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 `1 = `1(N) whose norm is invariant under permutations. Let H be a Hilbert space. We define an associated Schatten ideal 1 SE ⊂ B(H) by T 2 SE () (sn(T ))n=1 2 E with a (quasi-)norm kT k = k(s (T ))1 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 k1; kYX kE ≤ kX kE kY k1; 8X 2 E; Y 2 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. 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 Calkin algebra 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 `1 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 `1=c0.\The Calkin algebra is important because it is the repository of all asymptotic information about operators on H". (Arveson). 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 2.9 from J.W. Calkin's paper 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 2 B(H) is such that AB − BA 2 J for all B 2 B(H), then A = T + λ1 for some T 2 J and λ 2 C. RESTATEMENT: D(B(H); J ) := fT 2 B(H):[T ; S] := TS − ST 2 J 8S 2 B(H)g = J + C1 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 Some notations Let A be a complex algebra. For x; y 2 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); 8x; y 2 A: If w 2 A, then the map δw : A!A given by δw (x) = [w; x], x 2 A is a derivation. A derivation of this form is called inner. 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 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 2 B(H) which is not in K + C1. Consider δw : B(H) ! B(H). It is trivial that δw (x) 2 K for any x 2 K. That is δ := δw jK is a derivation on K. However, δ is not inner on K ! Indeed, suppose there exists an element v 2 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 2= K + C1 and v 2 K. 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 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 2 M and [d; M] ⊂ J, where J is closed in the uniform norm. Obviously, du − ud 2 J for any unitary u 2 M. Equivalently u∗du − d 2 J for any unitary ∗ u 2 M. By the DAT, there exists a convex combination of ui dui converging to z 2 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 2 J, we obtain that the uniform norm limit z − d 2 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 = fx 2 B(H): xI ⊂ J g and D(I; J ) = fT 2 B(H):[T ; S] 2 J ; 8S 2 Ig: 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.