TAILIEUCHUNG - Đề tài " The derivation problem for group algebras "

If G is a locally compact group, then for each derivation D from L1 (G) into L1 (G) there is a bounded measure μ ∈ M (G) with D(a) = a ∗ μ − μ ∗ a for a ∈ L1 (G) (“derivation problem” of B. E. Johnson). Introduction Let A be a Banach algebra, E an A-bimodule. A linear mapping D : A → E is called a derivation, if D(a b) = a D(b) + D(a) b for all a, b ∈ A ([D, Def. ]). For x ∈ E, we define the inner derivation adx :. | Annals of Mathematics The derivation problem for group algebras By Viktor Losert Annals of Mathematics 168 2008 221-246 The derivation problem for group algebras By Viktor Losert Abstract If G is a locally compact group then for each derivation D from L1 G into L1 G there is a bounded measure ạ E M G with D a a ạ ạ a for a E L1 G derivation problem of B. E. Johnson . Introduction Let A be a Banach algebra E an A-bimodule. A linear mapping D A E is called a derivation if D ab aD b D a b for all a b E A D Def. . For x E E we define the inner derivation adx A - E by adx a xa ax as in GRW adx ỗx in the notation of D . If G is a locally compact group we consider the group algebra A L1 G and E M G with convolution note that by Wendel s theorem D Th. M G is isomorphic to the multiplier algebra of L1 G and also to the left multiplier algebra . The derivation problem asks whether all derivations are inner in this case D Question p. 746 . The question goes back to J. H. Williamson around 1965 personal communication by H. G. Dales . The corresponding problem when A E is a von Neumann algebra was settled affirmatively by Sakai Sa using earlier work of Kadison see D p. 761 for further references . The derivation problem for the group algebra is linked to the name of B. E. Johnson who pursued it over the years as a pertinent example in his theory of cohomology in Banach algebras. He developed various techniques and gave affirmative answers in a number of important special cases. As an immediate consequence of the factorization theorem the image of a derivation from L1 G to M G is always contained in L1 G . In JS with A. Sinclair it was shown that derivations on L1 G are automatically continuous. In JR with J. R. Ringrose the case of discrete groups G was settled affirmatively. In J1 Prop. this was extended to SIN-groups and amenable groups serving also as a starting point to the theory of amenable Banach algebras . In addition some cases of .

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