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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: Fakult¨atf¨urMathematik, Ruhr-Universit¨atBochum, Germany 2: Dipartimento di Matematica, Universit´adi Roma Tor Vergata, Italy 3: Dipartimento di Matematica, Universit´adi L'Aquila, Italy 4: SFB-237, Essen-Bochum-D¨usseldorf 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 `1(N) of all bounded sequences. The singular trace is then obtained evaluating on such elements a state on `1(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 `1(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; +1] Any weight can be extended by linearity on the natural domain given by the linear span of fT 2 R+jφ(T ) < +1g A weight τ which has the property: τ(T ∗T ) = τ(TT ∗) 8T 2 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 ≡ +1 are respectively the ideals R and f0g 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 fTα; α 2 Ig 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 propor- tional 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. 1 For T a compact operator on H, fµn(T )gn=1 will denote the non increasing sequence of the eigenvalues of jT j 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 1 sequence fSn(T )gn=0 which is an indefinite integral (w.r.t. the counting measure) 1 of fµn(T )gn=1; i.e. Sn(T ) − Sn−1(T ) = µn(T ); n ≥ 1; and such that 0 T2 = L1(H) S0(T ) ≡ −tr(T ) T 2 L1(H) 1 1 Notice that if T2 = L (H);Sn(T ) = σn(T ); n ≥ 1; while if T 2 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 jT j 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 2 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 jT j. Let us also notice that a trace τ is finite on jT j 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 1 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 fpkg such that lim S (T ) = k!1 pk 1 2.7 Definition. A compact operator T which satisfies one of the equivalent prop- erties 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(jT j) < +1. (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) τ (A) ≡ ;A 2 I(T ) (2:1) Sn(T ) where is a generic non normal state on `1(N) ≡ `1: Recall that the state is just a linear bounded positive functional with the prop- erty (f1g) = k k = 1. It is easy to see that for such a the function τ 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 to be 2-dilation invariant (therefore vanishes on the set c0 of the infinitesimal sequences, hence it is not normal): 1 (fang) = (fa2ng) 8a 2 ` : In this case the generalized eccentric operator T must satisfy the stronger condition: S (T ) lim 2n = 1; n2N 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: 1 (fang) = '(fank g) 8a 2 ` : (2:2) 1 where ' is an arbitrary non normal state on ` and the sequence nk is given by nk ≡ kpk with pk defined in Lemma 2.6.
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