On Type II noncommutative geometry and the JLO character

In non-commutative geometry, the guiding principle is that the topology of spaces is encoded in properties of their algebras of continuous functions. A theorem of Gelfand-Naimark [24] states that any commutative unital C *algebra is of the form C(X) for some compact Hausdorff space X. Therefore, the category of C * -algebras (or even more generally Banach * -algebras) is seen as an extension of the category of compact Hausdorff topological spaces, and a general C * -algebra is sometimes referred to as a non-commutative topological space. The geometric features on a C * -algebra A are incorporated by the concept of an an unbounded Fredholm module (ρ, B(H), D) over A, where ρ is a continuous representation of A onto the Hilbert space H, and D is an unbounded Fredholm operator on H that satisfies certain axioms. As the prototypical example, let A be the algebra of continuous functions on a closed Riemannian manifold, H the square integrable sections of a spinor bundle with its natural action of A, and D the associated Dirac operator. Geometric features on the manifold such as geodesics, dimension, integrations, and differential forms etc can be retrieved algebraically in terms of A, B(H), and D [21]. Connes gives a set of five axioms characterizing the Fredholm modules arising in this way [15]. Taking A to be non-commutative thus leads to a notion of a non-commutative manifold. This theory is summarized in Connes' famous book [14], further details and newer developments are described in [21] and [24]. Each Fredholm module assigns an integer, the Fredholm index, to a given element in the K-theory of A. The Fredholm index provides a Z-valued pairing between the K-homology of A and the K-theory of A. In the commutative setting, the index can be viewed as the index of the Dirac operator D, twisted by a vector bundle. Suppose the unbounded Fredholm module (ρ, B(H), D) is finitely summable, a condition that models finite dimensionality according to Connes' axioms, Jaffe-Lesniewski-Osterwalder [23] defined a cocycle Ch • JLO in the entire cyclic cohomology HE • (A), now known as the JLO character. Together with the K-theory character ch • : K • (A) → HE • (A), they intertwine the K-theoretical pairing given by the Fredholm index with the cohomological pairing between HE • (A) and HE • (A) [18,20]. Such a result was originally established in a more general setting for weakly θ-summable Fredholm modules, where weak θ-summability can be thought of a suitable notion of infinite-dimensionality. Consequently, the JLO character provides a formula for the Fredholm index in terms of entire cyclic (co)homology for infinite dimensional non-commutative manifolds, which was the original motivation of JLO's work [23]. Furthermore, the formula reduces to the index formula of Atiyah-Singer in the commutative setting [3,19]. The operator D of a Fredholm module plays the role of a Dirac operator, and is typically unbounded. However, there is a canonical way of passing from an unbounded Fredholm module to a bounded one (ρ, B(H), F ), essentially by taking bounded functions of D, and the latter are often easier to work with in practice. When the bounded Fredholm module (ρ, B(H), F ) is finitely summable, there is a character ch n due to Connes [13], which again is a cocycle in HE • (A), and ch n intertwines the K-theoretical pairing the same way as the JLO character [13,16]. When (ρ, B(H), F ) is the associated bounded module of a finitely summable unbounded Fredholm module (ρ, B(H), D), Connes-Moscovici proved that in fact the cocycle ch n (F ) of (ρ, B(H), F ) defines the same cohomology class as the cocycle Ch • 1 Breuer-Fredholm modules and Connes-Chern character The section starts by stating the definition of Breuer-Fredholm modules from [8]. With the notion of (e, f )-Fredholm from [12], we proceed to develop a suitable Fredholm theory by following [2]. Entire cyclic (co)homology will be introduced, followed by a discussion of the Chern character [20,18] on K-theory and Connes-Chern character [2] on K-homology. The Section ends by showing that the characters intertwine the K-theoretical pairing given by the Fredholm index, with the cohomological pairing. For a given semi-finite von Neumann algebra N ⊂ B(H) of bounded operators on a Hilbert space H, with a faithful semi-finite normal trace τ ,we denote by K N the ideal of τ -compact operators in N . A τ -compact operator is a (densely defined closed) operator affiliated with N , such that its generalized singular number µ x (T ) with respect to τ converges to 0. The definitions and properties of K N and µ x (T ) can be found in the Appendix. If (ρ, N , F ) is equipped with a Z 2 grading χ ∈ N such that all ρ(a) are even and F is odd, then we call (ρ, N , F ) an even Breuer-Fredholm module. If N = B(H) and τ is the standard operator trace, we drop the prefix Breuer. As Fredholm modules are representatives of K-homology classes in Kasparov's sense [22], they are also referred to as K-cycles. Technically speaking, Breuer-Fredholm modules do not define K-homology classes in the usual sense, however one can still consider its classes given by the equivalence relations in K-homology. i.e., up to degenerate modules, two such modules are equivalent if their Fredholm operators are connected by a norm continuous homotopy of Fredholm operators (in N ) (see for example [22] for a precise definition). We think of Breuer-Fredholm modules as representatives of elements in some semi-finite or Type II K-homology as [9,11] did. Whenever we write [(ρ, N , F )] ∈ K • (A), we implicitly mean that the K-homology is in the semi-finite sense. Recall that a densely defined closed operator T with polar decomposition T = U |T | is said to be affiliated with N if U ∈ N and also the spectral projections of |T | lie in N (see Appendix). The only unbounded operators we are dealing with here are densely defined closed operators, hence the properties of an unbounded operator being densely defined and closed are automatically assumed throughout this paper. In particular, when we speak of an operator T affiliated with N , we demand that T is densely defined and closed. Definition 1.2. Given two projections e, f ∈ N , a (possibly unbounded) operator T affiliated with N is called (e, f )-Fredholm if there is an operator S ∈ N , such that where K eN e denotes the set of τ -compact operators in eN e, likewise for K f N f . The operator S is called an (e, f )-parametrix for T . Example 1.1. 2 )-parametrix u -1 . • Suppose that (ρ, N , F ) comes equipped with a Z 2 grading χ and that the projection p ∈ N is even with respect to χ, then F is (p + , p -)-Fredholm with (p + , p -)-parametrix F again. The following Proposition can be found in [2]. We adopted it in the (e, f )-parametrix case. Proposition 1.1. Let T be a (e, f )-Fredholm operator, and P ker T and P ker(T * ) be the projections onto the kernels of T and T * respectively. Then eP ker T and P ker(T * ) f have finite trace with respect to τ . Proof. Let S be a (e, f )-parametrix of T as in Definition 1.2. We have (e-eSf T e)P ker T = eP ker T and P ker(T * ) (ff T eSf ) = P ker(T * ) f , and eP ker T and P ker(T * ) f projections onto ker(T ) ∩ e(H) = ker( f T e| e(H) ) and ker(T * ) ∩ f (H) = ker( eT * f | f (H) ) respectively. By the ideal property of K eN e , eP ker T is a τ -compact projection. As projections only have eigenvalue {0, 1}, τ -compactness forces the singular values of projections to have support in a bounded region, hence τ of any τ -compact projection must be finite, and τ (eP ker T ) < ∞. Likewise, τ (P ker(T * ) f ) < ∞. Definition 1.3. The (e, f )-index Ind τ (f T e) of an (e, f )-Fredholm operator T is defined to be Ind τ (f T e) := τ (eP ker T )τ (P ker(T * ) f ) , where P ker T and P ker(T * ) are the projections onto the kernel of T and T * respectively. Given an even Breuer-Fredholm module (ρ, N , F ) over A, and a projection p ∈ A. It follows from Example 1.1 that F is a (ρ(p) + , ρ(p) -)-Fredholm operator. Thus it has a well-defined (ρ(p) + , ρ(p) -)-index, given by Ind τ (ρ(p) -F ρ(p) + ). For a given odd Breuer-Fredholm module (ρ, N , F ), and a unitary u Thus it has a well-defined (Q, Q)-index, given by Ind τ (Qρ(u)Q). Since the function Ind τ is locally constant [12], the (ρ(p) + , ρ(p) -)-index descends to a pairing between the K-homology class [(ρ, N , F )] ∈ K 0 (A) and the K-theory class [p] ∈ K 0 (A). Likewise, the (Q, Q)-index descends to a paring between the classes [(ρ, N , F )] ∈ K 1 (A) and [u] ∈ K 1 (A). We extend the pairing to a pairing between K-homology and K-theory of A with the following definition. To simplify our notation, whenever we mention an element a ∈ A, we think of it as an operator ρ(a) ∈ N represented on H, and will stop writing ρ. Similarly, when we have a ∈ M N (A), we think of it as an operator in M N (N ) represented on H N = H ⊗ C N with the obvious representation extended from ρ. Definition 1.4 ([8,9,11]). 1. Let (ρ, N , F ) be an even Breuer-Fredholm module over A, representing the K-homology class [(ρ, N , F )] ∈ K 0 (A), and p ∈ M N (A) be a projection , representing the K-theory class [p] ∈ K 0 (A). We define the even index pairing to be: where p -(F ⊗ 1 N )p + is an operator from p + H N to p -H N . 2. Let (ρ, N , F ) be an odd Breuer-Fredholm module over A, representing the K-homology class [(ρ, N , F )] ∈ K 1 (A), and u ∈ M N (A) be a unitary, representing the K-theory class [u] ∈ K 1 (A). We define the odd index pairing to be: 2 is a projection in M N (N ), and QuQ is an operator from QH N to QH N . Our goal is to construct characters from K-homology to another cohomology theory that intertwine the above K-theoretical pairing with the cohomological pairing. The target space of both the Connes-Chern character and the JLO character to be introduced next section is the entire cyclic (co)homology. Entire cyclic homology is not as well-known as its cohomology counterpart. We adopt the bicomplex construction from [18] and use the entire growth control given in [26]. Under this definition, the homology theory is precisely (pre-)dual to the cohomology counterpart [20] in the sense that their pairing produces a finite value. If B is a topological unital algebra over C, define (-1) j (a 0 , . . . , a j a j+1 , . . . , a n ) n-1 + (-1) n (a n a 0 , a 1 , . . . , a n-1 ) n-1 , B(a 0 , . . . , a n ) n := n j=0 (-1) nj (1, a j , . . . , a n , a 0 , . . . , a j-1 ) n+1 . Simple calculations show that b 2 = 0, B 2 = 0, and Bb + bB = 0. Therefore (b + B) 2 = 0 and we get the following bicomplex: . . . . . . . . . . . . ? ? . The space . We get a chain complex (C • (B), b + B) with the odd boundary map b + B. However, the homology of this chain complex is trivial [26]. In order to make it nontrivial, we need to control the growth of a chain as n varies. The following definition is taken from [18,26]. Definition 1.5. Define the space of entire chains is equipped with the obvious group structure inherited from the addition on Hom(C n (B), C). It is known that the de Rham homology (over C) on a closed manifold M is a summand of the entire cyclic cohomology of the algebra C ∞ (M ). They are expected to be equal, however it is not proved except for the case when M is one-dimensional [19]. Let Tr : where (m k ) ij denotes the entries of the matrix m k . Definition 1.7. where For convenience, we often write Tr (m 0 , m 1 , . . . , m n ) simply as (m 0 , . . . , m n ). Lemma 1.2 ( [20,18]). The Chern characters ch + (p) and ch -(u) are entire cyclic cycles in HE + (A) and HE As a result of Lemmas 1.2, the Chern character ch • descends to a map from K • (A) to HE • (A). It is easy to see that ch • respects group additions, hence it is a group homomorphism. The Connes-Chern character is a cohomological Chern character due to Connes that assigns to a Breuer-Fredholm module a cocycle in entire cyclic cohomology. However, not every Breuer-Fredholm module lies inside the domain of the Connes-Chern character. To characterize those that are within the domain, we need the following summability condition. For 0 < p < ∞, let L p N be the set of p-summable operators in N . That is, an operator T is in L p N if T ∈ N and its p-norm T p with respect to τ is finite. More details can be found in the Appendix. The Connes character for the Type II setting first appeared in the work of Benameur and Fack in [2]. Let A be a Banach * -algebra. Definition 1.9. 1. Recall that χ is the grading operator that anti-commutes with F . Define the even Connes character ch n (F ) of an even p-summable Breuer-Fredholm module (ρ, N , F ) to be the linear functional on C n (A) given by where n is an even integer greater than p and (•, •) denotes the pairing between cochains and chains. 2. Define the odd Connes character ch n (F ) of an odd p-summable Breuer-Fredholm module (ρ, N , F ) to be the linear functional on C n (A) given by where n is an odd integer greater than p and (•, •) denotes the pairing between cochains and chains. Theorem 1.3. For n > p, the even/odd Connes character ch n (F ) of an even/odd p-summable Breuer-Fredholm module (ρ, N , F ) defines an entire cyclic cocycle, its cohomology class is independent of n with the same parity. Proof. It is clear that ch n (F ) entire. Since Bch n (F ) = 0, we only need to show that bch n (F ) = 0, and that where the entire cochain ψ n+1 (F ) is given by Suppose that F t is a norm continuous family of Fredholm operators parametrized by t so that (ρ, N , F t ) defines 1-parameter family of p-summable Breuer-Fredholm modules. Theorem 1.4. The entire cyclic cohomology class defined by the Connes character ch n (F t ) is independent of t. More explicitly where the entire cochain ι( Ḟt )ch n-1 (F t ) is given by with the convention that χ = 1 in the odd case. Proof. Ḟt being bounded implies that ι( Ḟt )ch n-1 (F t ) is entire. It is clear that Let T k ∈ C n-1 (A) be defined by the equation By expanding the term [F t , a j a j+1 ] = a j [F t , a j+1 ] + [F t , a j ]a j+1 , we see that the above becomes a telescope sum and most of the terms cancel. Using the identity we simplify further and obtain By construction, The proof is complete. In fact, the way we obtain the transgression formula in Theorem 1.4 is by taking limits of the transgression formula in Theorem 3.4 below. for 2m > p. The following theorem shows that the characters ch n and ch • intertwine the K-theoretical pairing with the (co)homological pairing of entire cyclic (co)homology. ). 1. Let (ρ, N , F ) be an even p-summable Breuer-Fredholm module and p ∈ M N (A) be a projection, then for n > p even 2. Let (ρ, N , F ) be an odd p-summable Breuer-Fredholm module and u ∈ M N (A) be a unitary, then for n > p odd This section repeats the language in Section 1 for unbounded Breuer-Fredholm modules. It starts with the definition of unbounded Breuer-Fredholm modules from [8] and its pairing with K-theory. The JLO character is defined and a proof of its homotopy invariance is shown according to [20]. The section concludes by showing that the JLO character computes the index. Much of the work in this section is taken directly from [20] with minor modifications. Nonetheless, we give full details to illustrate the changes made in this Type II setting. 3. If (ρ, N , D) is equipped with a Z 2 grading χ ∈ N such that all ρ(a) are even and D is odd, then we call (ρ, N , D) an even unbounded Breuer-Fredholm module. If N = B(H) and τ is the standard operator trace, we drop the prefix Breuer. To avoid confusion, we will sometimes refer to the Breuer-Fredholm module from Definition 1.1 as bounded. Similar to its bounded counterpart, an unbounded Fredholm module is sometimes called an unbounded K-cycle. The term (semi-finite) spectral triple seems to be popular among physicists. It is a convenient term for the package consisting of the algebra A and an unbounded (Breuer-)Fredholm module. In this thesis, our algebra A is always fixed and we view the JLO character and Connes character as maps from K-homology classes to some cohomology classes that respect group additions. Hence, the term unbounded Breuer-Fredholm module is more convenient and suitable in our settings. An example of an unbounded Breuer-Fredholm module is given by the semi-finite spectral triple over a space of G-connections due to Aastrup, Grimstrup, and Nest. Similar to the Breuer-Fredholm module case, we think of an element a ∈ A as an operator ρ(a) ∈ N represented on H, and will stop writing ρ. In Sections 3.2, we will explain in details how we would associate a bounded Breuer-Fredholm module to an unbounded one. Definition 2.2. 1. For a given even unbounded Breuer-Fredholm module (ρ, N , D) over A, define its pairing with the even K-theory K 0 (A) of A given by the index: 2. For a given odd unbounded Breuer-Fredholm module (ρ, N , D) over A, define its pairing with the odd Ktheory K 1 (A) of A given by the spectral flow: The JLO character is a cohomological Chern character due to Jaffe, Lesniewski, and Osterwalder that assigns cocycles in entire cyclic cohomology to unbounded Breuer-Fredholm modules satisfying an appropriate summability condition. We begin by defining the summability conditions of main concern. Definition 2.3. An unbounded Breuer-Fredholm module (ρ, N , D) over A is: ) < ∞ for all t > 0; (c) weakly θ-summable if τ (e -tD 2 ) < ∞ for some 0 < t < 1. Observe that p-summability implies θ-summability, which in turn implies weak θ-summability. Example 2.2. The unbounded Breuer-Fredholm module given by Aastrup-Grimstrup-Nest's noncommutative space of connections is weakly θ-summable if the sequence {a j } in its definition diverges sufficiently fast [1]. The following Lemma was proved in [16] in the Type I case. Lemma 2.1. If (ρ, N , D) is p-summable for any finite p, then it is also θ-summable, and τ (e -tD 2 ) = O(t -p/2 ) as t ց 0. Proof. We can write e -tD 2 = (1 t -p/2 e t τ ((1 + D 2 ) -p/2 ) , which proves the lemma. To make the JLO character and other useful formulas easier to write down, we will define the JLO character in two steps. We start with the following definition. Let ∆ n := {(t 1 , . . . , t n ) ∈ R n ; 0 ≤ t 1 ≤ • • • ≤ t n ≤ 1} be the standard n-simplex and d n t = dt 1 • • • dt n is the standard Lesbeque measure on ∆ n with volume 1 n! . Definition 2.4. Let (ρ, N , D) be a weakly θ-summable unbounded Breuer-Fredholm module over A. Given F 0 , . . . , F n operators affiliated with N , define where χ = 1 when D is even. Let T be an operator affiliated with N , denote by |T | χ the degree of T with respect to χ. Any operators that we will consider will be either even or odd. From here and on, the commutator [ , ] is always graded with respect to χ. Lemma 2.2. Let F 0 , . . . , F n be operators affiliated with N that are either even or odd, then Proof. 1. The statement follows from τ (χ[X, Y ]) = 0 for X, Y operators affiliated with N . 2. The left hand side can be regarded as du by introducing a trivial extra integration; the polyhedron ∆ n × [0, 1] can be subdivided by the inequalities t j ≤ u ≤ t j+1 into n + 1 simplices, each of which is a copy of ∆ n+1 ; integration over these simplices yield the terms on the right hand side. equality follows. Replacing D 2 by (t j+1t j )D 2 and using the substitution u = (t j+1t j )s + t j , we obtain 0 = [e -(tj+1-tj )D 2 , X] + JLO (D) is entire. Whenever we have an operator affiliated with N , we demand that it is either even or odd with respect to χ. Lemma 2.4. Let (ρ, N , D) be a weakly θ-summable unbounded Breuer-Fredholm module over A. If F j and R j are operators in N for j = 0, . . . , n, and at most k of the operators F j are non-zero, then for ε ∈ [0, 1), where 0 < δ < 1 2e . For the purpose of future applications, Lemma 2.4 is slightly strengthened from the one in [20]. The proof in [20] carries through to our setting with minor modications. Proof. From the generalized Hölder's inequality, Theorem A.4(1), the following estimate holds: s -1 , observe that by using Proposition A.3 and functional calculus and that . Since the function |x| 1+ε e -δsx 2 is bounded by 1+ε 2δes 1+ε 2 and e -sδx 2 is bounded by 1, we can put together the above terms using Theorem A.4(ii) and get that Keeping in mind that at most k of the F j 's are non-zero, we get Along with the estimates the proof is complete. The above norm estimate immediately implies that Ch n JLO (D) We adopted the computation in [23] to the Type II case. We compute Ch n JLO (D) paired with b(a 0 , . . . , a n+1 ) n+1 . (-1) j a 0 , . . . , [D, a j a j+1 ], . . . (-1) j-1 a 0 , . . . , [D, a j-1 ]a j , . . . + a 0 , . . . , a j [D, a j+1 ], . . . The last term forms a telescope sum and reduces to The proof is complete. As a result, the JLO character defines an entire cyclic cohomology class called the JLO class. In this section, we will show that the cohomology class given by the JLO character is homotopy invariant. As a consequence, the JLO character descends to a well-defined map from (semi-finite) K-homology to entire cyclic cohomology. We follow closely to work by Getzler and Szenes [20]. Definition 2.6. Let V be an operator affiliated with N . Define the contraction ι(V ) by V to be Definition 2.7. Let V be an operator affiliated with N such that it has the same degree as D, i.e. |D| χ = |V | χ . Define Ch • JLO (D, V ) to be given by the equation Theorem 2.5. Let (ρ, N , D) be a weakly θ-summable unbounded Breuer-Fredholm module. 1. Let V be an operator affiliated with N such that it has the same degree as D, i.e. where α n (D, V ) is defined to be Proof. 1. From Lemma 2.4 we have that Denote by E j the cochain First we compute E j paired with b(a 0 , . . . , a n ) n : By expanding the [D, a k a k+1 ] terms using the Leibniz rule and re-ordering the sum, we get We are now in the setting to apply Lemma 2.2(4) to obtain A is a positive operator, and to see that B is also positive, we use the fact that Therefore, and the result is obtained. This section will show that the JLO character for a weakly θ-summable even unbounded Breuer-Fredholm module produces an index formula. For the odd case, we refer to a paper by Carey and Phillips [8], who developed the JLO character in the Type II setting. Theorem 2.9 ([8]). Let (ρ, N , D) be an odd weakly θ-summable unbounded Breuer-Fredholm module over A and u ∈ M N (A) be a unitary, then , where the angle bracket on the left is the spectral flow pairing [8] and the round bracket on the right is the (co)homology pairing. Theorem 2.10. Let (ρ, N , D) be an even weakly θ-summable unbounded Breuer-Fredholm module over A and p ∈ M N (A) be a projection, then where the angle bracket on the left is the index pairing and the round bracket on the right is the (co)homology pairing. Proof. It suffices to prove that It follows from the definition of (co)homology that the above equality will descend to the result stated in the theorem. For any projection p ∈ A, one can deform D to (pDp + (1p)D(1p)) via the homotopy D t = D + t(2p -1)[D, p] where t ∈ [0, 1]. As Ḋt = (2p -1)[D, p] is odd and in N , by Proposition 2.8, (ρ, N , D t ) is a differentiable family of weakly θ-summable unbounded Breuer-Fredholm modules. By Theorem 2.7, Ch + JLO (D) and Ch + JLO (pDp + (1p)D(1p)) are cohomologous. Specifically, Therefore, and The norm of τ e -(1-δ)t 2 D 2 /2 is uniformly bounded for t ∈ [1, ∞), and the term u n+1-p e -(1-δ)u 2 λ 2 /2 is integrable from 0 to ∞, therefore the limit for t → ∞ exists in norm for Ch r JLO (tD) and t 0 Ch n+1 JLO (uD, D)du. In particular, lim t→∞ Ch ≤n JLO (tD) = 0 in norm. Thus, From the proof of Proposition 3.1, For every p-summable unbounded Breuer-Fredholm module there is a canonically associated p-summable bounded Breuer-Fredholm module. We will go through a concrete construction of such a bounded Breuer-Fredholm module from an unbounded one when D is invertible, and remove the invertibility assumption at the end of the section. Most of the work in this section is adopted from [16] and [28]. Given an unbounded Breuer-Fredholm module (ρ, N , D) with D invertible, there is an associated bounded Breuer-Fredholm module (ρ, N , F ) by taking F = D|D| -1 . We will follow a technique in [28,21] to show that if (ρ, N , D) is p-summable, then so is (ρ, N , F ). N . Using the spectral formula for the power α ∈ (0, 1] of a positive operator x -α 2 dx, we compute for H = D 2 and b in a Banach * -algebra A, and and Now for any Lemma 3.7. Let H be a positive operator. Then By differentiating both sides with respect to α, the above turns into where the integrals converge as long as α > 0. Now using functional calculus to substitute H in y to get which is the desired equation. Proposition 3.8. Let D be invertible and F = D|D| -1 . For any a ∈ A, the commutator [F ln|D|, a] is bounded. Proof. By applying Lemma 3.7 for H = D 2 and α = 1, one obtains and Therefore, in the end we obtain x 1+ε e -δs k (ux) 2 The same proof will show that Ch ≤n-1 JLO (tD α , t Ḋα ) is entire. Since we do not need to integrate with respect to u, a term like Equation (15) does not appear. There does not need to be a lower bound for n, so by the same estimate techniques deployed in the proof of Proposition 3.9, it is easy to see that Ch ≤n-1 JLO (tD α , t Ḋα ) is entire. semi-finite if the * -subalgebra generated by positive elements with finite value under the functional is σ-weak dense in the von Neumann algebra [4]. A von Neumann algebra is called semi-finite if it admits a faithful, semi-finite normal trace. A von Neumann algebra is Type I if it is semi-finite and every projection contains a minimal sub-projection; Type II if it is semi-finite but not Type I [4]. Let N be a semi-finite von Neumann algebra with underlying Hilbert space H and a faithful semi-finite normal trace τ . From now on, when we say that an operator T is affiliated with N , we implicitly demand that T is densely defined and closed. Definition A.2. For an operator T affiliated with N and x > 0, the generalized singular number µ x (T ) with respect to (N , τ ) is defined to be µ x (T ) := inf where the infimum is taken over projections E ∈ N . Definition A.3. Let T be an operator affiliated with N , 0 < p < ∞, and x > 0. Then T is said to be • τ -measurable if for each ε > 0 there exists a projection E ∈ N such that Ran(E) ⊂ Dom(T ) and τ (1 -E) < ε . Remark A.4. Anything in N is τ -measurable. If a self-adjoint operator T is affiliated with N and its resolvent is τ -compact, then T is τ -measurable [2]. Proposition A. 1 ([17]). Let T , S, R be τ -measurable operators. 1. The map: x ∈ (0, ∞) → µ x (T ) is non-increasing and continuous from the right. Moreover, JLO (D) is entire. What remains to check is that Ch