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Capacity and the Quasicentral Modulus CAPACITY AND THE QUASICENTRAL MODULUS DAN-VIRGIL VOICULESCU Abstract. We point out that the quasicentral modulus is a noncom- mutative analogue of a nonlinear rearrangement invariant Sobolev con- denser capacity. In the case of the shifts by the generators of a finitely generated group, the quasicentral modulus coincides with a correspond- ing nonlinear condenser capacity on the Cayley graph of the group. Some other capacities related to the quasicentral modulus are also dis- cussed. 1. Introduction The quasicentral modulus kJ(τ) where τ is an n-tuple of operators and (J, | |J) is a normed ideal is a number which is key in many questions about normed ideal perturbations. In particular kJ(τ) is an essential ingredient in generalizations of the Kato-Rosenblum theorem and of the Weyl-von Neumann-Kuroda theorem to n-tuples of commuting hermitian operators. For a particular choice of the normed ideal J there are also connections with the Kolmogorov-Sinai entropy. Recently studying the commutant E(τ; J) of τ mod J, it turned out that this algebra provides new structure that helps understand the ubiquity of kJ(τ). Here we continue the exploration and we point out a noncommutative analogy with capacity in nonlinear potential theory. Note that J ∩ E(τ; J), which reduces to J, but can also be viewed as a noncommutative first order Sobolev space with respect to a rearrangement invariant norm | |J, while the role of the gradient goes to the n-tuple of inner derivations [·, Tj ] with respect to the components of τ. arXiv:2107.11924v3 [math.FA] 15 Aug 2021 In the case of the regular representation of a finitely generated group and of the n-tuple of shifts in ℓ2 by the generators, the nonlinear potential theory analogy fits well with a connection to Yamasaki hyperbolicity and potential theory on the Cayley graph of the group which we had already noticed earlier. 2020 Mathematics Subject Classification. Primary: 46L89; Secondary: 31C45, 47L20. Key words and phrases. quasicentral modulus, noncommutative nonlinear condenser capacity, normed ideals of operators. 1 2 D.-V. VOICULESCU 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 Sobolev space 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 von Neumann algebra 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 Hilbert space 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 measure 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 norm 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).
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