The condenser quasicentral modulus

We introduced the quasicentral modulus to study normed ideal perturbations of operators. It is a limit of condenser quasicentral moduli in view of a recently noticed analogy with capacity in nonlinear potential theory. We prove here some basic properties of the condenser quasicentral modulus and compute a simple example. We also discuss some associated variational problems. Part of the results are in the more general setting of a semifinite von Neumann algebra.


Introduction
The quasicentral modulus ( [8], [9], [10]) plays a key role in the study of Hilbert space operators modulo normed ideals (see our surveys [11], [12]).This paper is a sequel to [13].In [13] we made the case that the quasicentral modulus is a noncommutative analogue of capacity in nonlinear potential theory, where the first order Sobolev spaces use general rearrangement invariant norms of the gradients.One consequence is that the quasicentral modulus becomes a limit quantity of condenser quasicentral moduli.Note that the condenser quasicentral moduli are usually finite and non-zero also in situations where the quasicentral modulus can take only the values 0 and ∞, like in the case of the p -classes when p > 1.Another new feature is that we will often deal with the more general case of separable semifinite factors or von Neumann algebras, that is, not only with the type I case of the algebra B(H) of bounded operators on the Hilbert space H and note also that even the case when H is finite dimensional is no longer a trivial case.
Concerning the nonlinear potential theory capacity with which we observed an analogy see [2] and more references are in [13].Prior to this, we had noticed connections with Yamasaki hyperbolicity ( [14]) and with the noncommutative potential theory based on Dirichlet forms [1].
Can the analogy with nonlinear potential theory be further extended ?A way to achieve this may be via noncommutative variational problems .We take a small first step in this direction introducing variational problems related to the condenser.We observe that computations in the case of the p-classes naturally lead to noncommutative analogues of the p-Laplace equation.
Besides the introduction and references there are eight more sections.Section 2 is about preliminaries and basic definitions.Section 3 contains some general properties of the condenser quasicentral modulus in the semifinite setting.In particular we prove that under certain conditions the the condenser quasicentral modulus is symmetric with respect to switching the projections which define the condenser.We also give a result about the behavior with respect to certain conditional expectations.Section 4 gives a lower bound for the condenser quasicentral modulus in the case of the algebra of bounded operators on a Hilbert space.This is analogous to the lower bound in [10] for the quasicentral modulus.Section 5 is the computation of an example arising from the bilateral shift operator.In section 6 we adapt and generalize to our semifinite setting the result in [9] about the largest reducing projection on which the quasicentral modulus vanishes.In the analogy with nonlinear potential theory, this is a special noncommutative polar set.Section 7 deals with variants of the quasicentral condenser modulus.Section 8 is about noncommutative variational problems.We make some general remarks about minimizers for the condenser problem.

Preliminaries and Definitions
We introduce here the framework in which we will work, especially related to normed ideal / symmetric operator space norms ( [3], [4], [5], [6]) and we recall the definition of the quantities we introduced in [13].
By (M, ρ) we will denote a von Neumann algebra M ⊂ B(H) where H is a separable complex Hilbert space and ρ is a faithful normal semifinite trace on M. We will assume that M is either atomic, that is it is generated by its minimal projections or that it is diffuse and ρ(I) = ∞, in particular there are no minimal projections in this case.Thus M could be for instance a type I or a type II ∞ factor with its trace, but it could also be for instance L ∞ (S, dσ) where the measure sigma has no atoms and is not finite or it could be ℓ ∞ (X) with the measure giving mass 1 to each singleton subset ( the traces are those corresponding to the measures).
We will denote by P roj(M) the selfadjoint projections in M and by P(M) or simply P the set of P ∈ P roj(M) so that ρ(P ) < ∞.By R we will denote the set of x ∈ M for which there is P ∈ P so that xP = x , that is the ideal of operators of finite ρ − rank in M. Further, R + 1 will stand for the positive contractions in R , that is {a ∈ R|0 ≤ a ≤ I}.It will also be convenient to introduce the set Λ If x ∈ M, the generalized singular values ( see [5]) are µ(t, x) = inf{ A(I − P ) |P ∈ P, ρ(P )let} where t > 0 and µ(x) will denote the function On L 1 (M, ρ) ∩ M we will consider a norm | • | J to which we will refer as the symmetric operator norm.We will assume that | • | J satisfies the folllowing conditions: 1 This condition is trivially satisfied if M is a type I factor, because it is meaningless there being no non-zero P n as above.The condition is equivalent to The condition can also be put in the form : there is an increasing function φ : (0, ∞) −→ (0, ∞) so that lim t→0 φ(t) = 0 for which we have ).Note also that if (M, ρ) is diffuse and the condition is not satisfied, then there must be a constant C ∈ (0, ∞) so that |x| J ≥ C x .4. Considering the von Neumann algebra M ⊗ M n endowed with the trace ρ ⊗ T r n , the norm | • | J on M ∩ L 1 (M, ρ), identified with a subspace of M ⊗ e 11 has an extension to a norm, we will still denote by satisfying the analogues of 1. -3. .We opted for this ad hoc way of introducing | • | J , instead of a discussion starting with operator spaces [5] which would have taken us farther than the more modest aim of this paper.We will also denote by J the completion of L 1 (M, ρ) ∩ M with respect to the norm | • | J , which is an M bimodule.In the case of B(H), J identifies with a normed ideal in B(H) consistent with the notation we used in previous papers (see [12] ).Note also that property 4. in the symmetric operator spaces setting , actually follows from the general relation of symmetric function spaces, symmetric sequence spaces and symmetric operator spaces (see 2.5 in [5] , Thm.2.5.3 and the discussion of Questions 2.5.4 and 2.5.5 in [5] ) Definition 2.1 Let P, Q ∈ P, P Q = 0 and let τ = (T j ) 1≤j≤n , T j ∈ M, 1 ≤ j ≤ n.The condenser quasicentral modulus with respect to the symmetric operator norm | • | J is the number: Similarly if α = (α j ) 1≤j≤n is a n-tuple of automorphisms of M which preserve ρ we define We remark that if Definition 2.2 Let P, Q ∈ P roj(M), P Q = 0 and let τ, α, | • | J be like in Definition 2.1 .We define Further we define k J (τ ; P ) = k J (τ ; P, 0) k J (α; P ) = k J (α; P, 0) and the quasicentral moduli of τ and of α k J (τ ) = k J (τ ; I) k J (α) = k J (α : I).If M needs to be specified we will write k J,M (τ ; P, Q) and so on.
In case (M, ρ) = (B(H), T r) , this definition of k J (τ ) is equivalent to the definition we used in our earlier work, as we already pointed out in [13].
It will also be useful to have a technical result about replacing R + 1 by the larger set Λ in the above definitions.Lemma 2.1 Let P, Q ∈ P, τ, α be as in Definition 2.1..We have Proof.The Lemma is an immediate consequence of the fact that Ξ = {A ∈ R + 1 |AP = P, AQ = 0} is a dense subset with respect to the topology defined by the norm | • | J of the set Θ = {A ∈ Λ|AP = P, AQ = 0}.This in turn is seen as follows: so that the proof reduces to the proof of the density of R + 1 in Λ .If X ∈ Λ then let P j be the spectral projection of X for [1/j, 1].We then have P j ∈ P and XP j − X → 0, |XP j − X| 1 → 0 so that XP j ∈ R + 1 and by condition 2. we have |X − XP j | J → 0.

Proof.
Since k J (•; P, Q) for general P, Q is the sup of such quantities with P, Q ∈ P, it suffices to prove the Proposition under the additional assumption that P, Q ∈ P.Moreover, by symmetry it clearly suffices to prove that k J (•; P, Q) ≥ k J (•; Q, P ) .If ǫ > 0 there is Then we have Clearly 0 ≤ I − A ≤ I, (I − A)P = 0, (I − A)Q = Q and we would be done if there weren't the problem that I − A is not in R in general.Since k J (τ ) = 0 we also have k J (τ ; P + Q) = 0, so that there is B ∈ R + 1 so that ) and ǫ > 0 being arbitrary, 3ǫ is as good as ǫ here.
The case of the n-tuple of automorphisms α is dealt along the same lines.
and (I − A)P = 0, (I − A)Q = Q.Choosing B as in the previous case, we consider F = B(I − A)B and we have This leads then to and so on.
The framework for the next result involves a von Neumann subalgebra N ⊂ M so that ρ|N is semifinite, in which case R(N) is weakly dense in N. Let E be the conditional expectation of M onto N with respect to ρ ( see [7] Prop 2.36).If the n-tuple of automorphisms α is so that α j (N) = N, 1 ≤ j ≤ n , then in view of the assumption ρ • α j = ρ, 1 ≤ j ≤ n we will have that α j |N • E = E • α j , 1 ≤ j ≤ n.We shall also assume that Since the symmetric operator norm | • | J was introduced in an ad-hoc way, without going into the theory of spaces of operators, we prefer to treat this as an assumption.Proposition 3.2 Let N be a von Neumann subalgebra of M so that ρ|N is semifinite and let E be the conditional expectation of M onto N with respect to ρ. Assume also that the n-tuple of automorphisms α, which preserve ρ is so that α j (N) = N, 1 ≤ j ≤ n .Let further P, Q ∈ P roj(N) be so that P Q = 0. Then we have Proof.We first prove the statement in case P, Q ∈ P(N).The inequality is obvious, because the LHS is an inf over a subset of the set the inf of which is the RHS, We have E(Λ(M)) = Λ(N), so that:

Thus, if
then we have This concludes the proof in the case of P, Q ∈ P(N).
Assume now that we only have P, Q ∈ P roj(N).Then we get that We still must prove that if Then we have We shall now use the fact that On the other hand In view of condition 3. satisfied by | • | J we have This in turn gives which concludes the proof .
There is also a similar result for n-tuples τ of self-adjoint operators instead of the n-tuple α of automorphisms.Since the proof is along the same lines, we will leave out many details.
Proposition 3.3 Let N be a von Neumann subalgebra of M so that ρ|N is semifinite and let E be the conditional expectation of M onto N with respect to ρ.Let further τ be a n-tuple of selfadjoint elements in N and let P, Q ∈ P roj(N) be so that P Q = 0. Then we have Sketch of Proof.We first deal with P, Q ∈ P(N) .Obviously we have On the other hand E(Λ(M)) = Λ(N) and if τ ; P, Q).This concludes the proof in case P, Q ∈ P(N).
If we only have P, Q ∈ P roj(N) then the preceding immediately gives k J,M (τ ; P, Q) ≥ k J,N (τ ; P, Q).
We still must show that if Since ρ|N is semifinite, there are P 2 , Q 2 ∈ P(N) so that This in turn shows that An appropriate choice of ǫ > 0 concludes the proof.
Taking P = I and Q = 0 in Prop.3.2.and Prop 3.3.we have the following.
for some Hilbert space H 2 , the preceding corollary gives that This shows in particular that the examples of quasicentral modulus of ntuples in B(H 2 ) give automatically examples of quasicentral modulus in type II ∞ factors by taking I ⊗ τ 2 in M 1 ⊗ B(H 2 ).One can proceed in a similar way for n-tuples of automorphisms ( for the type I ∞ case, the next proposition shows that this reduces to the quasicentral modulus for n-tuples of unitary operators) .
Let us also make a very simple observation about the case of n-tuples of unitary operators.If u = (U j ) 1≤j≤n is a n-tuple of unitary elements of M we denote by Adu = (AdU j ) 1≤j≤n the n-tuple of inner automorphisms where (AdU j )(x) = U j xU * j .Consider also a map ǫ : {1, ..., n} −→ {1, * } and let then u ǫ = (U ǫ(j) j This immediately implies the following Proposition.Proposition 3.4 Let u = (U j ) 1≤j≤n be a n-tuple of unitary elements of M , let ǫ : {1, ..., n} −→ {1, * } and let P, Q ∈ P roj(M), P Q = 0. Then we have

The lower bound
In this section we assume M = B(H) and that | • | J is the symmetric norm arising from a norming function Φ, so that we will write . By Φ * we will denote the dual norming function, so that if Then we have Proof.We shall first prove ≥ and then assuming k Φ (τ ; P, Q) > 0 we shall prove ≤, which will yield the equality stated above.We start with 0 ≤ B ≤ I − P − Q, B ∈ R + 1 and (X j ) 1≤j≤n ∈ Ω and we will show that We have T r(( On the other hand Assume now that k Φ (τ ; P, Q) > 0. To prove ≤ we shall consider the real Banach space (S Φ with the norm max 1≤j≤n |X j | Φ and two disjoint convex subsets of this Banach space.The first is the open ball centered at 0 of radius k Φ (τ ; P, Q).The second is: The two convex sets are disjoint and the first is open, so that there is separating the two and having norm 1.Thus we have which is the result we wanted to prove.
There is an analogue of Proposition 4.1 for unitary operators or equivalently for the corresponding inner automorphisms.The proof being along the same lines is left as an exercise for the reader.Proposition 4.2 Let u be a n-tuple of unitary operators and let P, Q ∈ P, P Q = 0. Let further Then we have: and if k Φ (u; P, Q) > 0 equality holds.

An example
Let U be the bilateral shift operator on H = ℓ 2 (Z), U e j = e j+1 where {e j } j∈Z is the canonical orthonormal basis.If f : Z −→ C is a bounded function we will denote by D(f ) the diagonal operator in H with respect to the canonical basis.In case f : R −→ C we will write D(f ) for D(f |Z).Moreover if ω ⊂ Z, P ω will denote the projection D(χ ω ) .In case of a singleton {j} we will write P j instead of P {j} .
Here we will compute k p (U ; P M , P N ) where M, N ⊂ Z are two disjoint finite nonempty subsets, By Proposition 3.2 and Proposition 3.4 this would be equivalent to a problem in ℓ ∞ (Z) , that is a problem on the Cayley graph of Z.We will not use this explicitly, though all our computations will be around two operators B = D(f ) and X = D(g) .
Let a = inf (M ∪N ) and b = sup(M ∪N ) and let a ≤ a b be so that the (a j , b j ) are the maximal open intervals so that (a j , b j ) ∩ (M ∪ N ) = ∅ and the endpoints a j and b j are in different sets of the partition of M ∪ N into M and N .Let also h ∈ N. We define a continuous function f : R −→ R as follows.First we require that Then on each open interval at the endpoints of which f has been defined we extend the definition by linearity on the interval.Thus f will be piecewise linear with respect to the partition because h being arbitrary we can take the limit as h → ∞.
To get the lower bound using Proposition 4.2 we construct an operator X = D(g).Let ǫ(j) = −1 if a j ∈ M and b j ∈ N and let ǫ(j) = +1 if a j ∈ N and b j ∈ M and observe that ǫ(j) = −ǫ(j + 1).If p > 1 we define We have From Proposition 4.2 we get the lower bound Since b j , a j ∈ (M ∪ N ) the second term in the lower bound is zero, so we need only compute T rP M Y P M .If ǫ(j) = −1 we have a j ∈ M and b j ∈ N , while if ǫ(j) = +1 we have a j ∈ N and b j ∈ M .This gives that T rP M (ǫ(j)(P b j − P a j )) = 1 for all indices j.It follows that and because p/q = 1/p we have where we used 1 − 1/q = 1/p.Thus the lower and the upper bound are equal if p > 1.
To obtain the lower bound when p = 1 we shall consider and X = D(g) .Then Again, the lower bound reduces to computing It is also easy to see that so that also in this case the lower and upper bounds we found for k Φ (Ad(U ); P M , P N ) are equal.Summing up and using Proposition 3.4, we have proved the following result.

The singular projection and the regular projection
We adapt and generalize to our semifinite setting the facts in [9] about the largest projection on which the quasicentral modulus vanishes.
The following Lemma is based on an argument we used in the proof of Proposition 3.2.Lemma 6.1 Let φ be the function in property 3. of the J-norm.
Then we have A − B ≤ 1 and |A − B| ≤ 6ǫ so that Taking into account the way we defined k J (τ ; P, Q) using a max over j and then an inf over X, this gives For automorphisms we use the same argument with A, B defined now to be Then, if F is the support projection of B ( i.e.E(B; (0, ∞))) we have [T j , F ] = 0, 1 ≤ j ≤ n and k J (τ ; F ) = 0.
Proof.Replacing B m by F B m F , M by F M F |F H , τ by τ |F H etc., it is easily seen that the proof reduces to the case when F = I that is KerB = 0. Recall also that k J (τ ; I) = k J (τ ) .So, we need to prove that k J (τ ; P ) = 0 if P ∈ P.
Remark also that we may assume that Indeed, we may pass from the initial B m 's to a subsequence so that Next, we show how to complete the proof when B is invertible , that is E(B; [0, ǫ)) = 0 for some ǫ > 0 and then go back to the general situation KerB = 0. Let h : R −→ [0, 1] be a C ∞ -function which is 0 on (−∞, 0] and 1 on [ǫ, ∞) .Then we have Returning to the general case, where only KerB = 0 is assumed, the result we have obtained thus far is easily seen to give that k J (τ ; E(B, (ǫ, B )) = 0 if ǫ > 0 .The proof is then completed by observing that given P ∈ P we can find P k ∈ P, P k ≤ E(B; (1/k, B )) so that lim k→∞ |P k − P | 1 = 0 and use Lemma 6.1 .We may take P k to be the left support projection of E(B; (1/k, B ))P , that is the projection onto the closure of the range of this operator.
There is an entirely analogous result for automorphisms which we record as the next Proposition, the proof of which is omitted, being only a slight variation on the preceding proof.
Corollary 6.1 Let P 1 , P 2 ∈ P. Then we have Proof.We shall prove only the first assertion, the proof of the second being completely analogous.If k J (τ ; Since AP 1 = P 1 , we have that P 1 H and KerA are orthogonal.Similarly P 2 H and KerC are orthogonal.On the other hand Ker(A + C) = KerA ∩ KerC because A and C are ≥ 0. Thus Ker(A + C) is orthogonal to (P 1 ∨ P 2 )H .Applying Proposition 6.1 to the sequence B m = A m + C m we get the desired result.Proposition 6.3 Given τ there exists a projection E 0 J (τ ) ∈ P roj(M) so that : The projection E 0 J (τ ) is unique, in particular if β is an automorphism of M which preserves ρ and β(τ ) = (τ ), then we have β(E 0 J (τ )) = E 0 J (τ ).Similarly, given α there exists a projection E 0 J (α) ∈ P roj(M) so that: P ∈ P roj(M), k J (α; P ) = 0 ⇔ P ≤ E 0 J (α).Moreover the projection E 0 J (α) is unique, in particular if β is an automorphism of M which preserves ρ and β Proof.We will only prove the first half of the statement, the arguments being very similarly for the two cases.Moreover, in view of the definition of k J (τ ; P ) when P ∈ P roj(M)) it is easily seen that what we must prove , is that The Hilbert space H being separable, there is a sequence P i , i ∈ N so that k J (τ ; We can replace the increasing sequence E i by a subsequence and assume that the B i 's are weakly convergent to some B .Then BE = E and we can apply Proposition 6.1 to infer that k J (τ ; We shall call E 0 J (τ ), E 0 J (α) the J-singular projection of τ and respectively α.We shall also use the notation and call E J (τ ), E J (α) the J-regular projection of τ and α respectively.
In [9], in the case of B(H) and of a normed ideal given by a norming function Φ we had called a projection P which is τ -invariant Φ-well-behaved if k Φ (τ |P H) = 0.This is equivalent to k Φ (τ ; P ) = 0 and we think that Φ -singular, the terminology we introduce here, is perhaps a better term for this.
Proof.Also here we will give only the proof of the first assertion, the proof of the second being along the same lines.
It is clear that it suffices to prove that if P ∈ P is so that k J (τ ; P ) = 0, then there is P ′ ∈ P roj(M) so that P ′ ≥ P, k J (τ ; P ′ ) = 0, [P ′ , T j ] = 0, 1 ≤ j, ≤ n.Indeed, since k J (τ ; P ) = 0 , there are B m ∈ R + 1 so that B m P = P and lim Proof.We will prove only the first assertion, the proof of the second being along the same lines.Let B m = E J (τ )A 2 m E J (τ ) so that we will have to prove that w − lim m→∞ B m = 0.

Variants
We briefly discuss here modifications of the definition of the condenser quasicentral modulus quantities along lines, which for the quasicentral modulus we already pointed out in [8] .We use instead of the max comparable devices in the definitions.This is in preparation of the next section where the variants may have some advantages.Thus Definition 2.1 is modified as follows: This is then extended also to Definition 2.2 and the further k J quantities are replaced by kJ quantities.Remark that in essence this amounts to replacing is the positive operator in the polar decomposition of the column operator matrix.We view here the 1 × n matrices with entries in M as a subspace in M ⊗ M n and use item 4. from the properties of | • | J in the preliminaries.
Though we will mostly use kJ in this paper, it is also quite natural to consider a Dirac operator construction.This produces Dirac-versions k D J (τ ; P, Q) etc. where with e 1 , . . ., e n denoting Clifford matrices.
We have k J (τ ; P, Q) ≤ kJ (τ ; P, Q) ≤ nk J (τ ; P, Q).This implies that a k J -condenser quantity is zero or infinity iff the corresponding kJ -condenser quantity is zero or respectively infinity.
It is also easy to see that Lemma 2.1., Proposition 3.1., Proposition 3.2.and Proposition 3.3 still hold if k J is replaced by kJ .Note however that it may not be the case that Proposition 3.4.remains valid when we pass to kJ .

Noncommutative variational remarks
The quasicentral modulus, in its different versions, is based on quantities for which minimization problems can be formulated: where τ = τ * throughout this section.If I(X) ∈ [0, ∞] denotes any of the above, remark that it is a differential seminorm with additional properties: and with the exception of Ĩτ and Ĩα we also have If X = X * then Ĩτ and Ĩα can also be written as follows: Actually X = X * is a quite natural condition when we set up variational problems.
Euler equations in case J is the p-class , 2 ≤ p < ∞ can be found for the power-scaled I p when I does not include a max .These equations can be viewed as analogues of the p-Laplace equation.More precisely let X = X * be such that: for all B = B * ∈ R , where I(X) < ∞.
In case Similarly in case I = Ĩα we have where With this notation we have Similar computations can be carried out in the Dirac case.
Remark 8.1 It is natural to view solutions of ( * ) as τ −p-harmonic elements and solutions of ( * * )as α − p -harmonic elements.A possible technical problem which may appear is that in order not to limit considerations to "bounded p-harmonic" elements it may be necessary to be able to handle the situation when X is an unbounded operator affiliated with M.
Remark 8.2 In the case of automorphisms, if I ∈ N ⊂ M is a von Neumann subalgebra so that ρ|N is semifinite and α j (N) = N, 1 ≤ j ≤ n let E be the conditional expectation of M onto N so that ρ • E = ρ.If I(X) stands for I α (X), Ĩα (X) or I D α (X) , it is easily seen that I(X) ≥ I(EX).
The definitions of the condenser quasicentral moduli k J (τ ; P, Q), k J (α; P, Q), kJ (τ ; P, Q), kJ (α; P, Q), k D J (τ ; P, Q), k D J (α; P, Q) where P, Q ∈ P, P Q = 0 suggest corresponding variational problems for the I(X) quantities involving the convex sets The inf of I(B) when B ∈ C 0 P Q gives the condenser quasicentral moduli, while C 0 P Q is weakly dense in C P Q , which is a weakly compact convex set.
In view of the weak lower semicontinuity property of I(X) we have that the inf of I(X) over C P Q is attained at some point of C P Q .Note however, that we only know that inf{I Let X m ∈ C 0 P Q , m ∈ N be a sequence so that lim m→∞ I(X m ) = inf {I(X)|X ∈ C 0 P Q } and which is weakly convergent which can be arranged by passing to a subsequence.We have X ∞ ∈ C P Q .
More can be said when (J, | • | J ) is the p -class, 1 < p < ∞ because then (J ⊗ M k , | • | J ) is a uniformly convex Banach space.Assume moreover I(X) is one of Ĩτ (X), Ĩα (X), I D τ (X), I D α (X) i.e. there is no max in the definition of I(X).Then I(X) = |∂(X)| J , where in each of the four cases ∂(X) is : The preceding remark leaves open the question about equality of the infimum of I over C 0 P Q and C P Q .We can answer this in case k J (τ ) = 0 or respectively k J (α) = 0.There is not a "no max " restriction on I(X) for this.In this situation there is a sequence B k ∈ R Similarly if ǫ ∈ [0, 1] we have where Similar computations can be carried out in the Dirac case.
Summarizing we have proved the following.
Remark 8.5 Assume that J is the p -class , 2 ≤ p < ∞ and that X is a minimizer of I(•) in C P Q .If I is Ĩτ then X satisfies ( * * * ) and if I is Ĩα then X satisfies ( * * * * ).These conditions are compressions of noncommutative τ − p -Laplace and respectively α − p -Laplace inequalities.
The intervals on which this function is not constant are the [a j , b j ] and possibly also [a − h, a] and [b, b + h] depending on whether a, b ∈ M or not.Thus if B = D(f ) the list of non-zero singular values of B − Ad(U )(B) consists of b j − a j times the number (b j − a j ) −1 for each interval [a j , b j ] and each of the intervals [a − h, a], [b, b + h] may contribute h times the number h −1 depending on whether a, b ∈ M .This gives: |B − Ad(U )(B)| p p = j (b j − a j ) 1−p + h 1−p • ♯({a, b} ∩ M ).Since |B − Ad(U )(B)| p ≥ k p (Ad(U ); P M , P N ) we get the following upper bound.If p = 1 , we have
(B m ), T j ]| J = 0. Hence, replacing B m by h(B m ), we may assume that B m ∈ R + 1 and B m converges strongly to I. Let P ∈ P .Remark then that lim m→∞ |B m − (P + (I − P )B m (I − P )| 1 = 0.This follows from |P − P B m P | 1 → 0 and |(I − P )B m P | 1 → 0 which in turn follow from the strong convergences P − P B m P → 0 and (P B m (I − P )B m P ) 1/2 → 0 in the finite von Neumann algebra P MP |P |H endowed with the finite trace which is the restriction of ρ.Since also clearly B m − (P + (I − P )B m (I − P )) ≤ 2 we infer that also |B m − (P + (I − P )B m (I − P )| J → 0 as m → ∞.This then gives |[T j , P + (I − P )B m (I − P )]| J → 0 which then finally implies k J (τ ; P ) = 0.
j , B m ]| J = 0. Passing to a subsequence, we can assume that w − lim m→∞ B m = B and we will then have BP = P and [B, T j ] = 0, 1 ≤ j ≤ n .It follows from Proposition 6.1 that P ′ = E(B; (0, ∞)) has the desired properties.Proposition 6.4 If A m = A * m ∈ R are so that A m ≤ C for all m ∈ N and lim m→∞ max 1≤j≤n |[A m , T j ]| J = 0 then we have s − lim m→∞ A m E J (τ ) = 0. Similarly, if A m = A * m ∈ R are so that A m ≤ C for all m ∈ N and lim m→∞ max 1≤j≤n |α j (A m ) − A m | J = 0 then we have s − lim m→∞ A m E J (α) = 0