. We can now state the main result of this section.

2.9. Theorem. Let T be a compact operator. Then the following are equivalent: (a) There exists a singular trace τ such that 0 < τ(|T |) < +∞. (b) T is generalized eccentric. As we have already mentioned the proof of the theorem can be found in [AGPS3]. Nevertheless we are going to discuss here some possible procedures to construct singular traces on K(H) because it is important for the non standard part of this work. 5

Basically, all singular traces in question can be represented as follows: Sn(A) τψ(A) ≡ ψ ,A ∈ I(T ) (2.1) Sn(T ) where ψ is a generic non normal state on `∞(N) ≡ `∞. Recall that the state is just a linear bounded positive functional with the property ψ({1}) = kψk = 1. It is easy to see that for such a ψ the function τψ has all properties of traces with the only exception of additivity which requires some more conditions to be imposed upon. In the literature there are two ways to do this. Although both of them were introduced in more particular situations than we are dealing with they can be adapted to our situation too. The ﬁrst way is due to Dixmier [D] and consists in choosing ψ to be 2-dilation invariant (therefore ψ vanishes on the set c0 of the inﬁnitesimal sequences, hence it is not normal):

∞ ψ({an}) = ψ({a2n}) ∀a ∈ ` .

In this case the generalized eccentric operator T must satisfy the stronger condition:

S (T ) lim 2n = 1, n∈N Sn(T )

(compare with the conditions given in Lemma 2.6). The second way of getting additivity was suggested by Varga [V] who considered non normal states supported by ”fast” sequences, i.e. the states of the form:

∞ ψ({an}) = ϕ({ank }) ∀a ∈ ` . (2.2)

∞ where ϕ is an arbitrary non normal state on ` and the sequence nk is given by nk ≡ kpk with pk deﬁned in Lemma 2.6. We now formulate brieﬂy two results on the existence of singular traces.

2.10 Theorem. Let T be a generalized eccentric operator. The functional τψ deﬁned in (2.1) and (2.2) linearly extends to a singular trace on the ideal I(T ).

S (T ) 2.11 Theorem. If ψ is a two-dilation invariant state and lim 2n = 1, then Sn(T ) n∈N τψ is a trace on I(T ). Moreover, in this case formula (2.1) gives rise to a singular trace (which will be denoted by τψ as well) even on the (larger) ideal Sn(A) ∞ Im(T ) ≡ A ∈ K(H)| ∈ ` Sn(T )

1 We would like to point out that, when T 6∈ L (H), the ideal Im(T ) is a maximal norm ideal in the sense of Shatten [S] (see also [GK]).

2.12 Remark. We should stress that we did get new types of singular traces. Of course, if the operator T/∈ L1(H) then the traces described in Theorems 2.10 and 2.11 are exactly the traces discussed by Dixmier in [D2] and by Varga in [V] 6 respectively, and they vanish on L1(H).. In the case T ∈ L1(H) our Theorems 2.10 and 2.11 produce new classes of singular traces, which in a sense are the inverse images inside L1(H) of the class of Varga-type traces and of the class of Dixmier- type traces respectively. We notice also that if the trace τ2 in the decomposition of proposition 2.3 does not vanish, it should be of this new type, i.e. it should not vanish on L1(H). The existence of new Dixmier-type traces was announced in [AGPS 2].

Section 3. Two-dilation invariant states and ergodicity

The main problem we are going to discuss in this section concerns extremal (ergodic) states which give rise to singular traces. In our opinion it is non standard analysis which supplies us the most convenient tools for this purpose. We ﬁrst brieﬂy discuss extremal states corresponding to the Varga-type traces described in Theorem 2.11.

Let us denote by ∆{nk} the set of all extremal points in the set of non normal states of the form (2.2).

3.1 Proposition. The set ∆{nk} consists of the states

◦ ∗ ψ({an}) ≡ ( anm )

∗ ∗ for some m ∈ N∞ = N − N. Proof. It immediately follows from the fact that extremal states on `∞ are all of the form ◦ ∗ ∗ ϕ({an}) = ( am), n ∈ N. (3.1)

Since additionally ϕ must vanish on the set c0 then m becomes inﬁnitely large.

3.2 Remark. Of course, instead of infinitely large numbers one can equivalently use the Stone-Cechˇ compactification N of N and the isomorphism `∞(N) ' C(N) given by the Gelfand transform in order to describe extremal states in Proposition 3.1. Namely, they will be given by Dirac measures supported by the set N − N. On the contrary, the classification of ergodic 2-dilation invariant states does require NSA (see e.g. the next remark). Now we come to the much more difficult problem of classification of two dilation invariant states. First, we remark that in order to prove the existence of such states, Dixmier invoked the amenability of the affine group. As promised, here we shall adopt an alternative point of view which relies on the use of NSA and related methods (see e.g. [HL], [AFHKL]).

∗ 3.3 Theorem. The map ω → ϕω, ω ∈ N∞, deﬁned by ω ◦ 1 X ∗ ϕ (a) ≡ a k (3.2) ω ω 2 k=1 takes values in the convex set of 2–dilation invariant states over `∞. 7

1 Pn Proof. Let bn ≡ n k=1 an. Since {an} is bounded {bn} is also bounded so that ◦ ∗ ϕω(a) = ( bω) is well deﬁned for all ω. Obviously, ϕω is a state. It is also 2–dilation invariant since: ◦ 1 ∗ ∗ ϕ ({a }) − ϕ ({a }) = ( a ω − a ) = 0. ω 2n ω n ω 2 +1 2

A consequence of this theorem is that an explicit formula for the previously introduced traces can easily be given.

S2n 3.4 Corollary. If T is an operator verifying limn = 1 and ω is an inﬁnite Sn hypernatural then

ω ∗ ◦ 1 X S2k (A) τω(A) ≡ ∗ A ∈ Im(T ) (3.3) ω S k (T ) k=1 2 is one of the singular traces described in Theorem 2.12. The proof of this corollary follows immediately from Theorems 2.12 and 3.3. There is a simple generalization of the formula (3.2) which describes 2-dilation ∗ ∗ invariant states. If j ∈ N and n ∈ N∞ the map n ◦ 1 X ∗ {a } → a i (3.4) k n j2 i=1 is a 2–dilation invariant state over `∞ and therefore gives rise to a singular trace. Since any hypernatural j can be written in a unique way as a product of an odd number and a power of 2, j = (2m − 1)2k−1, we may rewrite the previous states as

k+n ◦ 1 X ∗ ∗ ∗ ϕ (a) = a i−1 k, m ∈ , n ∈ (3.5) k,m,n n (2m−1)2 N N∞ i=k+1

In the rest of the section we shall study states of the form (3.5) in relation to the problem of ergodicity. Let ∆ : N → N the multiplication by 2, ∆∗ the corresponding morphism on ∞ ` (N), ∆∗({an}) = {a2n}, we shall say that the state ϕ is ∆-invariant if ϕ◦∆∗ = ϕ. We shall give necessary conditions for extremality in the (convex compact) set of ∆-invariant states in terms of NSA. It is known (see e.g. [E, p.113]) that the states on `∞(N) can be identiﬁed with the ﬁnitely additive probability measures on N, therefore we shall denote any such a state by µ, and the notation µ(A) with A ⊂ N makes sense. Moreover extremality of a ∆-invariant state µ can be expressed in terms of ergodicity of µ seen as a measure, i.e. for each A ⊂ N such that ∆A = A, µ(A) = 0 or 1.

3.5 Remark. Using the Stone-Cechˇ compactification N of N and the isomorphism `∞(N) ' C(N) given by the Gelfand transform once again we get an 8 identification of the states on `∞(N) with the σ-additive probability Radon measures on N. On the other hand a transformation on N extends to a continuous transformation on N. We shall denote with µ, ∆ the measure and the transformation on N induced by µ and ∆ respectively. It turns out that ergodicity of µ is equivalent to ergodicity of the finitely additive measure µ. This equivalence can be shown using well known criteria for ergodicity (see e.g. [Ma]).

3.6 Remark. Let us consider the correspondence η : N × N → N deﬁned n−1 by (m, n) → (2m − 1)2 , which is a bijection. It induces an isomorphism: η∗ : `∞(N) → `∞(N × N) given by

(η∗a)m,n ≡ aη(m,n),

−1 ∆∗ ≡ η∗ ∆η∗ becoming the translation T in the second variable: ∆∗(m, n) = (m, n + 1) ≡ (m, T n). It might be thought that the isomorphism η∗ gives rise to the splitting of the dynamical system (N, ∆) in a product of two dynamical systems (N, id) and (N, T ), thus furnishing a standard approach to the considerations we shall make below. Unfortunately this is not true since the spaces N × N and N × N are not homeomorphic (see e.g. [G]). Our idea is to exploit the features of nonstandard analysis, in particular the nice functorial property ∗N × ∗N =∗ (N × N). ∞ Let M∆ denote the set of extremal ∆-invariant (i.e. ergodic) states on ` (N).

3.7 Proposition. Any µ ∈ M∆ coincides with one of the states ϕk,m,n for some ∗ ∗ k, m ∈ N, n ∈ N∞. Proof. Evidently, for each state µ and each ﬁnite dimensional E ∈ `∞(N) ≡ `∞ there exist numbers s(E) and λj(E) such that Ps(E) Ps(E) (i) µ(a) = j=1 λj(E)aj (∀a ∈ E), (ii) λj(E) ≥ 0, (iii) j=1 λj(E) = 1. For every hyperﬁnite dimensional space E (see e.g. [AFHKL, p.55]), `∞ ⊂ E ⊂ ∗`∞, there is therefore an internal set {λ1, . . . , λs} satisfying (ii) and (iii) and such that

s ∗ ∗ X ∗ ∞ µ(a) = µ( a) = λj aj (∀a ∈ ` ) j=1

If µ is 2-dilation invariant, then for all ﬁnite n

s n X 1 X ∗ ∞ µ(a) = λ a i−1 (∀a ∈ ` ). j n j2 j=1 i=1

As it can be easily checked, for each ﬁnite dimensional space F ⊂ `∞, and each n ∈ N we have ( s n ∗ X 1 X αn,∗F ≡ sup sup | µ(a) − λj aj2i−1 |+ ∗ n a∈ F,kak≤1 1≤j≤s j=1 i=1

n ) ∗ 1 X +| ϕ (a) − a i−1 | ≈ 0 (3.6) k,m,n n j2 i=1 9 where j = (2m − 1)2k−1. By saturation, there are a hyperﬁnite dimensional space ∞ ∗ ∞ ∗ F , ` ⊂ F ⊂ ` and a number n ∈ N∞ such that (3.6) remains valid. This means that s ∗ X ∗ k−1 µ(a) ≈ λj ϕm,k,n(a)(j = (2m − 1)2 ) (3.7) j=1 for all a ∈ `∞ and immediately implies that the set of all T −invariant states ∗ ∗ coincides with the closed convex hull coM of the set M ≡ {µk,n | k ∈ N, n ∈ N∞}. Finally, we show that M is closed which, according to [E, p.708], would im- ∗ 3 ply the inclusion M∆ ⊂ M. Consider a directed set Γ ⊂ N and assume that Γ ∞ ϕk,m,n→µ. Clearly, for every ﬁnite dimensional E ⊂ ` there exists a subsequence {(ki, mi, ni)} ∈ Γ such that

∗ ∗ ∗ ∗ ϕki,mi,ni (a) ≈ µ(a) for some i ∈ N∞ and for all a ∈ E, kak ≤ 1. If µ is ∆ invariant then by saturation the latter relation holds for a certain hyper- ∞ ∗ ∗ ﬁnite dimensional E ⊃ ` with k ∈ N and n ∈ N∞. This concludes the proof.

3.8 Corollary. The problems of description of extremal 2-dilation invariant and extremal translation invariant states are equivalent.

Proof. Recalling the map η∗ given by (η∗a)m,n = aη(m,n) where η∗(m, n) = (2m − 1)2n−1 and applying the proposition just proved we conclude that for each ﬁxed ∞ m the states ϕk,m,n ◦ η∗ coincide with the translation invariant states µk,n on ` deﬁned by k+n ◦ 1 X µ (a) ≡ ∗a , k,n n i i=k+1 −1 so that any µ ∈ η∗ (M∆) is contained in one of the sets {δm ⊗ ν|ν ∈ MT ∩ M} ◦ ∗ ∗ where δm is given by δm(a) ≡ ( am), m ∈ N, MT stands for the set of extremal ∞ ∗ ∗ translation invariant states on ` and M ≡ {µk,n|k ∈ N, n ∈ N∞}. On the other hand it is known (see [KM], [L], or also [AGPS1] where the result was proved independently) that MT ⊂ M which completes the proof.

3.9 Remark. Of course, the representation given in Proposition 3.7 is not unique due to Corollary 3.8 and the following trivial

k−l p−n 3.10 Proposition. If n ≈ 0 and n ≈ 0, then µk,n = µl,p. Now we formulate the main result in this section more precisely.

3.11 Theorem. If µ is an extremal 2-dilation invariant state on `∞ then ∗ µ = ϕk,m,n (with ϕk,m,n deﬁned by (3.5)) for some m ∈ N and inﬁnitely large n hypernaturals k and n such that k ≈ 0.

Proof. By virtue of Corollary 3.8 it suffices to show that if ν ∈ MT then ν = µk,n ∗ ∗ for some k, n ∈ N∞. We first prove that k ∈ N∞. Suppose it is not the case and k is finite. Without loss of generality we can assume k = 1, and, due to proposition 3.10, n = 2m. If we show that µ1,n 6= µ1,m then the representation 1 µ1,n = 2 (µ1,m + µm,m) implies µ1,n is not extremal. 10

−p ∗ o For b ≡ m2 , choose p ∈ N∞ such that 0 < b < ∞ and consider a non ∗ decreasing sequence {bj} such that b2p ≈ b. Deﬁne a sequence {cj} of natural j numbers by putting cj ≡ [2 bj] ([·] denotes the integer part). Since 1 1 1 c ≤ [2j+1b ] + 1 ≤ [2j+1b ] + 1 = c + 1, j 2 j 2 j+1 2 j+1 one gets cj+1 ≥ 2cj − 2. At the same time,

∗c − m ∗c − m 22p [22p∗b ] 1 o 2p = o 2p o = o 2p − b m 22p m 22p ob 1 ≤ o(∗b − b) = 0. ob 2p

∗ c2p+1−n By the same reason, n ≈ 0. Applying proposition 3.10 once again we obtain

∗ ∗ m c2p n c2p+1 1 X 1 X 1 X 1 X ∗a ≈ ∗a ; ∗a ≈ ∗a (∀a ∈ `∞) (3.8) m i ∗c i n i c i i=1 2p i=1 i=1 2p+1 i=1

∞ Now we introduce a set B = ∪q=1[c2q−1, c2q] ⊂ N and a sequence χ = {χi} where χi = 1 for i ∈ B and 0 otherwise. By (3.8),

∗  c2p+1  1 X µ (B) = o ∗χ = 1,n  ∗c i 2p+1 i=1  ∗ ∗  ∗ c2p c2p+1 o c2p o 1 X ∗ 1 X ∗ ∗  ∗ χi + ∗ χi = c2p+1 c2p c2p ∗ i=1 i= c2p+1 p∗ o [2 cp] 1 = p+1∗ µ1,m(B) ≤ µ1,n(B). [2 cp+1] 2

It remains to prove that µ1,m(B) 6= 0. In order to see this, let us observe that ∗ ∗ ∗ ∗ ∗ ∗ ∗ χi = 1 for i ∈ [ c2p−1, c2p] and that c2p − c2p−1 ≥ c2p−1 −2. Hence, ]{i | χi = 1 ∗ 1} ≥ 2 c2p − 1 and

∗  c2p  1 X 1 1 1 µ (B) = o ∗χ ≥ o ∗c − 1 = , 1,m  ∗c i ∗c 2 2p 2 2p i=1 2p

∗ n so that µ1,n is not extremal. We continue the proof assuming k ∈ N∞ and k 6≈ 0 o k or, equivalently, n < ∞. We have to show that again µk,n is not extremal. First we notice that √ √ √ −k + [ k + 1]2 2 k + 1 2 k 2k 0 ≤ ≤ ≈ = √ ≈ 0 n n n n k and analogously, √ √ √ −k + [ k]2 k + n − [ k + n]2 k + n − [ k + n + 1]2 ≈ 0, ≈ 0, ≈ 0. n n n 11

Applying now proposition 3.10, one can assume that k = (2r)2, k + n = (2s)2. ∞ 2 2 Let us introduce a set C ≡ ∪i=1[(2i − 1) , (2i) ] and observe that

]{i | i2 ∈ [k, k + n]} ]{i | 2r ≤ i ≤ 2s} µ (C4TC) ≤ o = o = 0. k,n n (2s)2 − (2r)2

Extremality of µ would imply, therefore, that µk,n(C) should have been equal to 0 or 1. On the other hand,

s ! 1 X (4s + 1 + 4r + 3)(r − s) 1 µ (C) = o (2i)2 − (2i − 1)2 = o = . k,n n 8r2 − 8s2 2 i=r+1

This contradiction implies the result.

3.12 Remark. The necessary conditions in theorem 3.11 are surely not suﬃcient. ∗ To see this, one can consider a state µ ≡ µ4p−n,n for arbitrary p, n ∈ N∞. If n ≥ 22p−1, then non extremality of µ follows from the theorem. Otherwise, we may ∞ 2q−1 2q introduce a set A ≡ ∪q=1(2 , 2 ] which is easily seen to be µ−a.e. T −invariant, 1 but µ(A) = 2 . Now we give a Corollary which relates the results of this section with the description of the Dixmier-type traces (for the proof cf. [AGPS1]).

3.13 Corollary. Let τ be a Dixmier-type trace on the ideal Im(T ) (see section 2). Then τ is in the closure of the convex hull of the family

n {τ | m ∈ ∗ , k, n ∈ ∗ , ( ) ≈ 0}, k,m,n N N∞ k

where τk,m,n is the trace associated with the state τϕk,m,n via formula (2.1) on the same domain Im(T ).

3.14 Remark. The states µk,n can be seen intuitively as averages on intervals of the set ∗N. This suggests to call ergodic all the intervals associated with ergodic states. Then it is easy to show that if the interval I is ergodic and a subinterval J is |I| such that |J| 6≈ 0 (where |I| denotes the length of I), then µI = µJ . A sketch of the proof is the following: let I = I0 ∪ J ∪ I1 be a partition of I into subintervals. It turns out that

|I | |J| |I | o 0 µ + o µ + o 1 µ = µ , |I| I0 |I| J |I| I1 I hence, by the ergodicity of I, µI = µJ .

Section 4. A computational example

We shall now discuss some advantages of representing singular traces by means of NSA. 12

A remarkable advantage lies, in our opinion, in the increased computability of the value of a singular trace on a given operator when such a trace is parameterized by some infinite number. In what follows, we shall work out an example in which we explicitly calculate the value of the Dixmier trace of an operator, even though it depends on the nonstandard parameter. To this aim we shall make use of formula (3.3) choosing a compact operator T such that Sn(T ) = log n. The choice of ”summing” logarithmic divergences has extensively been used by Connes in some applications to non-commutative geometry [C]. Let q ≥ 1 be a fixed natural number, we consider a positive compact operator Aq whose sequence of eigenvalues (λn | n = 3, 4.... ) is defined in the following way: let (nk | k = 0, 1,... ) be an unbounded increasing sequence of natural numbers (with n n n0 ≡ 1) whose explicit dependence on q will be given below. For n ∈ (2 k , 2 k+1 ], we define nk+1 − nk λn := (4.1) 2nk+1 − 2nk

2m P For m ≥ 2 we consider the sum σ2m := λj. j=3

Let nk < m ≤ nk+1, then we have

2m − 2nk σ2m = nk + · (nk+1 − nk) − 1 (4.2) 2nk+1 − 2nk since k−1 2nr+1 2m X X X σ2m = λj + λj. r=0 j=2nr +1 j=2nk +1

Now let p > 1 and hence ns < p ≤ ns+1 for some s, we have

p s−1 nk+1 p ! 1 X σ2m 1 1 X X σ2m X σ2m = · + (4.3) p log 2m log 2 p m m m=1 k=0 m=nk+1 m=ns+1

We now proceed to estimate the sums appearing on the r.h s. of (4.3). By means of (4.2) we have

nk+1 nk+1 nk X σ2m 2 X 1 = nk − 1 − (nk+1 − nk) m 2nk+1 − 2nk m m=n +1 m=n +1 k k (4.4) nk+1 m nk+1 − nk X 2 + · 2nk+1 − 2nk m m=nk+1

We notice that the following equalities hold

nk+1 X 1 nk+1 1 1 = log + O − (4.5a) m nk nk nk+1 m=nk+1 13

nk+1 X 2m 1 2nk+1 2nk 1 = − 1 + O (4.5b) m log 2 nk+1 nk nk m=nk+1 from which it follows

nk+1 X σ2m nk+1 nk 1 = nk log + O 1 − 1 + O (4.6) m nk nk+1 nk m=nk+1

1 2nk under the assumption O ≥ O nk+1 . nk 2

To verify such a condition we ﬁx the initial sequence (nk|k = 0, 1,... ) to be of kq the form nk := 2 , where q ∈ N. Formula (4.6) takes then the form:

nk+1 X σ2m = 2kqq log 2 + O(1) 1 + O(2−kq) (4.7) m m=nk+1

Therefore we obtain

s−1 nk+1 sq 1 X X σ2m 1 2 − 1 = p log 2 + O(s) (4.8) p log 2 m q log 2 2q − 1 k=0 m=nk+1

∗ Now, by deﬁnition, taking p ∈ N∞, we have p ! 1 X σ2m τ Dix(A ) = ◦ ∗ p q p log 2m m=1 s−1 nk+1 ! 1 X X σ2m = ◦ ∗ + p log 2 m k=0 m=nk+1 p ! (4.9) 1 X σ2m + ◦ ∗ p log 2 m m=ns+1 sq p ! 2 q 1 X σ2m = ◦ + ◦ ∗ p 2q − 1 p log 2 m m=ns+1

◦ s where, in the last equality, we have used (4.8) and the fact that p = 0. To end the computation of the Dixmier trace of Aq we need to evaluate the second term in the r.h.s. of (4.9). We have, for p ∈ N,

p 1 X σ2m = p log 2 m m=ns+1 1 n 1 p 1 = s 1 + O log + O + (4.10) log 2 p ns ns ns p ns 1 ns+1 − ns 2 2 1 2 n n − 1 + O p(log 2) 2 s+1 − 2 s p ns ns 14

∗ Hence, by estimates similar to the previous ones, and taking p ∈ N∞ we obtain

p ! sq 1 X σ2m 1 2 p ◦ ∗ = ◦ log ◦ (4.11) p log 2 m log 2 p 2sq m=ns+1

From (4.9) and (4.11) it follows

q τ Dix(A ) = t − log (t) (4.12) p q 2q − 1 2

◦ 2sq where t := p . In general t can take any value in the interval [2−q, 1] . In particular, in the case p = 2sq+r, 1 ≤ r ≤ q, formula (4.12) becomes

q τ Dix(A ) = 2−r + r (4.13) p q 2q − 1

References

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