Commutator Estimates in W -Algebras and Applications

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Prehistory: classical results (von Neumann and Calkin) Derivations: some classical facts. M.J. Hoﬀman’s extension of Calkin’s Theorem 2.9 Main questions in the bounded setting Main result in the bounded setting Comments and corollaries The outline of the proof of Theorem

Commutator estimates in W ∗-algebras and applications

Fedor Sukochev

University of NSW, AUSTRALIA

April 25, 2020

F.Sukochev Commutator estimates and derivations Prehistory: classical results (von Neumann and Calkin) Derivations: some classical facts. M.J. Hoﬀman’s extension of Calkin’s Theorem 2.9 Main questions in the bounded setting Main result in the bounded setting Comments and corollaries The outline of the proof of Theorem J. von Neumann, Some matrix inequalities and metrization of matric-space, Rev. Tomsk Univ. 1 (1937), 286–300.

In 1937, J. von Neumann showed that if k·kE is a symmetric norm n on R then one can deﬁne a norm on the space of n × n matrices by

kAkE = k(s1(A),..., sn(A))kE ,

where s1(A),..., sn(A) are the singular values of A (i.e. the eigenvalues of (A∗A)1/2) in decreasing order. Inﬁnite-dimensional development of this pioneering result by von Neumann is in highly inﬂuential books by R. Schatten, Gohberg&Krein and B. Simon. All classical Banach space geometry (books by Lindenstrauss&Tzafriri) is strongly allied with this object.

F.Sukochev Commutator estimates and derivations Prehistory: classical results (von Neumann and Calkin) Derivations: some classical facts. M.J. Hoﬀman’s extension of Calkin’s Theorem 2.9 Main questions in the bounded setting Main result in the bounded setting Comments and corollaries The outline of the proof of Theorem N. Kalton and F.S.,Symmetric norms and spaces of operators, J. Reine Angew. Math. 621 (2008), 81–121.

Symmetric Banach sequence space E is a Banach ideal of the space `∞ = `∞(N) whose norm is invariant under permutations. Let H be a Hilbert space. We deﬁne an associated Schatten ideal ∞ SE ⊂ B(H) by T ∈ SE ⇐⇒ (sn(T ))n=1 ∈ E with a (quasi-)norm kT k = k(s (T ))∞ k . Recall, that a Banach space (E, k · k ) SE n n=1 E E which a linear subspace in B(H) is said to be a Banach ideal in B(H) if its norm k · kE satisﬁes the following estimates

kXY kE ≤ kX kE kY k∞, kYX kE ≤ kX kE kY k∞, ∀X ∈ E, Y ∈ B(H).

MAIN RESULT of [KS]: (S , k·k ) is a Banach ideal in B(H). E SE

F.Sukochev Commutator estimates and derivations Prehistory: classical results (von Neumann and Calkin) Derivations: some classical facts. M.J. Hoﬀman’s extension of Calkin’s Theorem 2.9 Main questions in the bounded setting Main result in the bounded setting Comments and corollaries The outline of the proof of Theorem J.W. Calkin, Two-sided ideals and congruences in the ring of bounded operators in Hilbert space, Ann Math. 42 (1941), 839-873.

From Introduction The developments of the present paper center around the observation that the ring B(H) of bounded everywhere defined operators in Hilbert space contains non-trivial two-sided ideals. This fact, which has escaped all but oblique notice in the development of the theory of operators, is of course fundamental from the point of view of algebra and at the same time differentiates B(H) sharply from the ring of all linear operators over a unitary space with finite dimension number.

Let H be a separable Hilbert space and let K be the C ∗-algebra of all compact operators on H. K is a noncommutative analogue of the space c0 of all vanishing sequences. The ideal K is closed in B(H), which is a noncommutative analogue of the algebra `∞ of all bounded sequences. The quotient C ∗-algebra C = B(H)/K is called the Calkin algebra. It is a noncommutative analogue of the quotient algebra `∞/c0.“The Calkin algebra is important because it is the repository of all asymptotic information about operators on H”. (Arveson).

THEOREM 2.9. Let J be an ideal in B(H), J 6= B(H). Then the center B(H)/J , that is the set of all elements from B(H)/J which commute with every element of B(H)/J , is the set of all elements λ1, where λ is a complex number. THE GIST: If A ∈ B(H) is such that AB − BA ∈ J for all B ∈ B(H), then A = T + λ1 for some T ∈ J and λ ∈ C. RESTATEMENT: D(B(H), J ) := {T ∈ B(H):[T , S] := TS − ST ∈ J ∀S ∈ B(H)} = J + C1

Let A be a complex algebra. For x, y ∈ A, the commutator [x, y] is deﬁned by setting [x, y] := xy − yx. A linear map δ : A → A is called a derivation if

δ(xy) = δ(x)y + xδ(y), ∀x, y ∈ A.

If w ∈ A, then the map δw : A → A given by δw (x) = [w, x], x ∈ A is a derivation. A derivation of this form is called inner.

Not every derivation on a C ∗-algebra is inner. Let K be the algebra of compact operators on H. Example (Sakai, C ∗-algebras and W ∗-algebras, Springer, 1971, see also Ber-Huang-Levitina-S.-JFA-2017 fore more examples) Take an arbitrary element w ∈ B(H) which is not in K + C1. Consider δw : B(H) → B(H). It is trivial that δw (x) ∈ K for any x ∈ K. That is δ := δw |K is a derivation on K. However, δ is not inner on K ! Indeed, suppose there exists an element v ∈ K such that δ = δv . Then since δv = δw on K, we immediately conclude that v − w belongs to the center Z(K). Hence, w = v + λ1, which is a contradictions, since w ∈/ K + C1 and v ∈ K.

However, we claim that every derivation δw : B(H) → K is inner (see Johnson–Parrott–Popa Theorem (Popa, JFA, 1985) and Ber–Huang–Levitina–S. for more general results). We provide here a very short proof of a slightly more general result. Let M be a von Neumann algebra, d ∈ M and [d, M] ⊂ J, where J is closed in the uniform norm. Obviously, du − ud ∈ J for any unitary u ∈ M. Equivalently u∗du − d ∈ J for any unitary ∗ u ∈ M. By the DAT, there exists a convex combination of ui dui converging to z ∈ Z(M) in norm. Obviously, the same convex ∗ combination of ui dui − d converging to z − d in norm. Since J is ∗ norm closed and since ui xui − d ∈ J, we obtain that the uniform norm limit z − d ∈ J (≈ Theorem 2.9). Finally, [d, ·] = [d − z, ·]. That is δd = δd−z .

F.Sukochev Commutator estimates and derivations Prehistory: classical results (von Neumann and Calkin) Derivations: some classical facts. M.J. Hoﬀman’s extension of Calkin’s Theorem 2.9 Main questions in the bounded setting Main result in the bounded setting Comments and corollaries The outline of the proof of Theorem M.J. Hoﬀman, Essential commutants and multiplier ideals, Indiana Univ. Math. J. 30 (1981), no. 6, 859–869.

Replace B(H) with an arbitrary ideal I. That is, ﬁx two self-adjoint ideals J , I in B(H). We set

J : I = {x ∈ B(H): xI ⊂ J }

and D(I, J ) = {T ∈ B(H):[T , S] ∈ J , ∀S ∈ I}.

MAIN RESULT: D(I, J ) = J : I + C1. If I = B(H), then J : I = J : B(H) = J and the assertion above yields Calkin’s Th.2.9.

F.Sukochev Commutator estimates and derivations Prehistory: classical results (von Neumann and Calkin) Derivations: some classical facts. M.J. Hoﬀman’s extension of Calkin’s Theorem 2.9 Main questions in the bounded setting Main result in the bounded setting Comments and corollaries The outline of the proof of Theorem Theorem 4.1.6. from Sakai’s book “C ∗ and W ∗-algebras”

Theorem Let δ be a derivation on a W ∗-algebra M. Then δ is inner, namely there exists an element a ∈ M such that δ(x) = [a, x], x ∈ M. Moreover, we can choose such an element a as follows: kak ≤ kδk. Remark Observe that what this theorem actually says is the following: given a ∈ M, there exists an element c ∈ Z(M) such that ka − ck ≤ kδak. Indeed, if for a, b ∈ M, the inner derivations δa and δb coincide, then we necessarily have δa−b = 0 on M, and therefore a − b ∈ M0. The latter implies immediately that a − b ∈ Z(M).

F.Sukochev Commutator estimates and derivations Prehistory: classical results (von Neumann and Calkin) Derivations: some classical facts. M.J. Hoﬀman’s extension of Calkin’s Theorem 2.9 Main questions in the bounded setting Main result in the bounded setting Comments and corollaries The outline of the proof of Theorem Questions motivated by Calkin and Hoﬀman’s work

Let M be a W ∗-algebra and let I be an ideal in M. We ﬁx this notation. Motivated by Calkin’s Th.2.9, we ask: “Let π : M → M/I be a canonical epimorphism. Do we have

π−1(center(M/I)) = Z(M) + I?”

Motivated by Hoﬀman’s “derivation viewpoint” on Calkin’s Th.2.9, we ask: “Let δ : M → I be a derivation. Does there exist an element a ∈ I, such that δ(·) = δa(·) = [a, ·]?” Remember, our ideal I is not necessarily uniform norm closed. The DAT is not applicable! Calkin’s and Hoﬀman’s techniques (which are heavily B(H)-type) are not applicable!

Let M be a W ∗-algebra and let Z(M) be the center of M. Fix a ∈ M and consider the inner derivation δa on M. It follows from Sakai’s Theorem 4.1.6, that there exists c ∈ Z(M) such that kδak ≥ ka − ckM. In view of this result, it is natural to ask whether there exists an element y ∈ M with kyk ≤ 1 and c ∈ Z(M) such that |[a, y]| ≥ |a − c|? Our main result (next slide) answers this as well as preceding questions.

Theorem (Ber-S) Let M be a W ∗-algebra and let a = a∗ ∈ M.

(i) ∗ There exists c0 = c0 ∈ Z(M), so that for any ε > 0 there ∗ exists uε = uε ∈ U(M) such that

|[a, uε]| ≥ (1 − ε)|a − c0 |. (1)

(ii) If M is a finite W ∗-algebra or else a purely infinite σ-finite ∗ ∗ W -algebra, then there exists c0 = c0 ∈ Z(M) and ∗ u0 = u0 ∈ U(M), such that

∗ |[a, u0 ]| = u0 |a − c0 |u0 + |a − c0 |, (2) where U(M) is the group of all unitary elements in M;

Observe that the equality (2) trivially yields the estimate (1) even for the case ε = 0. Nevertheless, the result of (i) is still sharp. Indeed, if M is an infinite semifinite σ-finite factor, then there exists a self-adjoint element a ∈ M such that for every λ ∈ C and u ∈ U(M) the inequality |[a, u]| ≥ |a − λ1| fails. Hence, the multiplier (1 − ε) in the part( i) of Theorem ?? can not be omitted.

Our Main Theorem yields a completely diﬀerent proof of Calkin’s Th.2.9. It also answers the questions stated above. Corollary (Ber-S) Let M be a W ∗-algebra and let I be an ideal in M. Let δ : M → I be a derivation. Then there exists an element a ∈ I, such that δ = δa = [a, ·].

Proof. Since δ is a derivation on a W ∗-algebra, it is necessarily inner (Sakai’s [Theorem 4.1.6]). Thus, there exists an element d ∈ M, such that δ(·) = δd (·) = [d, ·]. It follows from our hypothesis that [d, M] ⊆ I. Without loss of generality, let d be self-adjoint. It follows now from Ber-S Theorem, that there exist c ∈ Z(M) and u ∈ U(M), such that |[d, u]| ≥ 1/2|d − c|. This implies obtain d − c ∈ I. Setting a := d − c, we deduce that a ∈ I and δ = [a, ·].

Corollary Let M be a W ∗-algebra, let I be an ideal in M and let π : M → M/I be a canonical epimorphism. Then, π−1(center(M/I)) = Z(M) + I. Proof. Let a ∈ π−1(center(M/I)). Then [a, x] = ax − xa ∈ I for any x ∈ M. By the preceding Corollary applied to δa, we obtain a + c ∈ I for some c ∈ Z(M). Therefore a ∈ Z(M) + I, that is π−1(center(M/I)) ⊂ Z(M) + I. The converse inclusion is trivial.

Let us ﬁx a W ∗-algebra M and two self-adjoint ideals I, J in M. We set J : I = {x ∈ M : xI ⊂ J } and D(I, J ) = {x ∈ M :[x, y] ∈ J , ∀y ∈ I}. Observe that J : I is an ideal in M. In particular, (J : I)∗ = J : I = {x ∈ M : Ix ⊂ J }.

Corollary For any W ∗-algebra M and any ideals I, J in M we have D(I, J ) = J : I + Z(M).

Deﬁnition A linear subspace I in the von Neumann algebra M equipped

with a norm k·kI is said to be a symmetric operator ideal if (1) kSkI ≥ kSk∞ for all S ∈ I, ∗ (2) kS kI = kSkI for all S ∈ I, (3) kASBkI ≤ kAk∞ kSkI kBk∞ for all S ∈ I, A, B ∈ M. Observe, that every symmetric operator ideal I is a two-sided ideal in M, and therefore, it follows from 0 ≤ S ≤ T and T ∈ I that S ∈ I and kSkI ≤ kT kI .

Corollary Let M be a W ∗-algebra, let I be a symmetric operator ideal in M and let δ : M → I be a self-adjoint derivation. Then there exists an element a ∈ I, satisfying the inequality kakI ≤ kδkM→I and such that δ = δa = [a, ·].

Let M coincide with algebra Mn(C) of all n × n complex matrices. ∗ Fix a = a ∈ Mn(C). The claim of Theorem (Ber-S) ∗ |[a, u0 ]| = u0 |a − c0 |u0 + |a − c0 |, is invariant under the action of inner ∗-automorphisms and so, we can assume that

λ1 0 0

0 λ2 0

a = ...... ∈ Mn(C),

......

0 0 λn

where λ1 ≤ λ2 ≤ ... ≤ λn. F.Sukochev Commutator estimates and derivations Prehistory: classical results (von Neumann and Calkin) Derivations: some classical facts. M.J. Hoﬀman’s extension of Calkin’s Theorem 2.9 Main questions in the bounded setting Main result in the bounded setting Comments and corollaries The outline of the proof of Theorem The main idea in ﬁnite-dimensional setting-2

Let the unitary matrix u ∈ Mn(C) be counter-diagonal, that is

0 0 1

0 1 0

u = ...... ,

......

1 0 0 and observe that

λn 0 0

0 λn−1 0 ∗ u au = ......

......

0 0 λ1 F.Sukochev Commutator estimates and derivations Prehistory: classical results (von Neumann and Calkin) Derivations: some classical facts. M.J. Hoﬀman’s extension of Calkin’s Theorem 2.9 Main questions in the bounded setting Main result in the bounded setting Comments and corollaries The outline of the proof of Theorem The main idea in ﬁnite-dimensional setting-3

Therefore,

|λn − λ1| 0 0

0 |λn−1 − λ2| 0 ∗ |[a, u]| = |u au − a| = ......

......

0 0 |λ1 − λn|

If n is odd, then for all 1 ≤ k ≤ n we have

|λk − λn+1−k | = |λk − λ0| + |λn+1−k − λ0|

for λ0 = λ(n+1)/2.

If n is even, then for all 1 ≤ k ≤ n we have

|λk − λn+1−k | = |λk − λ0| + |λn+1−k − λ0|

for every λ0 ∈ [λn/2, λn/2+1]. Therefore, for every n ∈ N, we have

∗ |[a, u]| = u |a − λ01|u + |a − λ01|.

This completes the proof.

F.Sukochev Commutator estimates and derivations