Measure Theory in Noncommutative Spaces

The integral in noncommutative geometry (NCG) involves a non-standard trace called a Dixmier trace. The geometric origins of this integral are well known. From a measure-theoretic view, however, the formulation contains several difficulties. We review results concerning the technical features of the integral in NCG and some outstanding problems in this area. The review is aimed for the general user of NCG.


Introduction
The basic object of noncommutative geometry (NCG) is a spectral triple (A, H, D) [1], originally called a K-cycle [2, p. 546]. Here A is a countably generated unital non-degenerate * -algebra of bounded linear operators on a separable complex Hilbert space H, and D : Dom D → H is a selfadjoint linear operator with the properties [D, a] < ∞ for all a ∈ A and D −1 := (1 + D 2 ) − 1 2 is a compact operator. In Connes' quantised calculus the compact operator D −1 is the analogue of an infinitesimal length element ds [2, p. 545], [3, p. 157], and the integral of a ∈ A is represented by the operator-theoretic expression [2, p. 545], [3, p. 158], where n (assumed to exist) is the smallest value such that D −p is trace class for p > n, and the positive trace Tr ω (nominally a 'Dixmier trace') measures the log divergence of the trace of a D −n . The link between integration in the traditional sense and sums of eigenvalues of the product a D −n is not obvious. Segal, in [4], formalised an operator-theoretic view of integration theory. The subsequently developed theory 1 [5,6], through to [7,8,9], features the notion of a faithful normal trace τ on a * -algebra N of bounded linear operators on H that is closed in the weak operator topology (a von Neumann algebra). The trace τ is used to build an L p -theory with L ∞ (N , τ ) = N and L p (N , τ ) being the completion of the set {x ∈ N |τ (|x| p ) < ∞}. The trace (and the integration theory) is called finite if τ (1 N ) < ∞ where 1 N is the identity of N . The integration theory associated to the pair (N , τ ) has analogous machinery to measure theory, including Radon-Nikodym theorems, dominated and monotone convergence theorems 2 , etc. Without the condition of normality this machinery fails.
A commutative example of the theory (and the prototype) is a regular Borel measure space (X, µ). For the von Neumann algebra of complex functions f ∈ L ∞ (X, µ) acting on the separable Hilbert space L 2 (X, µ) by (almost everywhere) pointwise multiplication, the integral is a faithful normal trace. The noncommutative L p -spaces are the usual L p -spaces, i.e.
L p (L ∞ (X, µ), τ µ ) ≡ L p (X, µ). 1 The explanation here is for semifinite von Neumann algebras with separable pre-dual. 2 The condition of normality of a linear functional preempts the monotone convergence theorem. If N is a closed * -subalgebra of L(H) in the weak operator topology then a linear functional ρ ∈ N * is normal if ρ(S) = sup α ρ(Sα) for all increasing nets of positive operators {Sα} ⊂ N in the strong operator topology with l.u.b. S. This condition is equivalent to ρ( i Pi) = i ρ(Pi) for all sets {Pi} of pairwise orthogonal projections from N [10, p. 67], i.e. analogous to additivity of a measure on a σ-algebra of sets.
are the Schatten-von Neumann ideals of compact operators [11] ( [12] for finite dimensional H). A consequence of the Radon-Nikodym theorem in this context is that for any normal linear functional φ on a von Neumann algebra N ⊂ L(H) there is a trace class 'density' T ∈ L 1 (H) such that, [10, p. 55], φ(a) = Tr(aT ), ∀ a ∈ N .
Combining the examples, for every regular Borel measure µ on a space X there is a positive trace class operator T µ on the Hilbert space L 2 (X, µ) such that X f dµ = Tr(f T µ ), f ∈ L ∞ (X, µ). (1.4) This formula exhibits the standard association between integration and the eigenvalues of the product f T µ , a compact linear operator.
What are the differences between (1.1), the integral in NCG, and (1.3), the integral according to standard noncommutative integration theory? Let us mention basic facts about (1.1).
Firstly, since D −n belongs to the ideal of compact linear operators  [15, p. 280], [16, p. 47]. The fundamental difference between the functionals (1.3) and (1.6) is that Tr ω (S D −n ) = 0 for all positive finite rank operators S (see Section 2 below). In comparison Tr(ST ) > 0 for a trace class operator 0 < T ∈ L 1 (H) with trivial kernel and every finite rank operator S > 0. This property of vanishing on the finite rank operators, called singularity, is essential for the fundamental theorem of integration theory in NCG (Connes' trace theorem [17], and Theorem 4.1 in this text). The property of singularity, however, implies that the functional (1.6) is either trivial or else is a non-normal functional on L(H). To see this, let S n be a sequence of positive finite rank operators converging upward strongly to the identity 1 of L(H) and suppose (1.6) is a normal linear functional on L(H), then 0 = sup n Tr ω (S n D −n ) = Tr ω ( D −n ).
The property of singularity does not exclude the trace 0 < Φ ω ∈ (A ′′ ) * (the restriction of (1.6) to the von Neumann algebra A ′′ ⊂ L(H)) from being a normal linear functional on A ′′ . There are spectral triples where Φ ω is not a normal linear functional on A ′′ (evidently when A ′′ contains a finite rank operator), and where Φ ω is a normal linear functional on A ′′ (see Theorem 4.3).
Currently we lack, in general, a characterisation of the relationship between D and A that implies Φ ω ∈ (A ′′ ) * is a normal linear functional. All we know of is a sufficient condition, due to the authors, see Theorem 6.2 in Section 6.2. It has been suggested that A ′′ containing no finite rank operators is a necessary and sufficient condition for normality of Φ ω . This conjecture is open. Non-normality of the linear functional on L(H) defined by (1.6) can have advantages, as indicated by Connes [2, p. 326]. However, normality of the trace 0 < Φ ω ∈ (A ′′ ) * is coincident with our ability to apply the standard tools of integration theory to the formula (1.1).
In general, the value of the trace 0 < Φ ω ∈ (A ′′ ) * on the projections of A ′′ , In this review, we discuss the known ways to measure the 'log divergence of the trace', including residue formulas, heat kernels and Dixmier traces. Each of these ways may result in different functionals in (1.1). It is desirable to know when all these ways agree, and that (1.1) defines a unique linear functional on A. There are a variety of difficulties behind this problem. For example, if V denotes a set of linear functionals on M 1,∞ 'measuring the log divergence of the trace', the set is not an ideal [19,Remark 3.6]. Therefore, D −n ∈ K V will not imply a D −n ∈ K V for all a ∈ A in general. Specifying D −1 alone is not sufficient for a unique value from the formula (1.1). Additionally, if one picks smaller sets, V, or larger sets, V ′ , of linear functionals, it may be that K V = K V ′ (discussed in Section 3). Allied to this problem is the fact that, if a ∈ A is a selfadjoint operator and a D −n ∈ K V it is not necessarily true that the decompositions a + D −n , a − D −n ∈ K V , where a + and a − are the positive and negative parts of a.
Currently, in general, there is no characterisation, given D −n , what relationship with a * -algebra A is necessary so that Φ ω (a) = Tr ω (a D −n ) specifies a unique trace (meaning the value of Φ ω (a) is independent of the linear functional Tr ω ∈ V). This is an open problem. Checking uniqueness is a case by case basis which requires identifying Tr ω (a D −n ) with a known functional, e.g. the Lebesgue integral for a compact Riemannian manifold [17,16], or the Hausdorff measure of Fuchsian circles and self-similar fractals [2,IV.3], [20], or estimating the singular values of the product a D −n .
In the review we shall provide some sufficient conditions for normality of the trace 0 < Φ ω ∈ (A ′′ ) * . We review the demonstration of normality for the case of a compact Riemannian manifold (in Section 4) and the noncommutative torus (in Section 6).
Finally, we discuss the role of other positive singular traces on M 1,∞ , besides Dixmier traces, in the formula (1.1). We show that, even in Connes' trace theorem [17, Theorem 1], there exist other positive singular traces ρ on the compact operators, which are not Dixmier traces, such that ρ(P ) = Res(P ) (up to a positive constant) for a classical pseudo-differential P of order −n on a n-dimensional compact Riemannian manifold, where Res is the Wodzicki residue.

Dixmier traces
For a separable complex Hilbert space H, denote by µ n (T ), n ∈ N, the singular values of a compact operator T (the singular values are the eigenvalues of the operator |T | = (T * T ) 1/2 arranged with multiplicity in decreasing order [11, § 1]). Define the logarithmic average defines a Banach ideal of compact operators whose sequence of partial sums of singular values is of order logarithm. We refer to the recent paper of Pietsch [21], discussing the origin of this object in mathematics. In [22], J. Dixmier constructed a non-normal semifinite trace (a Dixmier trace) on L(H) using the weight where ω is a state on ℓ ∞ associated to a translation and dilation invariant state on R. The sequence (2.1), while bounded for T ∈ M 1,∞ , is not generally convergent. There are operators 0 < T ∈ M 1,∞ for which the sequence α({µ n (T )} ∞ n=1 ) ∈ c (called Tauberian operators), and for the Tauberian operators we want a Dixmier trace to be equal to the limit lim k→∞ α({µ n (T )} ∞ n=1 ) k . However, the linear span of the set of Tauberian operators does not form an ideal of M 1,∞ . For the semifinite domain of a Dixmier trace to be an ideal of compact operators, the use of a state ω on ℓ ∞ such that ω(a k ) = lim k a k when a k ∈ c (known as a generalised limit) cannot be avoided. Note that |ω(a k )| ≤ a k ∞ , hence

4)
A Dixmier trace Tr ω is positive by construction and continuous by (2.4), i.e. 0 < Tr ω ∈ M * 1,∞ . That it is a trace follows from the identity µ n (U * T U ) = µ n (T ) for any unitary U ∈ L(H). That it is singular (vanishes on finite rank operators, in fact on any trace class operator) follows from α({µ n (S)} ∞ n=1 ) ∈ c 0 for any finite rank operator S (resp. S ∈ L 1 ). The basic facts stated in the introduction now follow (for the fourth see the cited references). For the third notice that The non-trivial feature of Tr ω is linearity. Dixmier's specification that ω is constructed from a translation and dilation invariant state on R provides sufficient conditions for linearity. Later formulations of the Dixmier trace considered wider or narrower specifications suited to their context.
Unfortunately, the specifications are a mess. We will now introduce a wide range of sets of linear functionals, all of which play some part in known results concerning the Dixmier trace.
Indeed, as a result of [27] and [29], one can write any Dixmier trace (up to a constant) as T > 0 for some Banach limit ω, and any Connes-Dixmier trace using the same formula except with a Cesàro-Banach limit ω ∈ CBL. The sets V D(C) [1,∞) and V M DL [1,∞) were introduced with a different intention. The paper [25] by A. Carey, J. Phillips, and the second author, demonstrated the residue and heat kernel formulations of the Dixmier trace, and did so with the type I von Neumann algebra (L(H), Tr) replaced with any semifinite von Neumann algebra (N , τ ) with faithful normal semifinite trace τ , see also [19,31,16]. The generalised s-numbers ρ s (T ) of a τ -compact operator T in N , [8], provide a generalisation of singular values, such that ρ s (T ) = p({µ n (T )} ∞ m=1 ) for the pair (L(H), Tr). To use the generalised s-numbers, one replaces the sequence (2.1) by the function (2.30). For the semifinite pair (N , τ ) only the functionals V DL[1,∞) (and all continuous variants in (2.33) and (2.35)) can be defined.
The set M DL[1, ∞), which is non-empty [25, p. 74], was a constraint used in [25] to identify the Dixmier trace with residues of a zeta function and heat kernel asymptotics (see Section 5 below). Later results on zeta functions [15] enabled a residue formulation for all Dixmier traces in the set V P DL [1,∞) . It is still open whether a residue formulation exists for all functionals in the set V DL[1,∞) . While residue formulations were able to drop the Cesàro invariance condition (which is unnecessary for the weak * -Karamata theorem [15, p. 271]), a later heat kernel formulation due to A. Sedaev dropped the power invariance requirement (as [15,Proposition 4.3] is unnecessary for the heat kernel). In [32], heat kernel formulations were derived for Connes-Dixmier traces in the sets V CDLc[1,∞) and V D(C)c [1,∞) .
The set D(C)[1, ∞) was used in [16, § 1.3]. Finally, the set B(C) was introduced in the paper [34] with an application to measurability (see the next section).
It will be convenient to simplify the schematic (2.37) of Dixmier traces. Set The schematic (2.37) is then simplied to Dixmier traces (V 1 ), Connes-Dixmier traces (V 2 ) and Cesàro invariant Dixmier traces (V 3 ): . (2.42) It is known W B(C) W CBL W BL , which is not trivial, see [27] and [34]. Hence the inclusions are strict, i.e. V 1 V 2 V 3 . We do not know whether V 1 2 ⊃ V 1 is strict, i.e. anything is gained from the larger set of generalised limits. In some NCG papers Dixmier traces are defined with vague reference to the type of generalised limit being used, relying on the assumption of measurability (next section) to make the distinction irrelevant. To our knowledge, the classes of functionals in (2.42) represent the known relevant distinctions in measuring the 'log divergence of the trace' using the idea of Dixmier.

Singular symmetric functionals
It would be remiss to finish a section on Dixmier traces without mentioning singular symmetric functionals.
Consider the following well-known formula for the canonical trace, a special case of the Lidskii formula 3 : where f is a positive linear functional on the sequence space ℓ 1 in the case of Tr, and the sequence space where α is the sequence from (2.1) and x * is the sequence {|x n |} ∞ n=1 rearranged in nonincreasing order, in the case of Tr ω .
A Calkin sequence space j is a subset of c 0 such that x ∈ j and y * ≤ x * implies y ∈ j (called the Calkin property) [11, p. 18]. It is known that a subset J of compact operators is a two-sided ideal if and only if the sequence space A symmetric sequence space is a Calkin space e with norm · e such that y * ≤ x * implies y e ≤ x e ∀ x ∈ e. A symmetric ideal of compact operators E is a two sided ideal with a norm · E such that ST V E ≤ S T E V , ∀ S, V ∈ L(H), T ∈ E. It is known (actually, only recently when [38] extended the finite dimensional result of J. von Neumann in [12]) that E is a symmetric ideal if and only if j E is a symmetric sequence space and that These correspondences allow us to form the equation (2.44) for any symmetric ideal E and using 0 < f ∈ j * E . The functional defined on E + by (2.44) is positive, unitarily invariant, continuous (in the symmetric norm · E ), but not necessarily linear. To obtain linearity, and so derive a trace on E, one introduces the notions of symmetric and fully symmetric functionals. If e is a symmetric sequence space, then a functional 0 < f ∈ e * is called symmetric if y * ≤ x * implies f (y * ) ≤ f (x * ). If x, y ∈ e, we write y ≺≺ x (and say y is submajorised by x) when A symmetric sequence space e is called fully symmetric if x ∈ e and y ≺≺ x implies y ∈ e (the Calkin property with rearrangment order replaced by Hardy-Littlewood-Pólya submajorisation [39]). A functional 0 < f ∈ e * on a fully symmetric sequence space e is called fully symmetric if y ≺≺ x implies f (y * ) ≤ f (x * ). Both ℓ 1 and m 1,∞ are fully symmetric sequence spaces. The ideal E is called fully symmetric if j E is a fully symmetric sequence space. Finally, a symmetric functional 0 < f ∈ e * is called singular if f vanishes on all finite sequences. In [28,29,30] and [40, p. 77] 4 , it was shown (singular) symmetric linear functionals 0 < f ∈ j * E correspond to (singular) positive linear unitarily invariant functionals on fully symmetric ideals E using the formula (2.44). Now, the relevant question is whether all Dixmier traces (V 1 ) on M 1,∞ are constructed from singular symmetric functionals 0 < f ∈ m * 1,∞ . The answer is yes. In fact, we now know f ({µ n (T )}) defines a Dixmier trace if and only if 0 < f ∈ m * 1,∞ is a singular fully symmetric functional (a result of the second author, N. Kalton and A. Sedaev, to appear). (Fully) symmetric singular functionals on (fully) symmetric ideals thus generalise the construction of Dixmier (other generalisations exist, see [31,41]

Measurability
Measurability, as defined by Connes [2, p. 308], is the notion that where B is some subset of generalised limits which defines a Dixmier trace (see the previous section). Specifically, Connes used the set With reference to the noncommutative integral (1.1), one desires that the products a D −n are (B-)measurable, so that the formula (1.1) is independent of the generalised limit used. That the projections in A ′′ (using the extension (1.6)) have a unique value corresponding to the 'infinitesimal' D −n parallels the idea that the Lebesgue measurable sets are those with a unique 'size' based on the metric.
Define the Tauberian operators By construction, for a subset of generalised limits B ⊂ S ∞ (ℓ ∞ ), Tr ω (T ) = lim Hence the Tauberian operators are B-measurable for any B ⊂ S ∞ (ℓ ∞ ). We know from (2.42) in the previous section there are three (possibly five) main varieties of Dixmier trace. Then, a priori, there are three (possibly five) versions of measurability. Define and let K + i denote 0 < T ∈ K i . We concentrate on positive operators for the moment. Then, from (2.42), we know that where the inclusions may or may not be strict. In [27], our main result was that K + 2 = T + , identifying Connes' definition of measurability in [2] for positive operators with the Tauberian operators. Specifically we introduced the notion of the Cesàro limit property [27, p. 97] and any subset of Banach limits B satisfying this property has B-measurable positive operators (using the equivalence V DL[1,∞) = W BL ) equivalent to Tauberian operators. The set CBL of Cesàro-Banach limits satisfy the Cesàro limit property. From the paper of E. Semenov and the second author [34], we now know that K + 3 = T + . From this, the schematic (3.4) can be reduced to , (3.5) so that the set of positive measurable operators according to Connes-Dixmier traces (and any larger set of Dixmier traces) is equivalent to the Tauberian operators, but the set of positive measurable operators for Cesàro invariant Dixmier traces is strictly larger than the Tauberian operators. The Cesàro limit property is sufficient for equality with the Tauberian operators, it is an open question as to whether the property is a necessary condition (the set B(C) of Cesàro invariant Banach limits does not satisfy the property and K + 3 = T + ). Whether K + P DL[1,∞) is equal to the Tauberian operators is also an open question.
It is interesting to note that some notions of measurability can exclude the Tauberian operators. From the last section on symmetric functionals, since the set of Dixmier traces corresponds to the set of fully symmetric functionals 0 < f ∈ m * 1,∞ , the Tauberian operators T + are the set of 0 < T ∈ M 1,∞ such that the (2.44) is independent of the fully symmetric functional f . Given (2.45), we could try to widen this notion. For example, let V + sym to be the set of all positive singular traces ρ on M 1,∞ such that ρ(T ) = f ({µ n (T )}) ∀T > 0 for some symmetric functional 0 < f ∈ m * 1,∞ . Then set We are informed by [40, p. 79] that T + ⊂ K + sym . So, widening the notion of measurable in this manner excludes some Tauberian operators. We are indirectly informed by this result that there exist singular traces on M 1,∞ that do not arise from a generalised limiting procedure applied to the sequence { 1 In summary, the best way to avoid difficulties with the notion of measurability when one is using Dixmier traces to define the noncommutative integral (1.1) is to have the Tauberian condition: However, we know of no general criteria involving the * -algebra A and the selfadjoint operator D that demonstrates ( . A unique trace state on A automatically implies the trace states in (3.7) have the same value for all ω ∈ D 2 . Hence, for all a > 0, a 1/2 D −n a 1/2 ∈ T + and condition (3.6) is satisfied. The principle of a unique (singular) trace is also the construct for measurability of pseudo-differential operators of order −n. So far as we know, these situations of unique trace, besides very strong geometric conditions involving Hochschild cohomology and axioms for noncommutative manifolds [2, p. 309], [3, p. 160] provide the only handle on (3.6).
The unsatisfactory feature of this situation, from a measure-theoretic point of view, is that a unique trace often enables more general traces than Dixmier traces on compact operators to be used in (1.1) to obtain the same result (see the sections below). Where is the definitive rationale that Dixmier traces should solely appear in connection with the noncommutative integral?

Decomposition of measurable operators
We concentrated in the previous section on identifying those positive compact operators for which Tr ω (T ) = const for various Tr ω in the classes V 1 , V 2 , V 3 . We know that general T ∈ M 1,∞ polarise, i.e.
Recent work by the second author, N. Kalton, and A. Sedaev, to appear, indicates that the proposition is false. In the next section on symmetric subideals we briefly describe the strictness of the inclusions in the following schematic: (3.10) The place of K 1 2 (and K P DL [1,∞) ) is open.
The strictness of the inclusionT K 1 and the result (3.5) for positive elements implies It is evident from f(T * ) = f(T ), f ∈ V i , i = 1 2 , 1, 2, P DL[1, ∞), that the sets K i are closed under the * -operation. It follows that T ∈ K i ⇒ |T | ∈ K i , i = 1, 2.
As a consequence, we lack a basic decomposition result of integration theory. Suppose T = T * with the standard unique decomposition into positive and negative operators Then, Compare this with the result for a Borel measure space (X, µ) (also true for the noncommutative variant of τ -measurable operators on (N , τ ) for a semifinite von Nuemann algebra N with faithful normal semifinite trace τ ) where L 0 (X, µ) denotes the set of (equivalence classes) of µ-measurable functions on X.

Closed symmetric subideals
Define symmetric subspaces of the symmetric sequence space m 1,∞ by The ideal of compact operators U 1,∞ and L 1,w , defined by are quasi-Banach symmetric ideals. Notice also Define, as before, New results of N. Kalton, A. Sedaev, and the second author, state that and that U 1,∞ is the maximal symmetric subideal E ⊂ M 1,∞ for which (3.14) holds. Thus, Their results also show It is an open question as to the existence of a maximal closed symmetric subideal U 2 such thatT ∩ U 2 = K 2 ∩ U 2 . Is L 1,w ⊆ U 2 ?  Tr ω (P ) = 1 n Res(P ) (4.1)

Origin of the noncommutative integral
for any Tr ω ∈ V 1 .
Here Res is the restriction of the Adler-Manin-Wodzicki residue to pseudo-differential operators of order −n [43,17]. The monograph [42, § 7] provides a detailed introduction to the residue Res and the origin and framework of the formula. We recall that, for classical pseudo-differential operators of order −n acting on the trivial line bundle (H = L 2 (M )), where dxdξ is the Liouville measure (volume form) on the cosphere bundle S 1 M (with fibres S 1 M x ∼ = {x} × S n−1 over x ∈ M where S n is the unit n-sphere) and σ −n is the principal symbol of P . On the left hand side of (4.1) one should read for P the unique extension of the negative order pseudo-differential operator C ∞ (M ) → C ∞ (M ) to a compact operator L 2 (M ) → L 2 (M ) [44]. Let ∆ denote the Hodge Laplacian on M and P a ∈ B(L 2 (M )) denote the extension of the zeroth order pseudo-differential operator associated to a function a ∈ C ∞ (S 1 M ). Since σ −n (P a (1 + ∆) −n/2 )| S 1 M = a formula (4.1) provides Tr ω P a (1 + ∆) −n/2 = 1 n(2π) n S 1 M a(x, ξ)dxdξ. Why is a singular trace needed in formula (4.1)? By the asymptotic expansion of classical symbols P − P 0 ≡ σ −n (P ) for a classical pseudo-differential operator P of order −n where P 0 is of order < −n and σ −n (P ) is the principal symbol of P [46, XVIII]. All pseudo-differential operators of order < −n on a compact n-dimensional manifold have trace class extensions [44].
Hence any singular functional f ∈ M 1,∞ applied to P , i.e. P → f(P ), since it vanishes on L 1 , is equivalent to a functional on symbols of order −n. Moreover the vanishing on L 1 permits that the statement need only be proved locally. Hence M may be taken to be homogeneous. If f is a positive trace this functional on symbols of order −n is equivalent to an invariant measure on S 1 M . Hence, up to a constant, it identifies with the Liouville integral.
The above argument, from the proof of Connes' trace theorem in [17], implies that any positive singular trace 0 < ρ ∈ M * 1,∞ can be substituted for the Dixmier trace in (4.1). Therefore we may write, using the above notation: and then define as the 'integral' we have the same claim to formulas (4.1) and (4.3). Of course, there are a great many formulas involving Dixmier traces besides (4.1) and (4.3), particularly in the development of (local) index theory in noncommutative geometry [47,16,19,48,49,50,51,52]. There are geometric consequences of the formulation involving the Dixmier trace and associations to pseudo-differential operators [ [58,59,20]. Are there any of these results for which the submajorisation property of the Dixmier trace is a necessity (recall Dixmier traces correspond to the fully symmetric singular functionals on m 1,∞ from Section 2.1)? Several current proofs depend on submajorisation, e.g. residue and heat kernel results are dependent on submajorisation (cf. proofs in [15] and [33]). It is open whether this dependence is a necessity for the results or an artifact of the proofs.

Normal extension
To make things simpler, we dispense with the constants for the moment. Define the noncommutative integral as Φ ρ (a) = ρ(a D −n ), a ∈ A, ρ ∈ V + sing , ρ( D −n ) = 1.
where dx is the normalised volume form. Recall Φ ρ is a state on L ∞ (M ). Hence, by uniform continuity of both the left hand side and right hand side of (4.9) Φ ρ (f ) = It was not until recently that an elementary argument 7 was found proving Φ ρ = φ for the ntorus. An earlier claim [42, p. 297], applied monotone convergence to both sides of (4.9) to extend from C ∞ -functions to L ∞ -functions. This method of proof is circular, monotone convergence can be applied to the right hand side, since the integral is a normal linear function on L ∞ (M ). To apply monotone convergence to the left hand side it must be known Φ ρ ∈ L ∞ (M ) * .
In [61] we used a different method than that for torii to show the following result. We remark that [61] concentrated only on Dixmier traces. We state the result for general singular traces here.
As a corollary, and we have an example where Φ ρ ∈ L ∞ (M ) * (i.e. the noncommutative integral is normal).
Very little is known about equivalent measure-theoretic results when (4.13) is not known to be normal (and hence the emphasis at present centres on characterising normality in (4.13), see Section 6).
Define the classical weak ℓ p -spaces (or equivalently the fully symmetric Lorentz sequence spaces ℓ p,∞ ): (4.14) Denote the corresponding closed fully symmetric ideals of compact operators by L p,w (i.e. ℓ p,w = j L p,w ). As is classically known, ℓ p,w is the p-convexification of ℓ 1,w , 1 ≤ p < ∞ [62]. Similarly, from [15] or [38], L p,w is the p-convexification of L 1,w . The p-convexification M p,∞ of the ideal M 1,∞ , denoted Z p in [15], is strictly larger that L p,w (see [15]) 8 . We do know, from F. Cipriani, D. Guido, and S. Scarlatti [14], and later Guido and T. Isola [20], that Dixmier traces satisfy a Hölder inequality. The result of [20,Theorem 5.1] applies to all singular traces ρ used in (4.13). . For T ∈ M p,∞ and V ∈ M q,∞ , p, q > 1, p −1 + q −1 = 1, and a singular trace 0 < ρ ∈ M * 1,∞ , we have T V ∈ M 1,∞ and the Hölder inequality: otherwise. The inequality also holds for T ∈ M 1,∞ and V ∈ L(H): The Hölder inequality is used to prove that ρ(a D −n ) is a trace on A when (A, H, D) is a spectral triple and ρ ∈ V 1 [14], [15,Theorem 6.1]. In [20] it was noted the same proof applies to any singular trace.

Zeta functions and heat kernels
Zeta functions and heat kernel asymptotics are alternative ways to measure 'the log divergence of the trace'.
The Wodzicki residue Res on classical pseudo-differential operators of order −n on a ndimensional compact Riemannian manifold M derives its name as the noncommutative residue from the zeta function formulation of the residue for positive elliptic pseudo-differential operators [43,63].
Explicitely, if 0 < Q ∈ Op d cl (M ), d > 0, is elliptic, then Q −s , s > n/d, is trace class [64,44,46]. It is known the function ζ Q (s) := Tr(Q −s ), s > n/d, called the zeta function and initially introduced for the Laplacian [65] has a meromorphic continuation with simple pole at s = n/d [64,66] and from [43] res s=n/d ζ Q (s) = lim The Wodzicki residue has a similar identification involving the heat kernel operator e −tQ , t > 0. The heat kernel operator is so named since the kernel K(t, x, y) ∈ C ∞ ((0, ∞)×M ×M ) associated to the trace-class family e −t∆ , t > 0, ∆ the Hodge Laplacian, is a solution to the (local) heat equation [65] ( where δ is the Dirac delta function and the limit is in the weak sense. The zeta function and the trace of the heat kernel operator are related by the Mellin transform Using the asymptotic expansion of the heat kernel operator [65,67,68] the Wodzicki residue is associated to the heat kernel by We detail only the latest known identifications between Dixmier traces and residues and heat kernels. For applications of the residue and heat kernel to index formulations in NCG see [47,19,49,50,52]. if either limit exists [2, p. 306]. Meromorphicity of the formula (5.3) is discussed in [47,69,66]. Generalisations of (5.4) appeared in [25] and later [15].
The zeta function of a positive compact operator T given by (5.3) relies on the assumption that there exists some s 0 for which T s is trace class s > s 0 (equally T ∈ L s for s > s 0 ). For convenience we assume s 0 = 1. The space of compact operators for which the zeta function exists and (s − 1)ζ T (s) ∈ L ∞ ((1, 2]) was studied in [15]. Define the norm where T s = Tr(|T | s ) 1 s . Then [15,Theorem 4.5] identified that and that where · 0 is the Riesz seminorm (4.12).
Moreover, lim s→1 + (s − 1) Tr(P T s P ) exists iff P T P is Tauberian and in either case Similarly, for any ξ ∈ BL ∩ DL, Moreover, if P T P is Tauberian for all projections P in the von Neumann algebra generated by A and A * , For the situation s 0 = p, see [15].

Characterising the noncommutative integral
The functional for some orthonormal basis {h m } ∞ m=1 . We now know that a similar characterisation exists for any functional Φ f,T as defined in (6.1).

Characterisation and singular traces
Let J be a (two sided) ideal contained in the compact operators L ∞ of the separable complex Hilbert space H. By a trace on J we mean a linear functional f : J → C such that f([a, T ]) = 0 for all a ∈ L(H), T ∈ J , i.e. f vanishes on the commutator subspace Com J of J . Note it is not assumed that f is positive or continuous.
The characterisation for (6.1) in Theorem 6.1 below was first shown for f ∈ V P DL[1,∞) , 0 < T ∈ M 1,∞ , in [70, Theorem 3.6] using residues (Corollary 5.1). The result below (to appear) is due to collaboration with N. Kalton and uses results on the commutator subspace and sums of commutators [71,72]. if and only if 0 < σ ∈ (ℓ ∞ ) * . The basis {h m } ∞ m=1 in the theorem is any basis of eigenvectors of T rearranged so that the sequence of eigenvalues λ m associated to h m is a decreasing sequence.
A singular trace is a trace on a two sided ideal J , containing the finite rank operators F , that vanishes on F ⊂ J . The theorem has the following corollary. for all a ∈ L(H).
Example 6.1. Let T n be the n-torus, R n /Z n , with Laplacian ∆. Let J be any two-sided ideal with trace f and g be a bounded Borel function g such that g(∆) ∈ J . Let f ∈ L ∞ (T n ) and denote by the action of f on L 2 (T n ) by pointwise multiplication coupled with the compact operator g(∆), c.f. [11, § 4]. Then, by Theorem 6.1, where m ∈ Z n (ordered by Cantor enumeration) and σ is a linear functional on ℓ ∞ . As Therefore, for any ideal J with a non-trivial trace f, by choosing the appropriate bounded Borel function g, the expression f(·g(∆)) may serve as the integral on the n-torus (up to a constant). Note the identification (6.6) holds for all f ∈ L ∞ (T n ).
Can the same statement as for the n-torus in the example above be made for the Lebesgue integral of a compact Riemannian manifold M , considering non-trivial positive singular traces instead of arbitrary traces?
We certainly know, from Corollary 6.1 and Theorem 4.3, that if the sequence for a basis of eigenvectors {h m } ∞ m=1 of the Hodge-Laplacian ∆ (ordered so that the eigenvalues associated to the eigenvectors are decreasing), then for any positive non-trivial singular trace f on an ideal J and with an appropriate choice of bounded Borel function g (so that g(∆) ∈ J ). The property (6.7) is a restricted form of a property for compact Riemannian manifolds known as Quantum Unique Ergodicity (QUE) [73,74]. It is not well known which manifolds have the QUE property.
The same result as for the n-torus applies to anything 'flat'. The next example involving the non-commutative torus first appeared (for f ∈ V P DL[1,∞) ) in [70]. Example 6.2. Consider two unitaries u, v such that uv = λvu, for λ := e 2πiθ ∈ S (the unit circle). Denote by F θ (u, v) the * -algebra of linear combinations (m,n)∈J a m,n u m v n , J ⊂ Z 2 is a finite set, with product ab = r,s ( m,n a r−m,n λ mn b m,s−n )u r v s and involution a * = r,s (λ rs a −r,−s )u r v s , a, b ∈ F θ (u, v). The assignment τ 0 (a) = a 0,0 is a faithful trace on F θ (u, v). Let (H θ , π θ ) denote the cyclic representation associated to τ 0 . The closure, C θ (u, v), of π θ (F θ (u, v)) in the operator norm is called a rotation C * -algebra, [75], or the noncommutative torus (λ = 1) [2,3,42]. Canonically, finite linear combinations of u m v n ֒→ H θ are dense in H θ . Define ∆ θ (u m v n ) = (m 2 + n 2 )u m v n . It can be shown that the 'noncommutative Laplacian' ∆ θ has a unique positive extension (also denoted ∆ θ ) ∆ θ : Dom(∆ θ ) → H θ with compact resolvent, see the previous citations. The eigenvectors h m,n = u m v n ∈ H θ form a complete orthonormal system. Note that h m,n , π θ (a)h m,n =: for any (m, n) ∈ Z 2 . Using the Cantor enumeration of Z 2 , from Theorem 6.1 we obtain for any ideal J with a non-trivial trace f, by choosing the appropriate bounded Borel function g such that g(∆ θ ) ∈ J . In fact, h m,n , bh m,n = τ (b) for any b in the weak closure C θ (u, v) ′′ of π θ (F θ (u, v)) and where τ denotes the normal extension of τ 0 . Hence

A dominated convergence theorem for the noncommutative integral
The following dominated convergence theorem appeared in [70] as a statement for Dixmier traces. We provide the general statement here. Let N be a weakly closed * -subalgebra of B(H). We recall that N * denotes the predual of N , or the set of all normal linear functionals on N .
We say a positive compact operator T is (N , h)-dominated if, for some orthonormal basis {h m } ∞ m=1 of eigenvectors of T , there exists h ∈ H such that P h m ≤ P h for all projections P ∈ N . We close with an example of (N , h)-dominated compact operators. Let H = L 2 (X, µ) for a σ-finite measure space (X, µ) and set N as multiplication operators of L ∞ -functions. Then 0 < T ∈ L ∞ (L 2 (X, µ)) being (N , h)-dominated is the same as statement |h m | 2 are dominated by some |h| 2 ∈ L 1 (X, µ) µ-a.e., where {h m } ∞ m=1 is any basis of eigenvectors of T . Hence any positive compact operator 0 < T ∈ L ∞ (L 2 (X, µ)) is (N , √ g)dominated if it has a basis of eigenvectors whose modulus squared are dominated by some positive integrable function g ∈ L 1 (X, µ). For instance, the negative powers of the Laplacian on the n-torus are dominated by the constant function 1 ∈ L 1 (T n ).

Summary
We summarise the main points of the review and list the open questions raised.

Summary
There are three main non-identical classes of Dixmier traces. They are, in descending order of inclusion: the (original) Dixmier traces (V 1 ); the Connes-Dixmier traces (V 2 ); and the Cesàro invariant Dixmier traces (V 3 ). The notion of measurable operator with respect to each of these classes are non-identical. The notion of Tauberian, (3.2), is the strongest notion with ascending order of inclusion None of these sets is an ideal of compact operators. The notion of measurable operator using Dixmier traces does not achieve T = K 1 , however T + = K + 1 . This has the consequence that the operators T ∈ K 1 do not decompose into unique measurable components. In short, K 1 does not have the property T ∈ K 1 ⇒ |T | ∈ K 1 .
Connes' trace theorem (Theorem 4.1) is valid for any ρ ∈ V + sing , (4.6). In particular, if M is an n-dimensional compact Riemannian manifold with Hodge Laplacian ∆ we have with the constant c ρ > 0 for the subset Θ n of V + sing such that ρ((1 + ∆) −n/2 ) > 0. The set Θ n of singular traces is larger than the set of Dixmier traces (V 1 ). Further, for ρ ∈ Θ n we have This defines a family of continuous positive traces on A (in the implied C * -norm) (Corollary 4.1).
Restricting the noncommutative integral to the consideration of ρ ∈ V 1 (the set of Dixmier traces or the set of Hardy-Littlewood-Pólya submajorisation ordered continuous functionals on m 1,∞ , Section 2.1) enables Lidksii type theorems, residue, and heat kernel theorems, see Section 5 and comments.
The functional where f is a trace on a two-sided ideal of compact operators J and 0 < T ∈ J , is the general format of noncommutative integration introduced by Connes. Such functionals can be characterised by linear functionals σ : ℓ ∞ → C applied to sequences of "expectation values" { h m , ah m } ∞ m=1 for some orthonormal basis {h m } ∞ m=1 of eigenvectors of T . In particular for all a ∈ L(H).
For flat torii, both commutative and noncommutative, this implies f(ag(∆)) = f(g(∆))τ (a) for any trace f on a two-sided ideal J , and g a positive bounded Borel function such that g(∆) ∈ J . Here ∆ is the Laplacian (resp. noncommutative Laplacian) on the torus and τ is the Lebesgue integral on commutative torii (resp. the unique faithful normal finite trace on the type II 1 noncommutative torii). If N ⊂ L(H) is a von Neumann algebra, a positive compact operator T is (N , h)-dominated if, for some orthonormal basis {h m } ∞ m=1 of eigenvectors of T , there exists h ∈ H such that P h m ≤ P h for all projections P ∈ N . If J is any two-sided ideal with positive singular trace f, and 0 < T ∈ J is (N , h)-dominated, then f(·T ) is a positive normal linear functional on N .

List of open questions
Let (A, H, D) be a spectral triple (as defined in the first paragraph of Section 1) such that D −n ∈ M 1,∞ (as defined at (1.5)).