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Besides the introduction and references there are six more sections. Sec- tion 2 recalls the definition of the quasicentral modulus. Section 3 provides background about condenser capacity in the nonlinear potential theory re- arrangement invariant context. Then in section 4 we explain the noncommutative analogy. The analogy in the case of finitely generated groups is the subject of section 5. Two capacities related to the quasicen- tral modulus are introduced in section 6. The brief concluding remarks in section 7 are about type II∞ factor and semifinite infinite framework, a possible role for the ampliation homogeneity property and the Hilbert-Schmidt class and linear noncommutative potential theory.

2. The quasicentral modulus Let H be a separable complex of infinite dimension and let B(H), K(H), R(H) denote the bounded ,the compact and the finite rank operators on H respectively. If (J, | |J) is a normed ideal ([10], [15]) and τ = (Tj)1≤j≤n is a n-tuple of bounded operators on H, kJ(τ) is defined as follows ([17], [19]. [20] or [22]).

kJ(τ) is the least C ∈ [0, ∞], such that there exist finite rank operators 0 ≤ Am ≤ I so that Am ↑ I and we have

lim max |[Am, Tj]|J = C. m→∞ 1≤j≤n − Particularly important is the case of the ideals Cp , 1 ≤ p ≤ ∞ our notation of the Lorentz (p, 1) ideal where

− −1+1/p |T |p = skk kX∈N ∗ 1/2 with s1 ≥ s2 ≥ . . . denoting the eigenvalues of (T T ) in decreasing order. − − If J = Cp we denote the quasicentral modulus kp (τ). Note that when p = 1 − , C1 = C1 . while if p > 1 the quasucentral modulus for the Schatten - von Neumann p-class can only take the values 0 and ∞ ([19]). In many − cases sharp perturbation results are for J = Cp , not for the Schatten - von Neumann class Cp ([19], [20], [22]).

3. Condenser capacity background Let Ω ⊂ Rn be an open set and let K ⋐ Ω be a compact subset. The p-capacity of K relative to Ω is

cap (K;Ω) = inf{ |∇u |p dλ|u ∈ C∞(Ω), 0 ≤ u ≤ 1, u| ≡ 1} p Z 0 K ∞ ∞ where ∇ denotes the gradient, dλ Lebesgue and C0 (Ω) the C functions with compact support in Ω. The definition is then extended to open subsets G ⊂ Ω by taking the sup over the compact subsets K ⋐ Ω and CAPACITYANDQUASICENTRALMODULUS 3 then further to general sets E ⊂ Ω by taking the inf over open sets G ⊃ E ([1], [9], [11], [13]). Note that a power - scaling, passing from capp(K; Ω) to 1/p capp(K; Ω) amounts to considering

∞ inf{k∇ukp|u ∈ C0 (Ω), 0 ≤ u ≤ 1, u|K ≡ 1} p where k kp is the L on Ω. There is a natural generalization to the Lorentz-space setting, where 1/p capp,q (K; Ω) is defined as the

∞ inf{k∇ukp,q|u ∈ C0 (Ω), 0 ≤ u ≤ 1, u|K ≡ 1} p,q where k kp,q is the norm of the Lorentz-space L (Ω) ([6], [7]) and of course instead of the Lorentz (p,q)-norm, general rearrangement invariant norms can be considered. A natural variant of the definition of p-capacity is with ∞ 1,p ∞ u ∈ C0 replaced by u ∈ W0 the closure of C0 in the first order Sobolev space of functions with gradients in Lp. There are also similar variants of the definition of capp,q(K : Ω) involving Sobolev-Lorentz spaces (for more on Sobolev-Lorentz spaces (see [4], [6], [7], [12], [16]). We should also add that a condenser is a more general object involving two disjoint subsets of Ω. Thus if K,L ⋐ Ω are compact subsets and K ∩ L = ∅ one considers cap (K,L;Ω) = inf{ |∇u |p dλ|u ∈ C∞(Ω), 0 ≤ u ≤ 1, u| ≡ 1, u| ≡ 0} p Z 0 K L 1/p Similarly one defines capp,q(K,L; Ω) to be

∞ inf{k∇ukp,q|u ∈ C0 (Ω), 0 ≤ u ≤ 1, u|K ≡ 1, u|L ≡ 0} There are corresponding variants and extensions to the general rearrange- ment invariant setting like in the case L = ∅, which was our previous dis- cussion.

4. The noncommutative extension We shall first bring to the forefront the algebras of operators on L2(Ω; dλ) 1/p in the definition of capp,q(K,L : Ω) or of a general Sobolev-rearrangement ∞ ∞ invariant analogue. Let A = L (Ω; dλ), B = C0 (Ω), B ⊂ A which we identify with the algebras of multiplication operators on L2(Ω; dλ) to which they give rise. Then A is a von Neumann algebra and B is a weakly dense *-subalgebra. For the sake of simplicity assume the two disjoint compact subsets K,L ⊂ Ω are equal to the closures of their interiors. They then give rise to the idempotents P = χK,Q = χL (the indicator functions of the sets). Thus we have P = P ∗ = P 2,Q = Q∗ = Q2,PQ = 0. Further let 4 D.-V. VOICULESCU

∞ δ(f)= k∇fkp,q if f ∈ C0 (Ω) or the same formula with some other rearrangement invariant norm in- stead of the (p,q)-Lorentz norm. Since ∇f has n components we take n max1≤j≤n k∂f/∂xj kp,q or use some other norm on C . Then δ has the property

δ(fg) ≤ kfkδ(g)+ kgkδ(f) that is, it is a differential seminorm ([3]) or Leibniz seminorm ([14]). In this 1/p simplified setting,the definition of (capp,q(K,L; Ω)) then becomes

inf{δ(u)|u ∈ B, 0 ≤ u ≤ 1, P ≤ u, uQ = 0} . The same construction can also be performed with noncommutative alge- bras. Let H be a separable complex Hilbert space and let A = B(H) and let B = R(H). Let further P,Q ∈ B be hermitian idempotents so that PQ = 0. If τ = (Tj)1≤j≤n is a n-tuple of operators in B(H) we define the differential seminorm

δ(X)= max |[Tj, X]|J 1≤j≤n where (J, | |J) is a normed ideal. We get a quantity

kJ(τ; P,Q)= inf{δ(X)|0 ≤ X ≤ I, X ∈ R(H), P ≤ X,XQ = 0} which is a noncommutative analogue of the condenser capacity with respect to the Sobolev rearrangement invariant space. A detail we would like to point out is that since 0 ≤ X ≤ I in the definition, P ≤ X is equivalent to XP = P. In case Q = 0 we get a quantity

kJ(τ; P )= kJ(τ; P, 0). Let P(H)= {P = P ∗ = P 2|P ∈ R(H)} . It is clear that

P1 ≤ P2 ⇒ kJ(τ; P1) ≤ kJ(τ; P2) since passing from P1 to P2 decreases the set over which the inf is taken. Fi- nally the quasicentral modulus is recovered from these quantities as kJ(τ; I) that is

kJ(τ)= sup{kJ(τ; P )|P ∈ P(H)}. CAPACITYANDQUASICENTRALMODULUS 5

Thus kJ(τ) is an analogue of a power-scaled Sobolev capacity of the whole ambient open set Ω. Obviously this construction can be performed with many other pairs A, B and differential seminorms δ, especially since we did not bother to put ad- ditional requirements on A, B, δ here.

5. Finitely generated groups

Let G be a group with a finite generator γ = {g1, ..., gn}. Then G is the set of vertices of its Cayley graph and (g, gj g) where g ∈ G, 1 ≤ j ≤ n are the edges. Let further (J, | |J) be a normed ideal and let ℓJ(G) be the corresponding function space on G, that is functions f : G −→ C with the 2 norm |f|J being the J-norm of the multiplication operator by f in ℓ (G). (That is the well-known bijection between normed ideals and symmetrically normed Banach sequence spaces.) By r(G) ⊂ ℓ∞(G) we denote the functions with finite support on G and we consider the following differential seminorm on r(G)

δJ(f)= max |f(gj·) − f(·)|J. 1≤j≤n If X ⊂ G is a finite subset, we define the J-capacity of X by

capJ(X)= inf{δJ(f)|f ∈ r(G), 0 ≤ f ≤ 1,f|X ≡ 1}. The definition is extended to general subsets Y ⊂ G by

capJ(Y )= sup{δJ(X)|X ⊂ Y, X finite}. The p-hyperbolicity of Yamasaki ([24]) of G, is the property

capp({e}) > 0

where e ∈ G is the neutral element and capp is capCp , where Cp denotes the Schatten-von Neumann p-class. It is easily seen that capJ({e}) > 0 is equivalent to

X 6= ∅⇒ capJ(X) > 0 and this is equivalent to capJ(G) > 0. In particular the Yamasaki p- hyperbolicity is equivalent to capp(G) > 0. Similarly Yamasaki p-parabolicity is the property capp({e}) = 0 , which is equivalent to capp(X) = 0 for all X ⊂ G and is also equivalent to capp(G) = 0. The capacity construction has also a more general version for condensers

capJ(X1, X2)= inf{δ(f)|f ∈ r(G), 0 ≤ f ≤ 1,f|X1 ≡ 1,f|X2 ≡ 0} 6 D.-V. VOICULESCU

where X1, X2 ⊂ G are disjoint finite subsets. In terms of algebras of operators these constructions correspond to A = ∞ ℓ (G), B = r(G) and the differential seminorm δJ. Potential theory on graphs or networks being a well-developed subject ( see for instance [2]) it is natural to pass from Cayley graphs of groups to more general graphs, which also should apply to the above remarks. Of course in view of this kind of generalization the definition in the case of Cayley graphs should be modified by moving the max1≤j≤n in the definition of δJ so that the modified quantity does not depend on a labelling of the edges, for instance

∗ δJ (f)= | max |f(gj·) − f(·)||J. 1≤j≤n On the other hand let λ be the left regular representation of G on ℓ2(G) and let λ(γ) be the n-tuple of unitary operators (λ(gj))1≤j≤n . If X ⊂ G is a finite subset, let χX ∈ r(G) be the indicator function, which being identified with the multiplication operator, is a hermitian projection in R(ℓ2(G)) ⊂ B(ℓ2(G)). We then have:

capJ({e})= kJ(λ(γ); χ{e})

capJ(X)= kJ(λ(γ); χX )

capJ(X1, X2)= kJ(λ(γ); χX1 ,χX2 )

capJ(G)= kJ(λ(γ)) where X1, X2 ⊂ G are disjoint finite subsets. Clearly the LHS’s are ≥ than the RHS’s because they involve infimums over subsets of the sets over which we take infimums in the RHS’s. For the converse one uses the projection of norm one

diag : B(ℓ2(G)) → ℓ∞(G) which amounts to taking the diagonal of the matrix of an operator with respect to the canonical basis and which decreases J-norms. Remark further that

−1 |[λ(g), A]|J = |λ(g)Aλ(g ) − A|J which is

−1 −1 ≥ |diag(λ(g)Aλ(g ) − A)|J = |λ(g)(diagA)λ(g ) − diagA|J CAPACITYANDQUASICENTRALMODULUS 7

so that if the matrix of A has only finitely many non-zero entries one sees that these kind of quantities in the infimums of the RHS’s majorize quan- tities in the infimums of the LHS’s. Combining this with an approximation argument involving compressions by projections corresponding to certain fi- nite sets, of the finite rank operators one obtains operators with matrices with finitely many non-zero entries which completes the proof.

6. Capacities related to the quasicentral modulus In case τ is a n-tuple of commuting hermitian operators, there are two capacities, in the usual sense, that arise in connection with the quasicentral modulus. Here, it will be suitable to describe the norm | |J by a norming function Φ, that is

|X|J = Φ(s1, s2, ...) ∗ 1/2 where s1 ≥ s2 ≥ ... are the eigenvalues of (X X) in decreasing order. Up to a power-scaling, the Hausdorff measure has a straightforward gen- eralization involving a norming function. If E ⊂ Rn is a bounded set, let

UΦ(E)= inf{Φ(r1, r2, ...)| B(xj; rj) ⊃ E} j[∈N and if ǫ> 0, let UΦ,ǫ be the above infimum restricted to coverings with balls of radius < ǫ. We then take the limit as ǫ → 0 of the UΦ,ǫ and we arrive ∗ at a quantity UΦ(E), which is a generalization of power scaled Hausdorff measure. Let τ be a n-tuple of commuting hermitian operators with a cyclic vector n ∗ ξ and let σ(τ) ⊂ R be its spectrum. Then UΦ(σ(τ)) provides an upper bound for kΦ(τ). We sketch below the argument, which is along the lines of [17], [22]. If σ(τ) ⊂ j∈N B(xj; rj), rj < ǫ, let (ωj)j∈N be a partition of σ(τ) into Borel sets soS that ωj ⊂ B(xj; rj). Let Pn ∈ P(H) be the projection onto C 1≤j≤n E(ωj)ξ, where E(·) denotes the spectral measure of τ. Then L

|[τ, Pn]|Φ ≤ 2Φ(r1, ..., rn, 0, ...) and

k[τ, Pn]k≤ 2ǫ,n →∞⇒kPnξ − ξk→ 0 (the norms for n-tuples are the max of the norms of the components). Let- ting ǫ → 0 we can find a sequence Qm ∈ P(H) so that

kQmξ − ξk→ 0, k[Qm,τ]k→ 0 8 D.-V. VOICULESCU and

∗ lim sup |[Qm,τ]|Φ ≤ 2UΦ(σ(τ)). m→∞

Note that if Q is a weak limit of Qm ’s we have Qξ = ξ, [Q,τ] = 0 which ∗ implies Q = I. This suffices to imply kΦ(τ) ≤ 2UΦ(σ(τ)) ([17]). In view of the results about kΦ in [19], the case of norming functions Φ defined from a sequence πj ↓ 0 as j →∞, by the formula

Φ(r1, r2, ...)= πjrj Xj where r1 ≤ r2 ≤ ... is of particular interest. It is also possible to proceed in the opposite direction, from the quasicen- tral modulus to a set function on Borel sets. Let µ be a Radon measure on Rn with compact support and let E ⊂ Rn 2 be a Borel set. On L (E,µ|E) we consider τE the n-tuple of multiplication operators by the coordinate functions. We define

Qµ,Φ(E)= kΦ(τE).

Note that Qµ,Φ(E) depends only on the absolute continuity class of µ, that is, if µ and ν have the same null sets, then

Qµ,Φ(E)= Qν,Φ(E). − Particular instances of this construction were used in [17], [22] to study kp . − If Φp is the corresponding norming function (i.e. the case of the sequence −1+1/p πj = j ), then if p = n we proved in [17] that

1/n Qµ,Φ(E)= C(λ(E)) when µ|E is equivalent to n-dimensional Lebesgue measure. We found a similar result in non-integer dimensions with Lebesgue measure replaced by the Hutchieson measure on certain selfsimilar fractals, on which it coincides with Hausdorff measure [22]. This idea of this construction can also be adapted to bounded metric measure spaces, but without some n-tuple of functions playing the role of the coordinate functions, it may be more natural to use the family of all real- valued Lipschitz functions with Lipschitz constant ≤ 1 and to consider kΦ of the infinite family of multiplication operators arising from these functions in L2(E,µ|E) where E is a Borel set. Note also that to prove that a given Qµ,Φ is > 0, in view of [19] requires more difficult analytic results. like in [8], [17]. CAPACITYANDQUASICENTRALMODULUS 9

7. Concluding remarks

A). Infinite semifinite von Neumann algebras, factors of type II∞ It is obvious that the quasicentral modulus can be defined in type II∞ factors. One takes A to be the factor, instead of B(H) and B the ideal of finite rank bounded operators with respect to the trace of the factor, that is operators in the factor which have support projections with finite trace. There are also normed ideals of such factors and the differential seminorm is also defined as the max of the ideal norms of the commutators with the components of the n-tuple of operators. The problem is to develop good examples which can be studied, part of the difficulties arising from the ex- istence non-isomorphic such factors, the multitude of non-conjugate masa’s etc On the other hand, the definition of the quasicentral modulus can be further extended to separable semifinite infinite von Neumann algebras with a specified normal semifinite faithful trace and n-tuples of trace-preserving automorphisms (αj)1≤j≤n. The differential seminorm for a bounded finite rank element A being

δ(A)= max |αj(A) − A|J 1≤j≤n where | |J is a corresponding normed ideal norm. Examples would include commutative von Neumann algebras A. This easily translates to n-tuples of measure-preserving automorphisms of measure spaces with an infinite σ- finite measure and a specified symmetric function space norm (assuming the measure has no atoms). Perhaps such invariants have the advantage of having many examples in sight when compared to factor situation. B). Power scaling The analogy of Sobolev capacities with respect to rearrangement invari- ant norms and the quasicentral modulus is up to a power scaling. One may wonder whether there are natural candidates of similar exponents for the quasicentral modulus. Introducing these exponents may improve the anal- − ogy. For instance in the case of kp , which corresponds to the Lorentz (p, 1) − − p ideal Cp one should consider (kp ) . The results of [17], [18], [23] for com- muting n-tuples of hermitian operators show certain advantages in the case of integer dimension and also in certain examples with non-integer dimen- sion. These results suggest that in case the normed ideal (J, | |J) has the ampliation homogeneity property [23]

1/α kJ(τ ⊗ Im)= m kJ(τ) the exponent should be α. C). Commutants mod Hilbert-Schmidt In ([21], 6.3) we found that commutants mod C2 of normal operators were objects which were related to the noncommutative potential theory based 10 D.-V. VOICULESCU on Dirichlet forms ([5]). One may see in this a certain similarity with what happens classically with p-capacity when p = 2.

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Department of Mathematics, University of California at Berkeley, Berke- ley, CA 94720-3840 Email address: [email protected]