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The derivation problem for group algebras. (English) Zbl 1171.43004

In this interesting substantial paper the author solves the so-called “derivation problem” firstly studied by B. E. Johnson in [Mem. Am. Math. Soc. 127 (1972; Zbl 0256.18014)].
Let \(L^1(G)\) and \(M(G)\) denote the convolution algebra of integrable functions and the convolution algebra of regular Borel measures on a locally compact group \(G\), respectively. A long-standing open problem (which is known as “derivation problem”) in the theory of cohomology of Banach algebras is whether every continuous derivation from \(L^1(G)\) into \(M(G)\) is inner, or equivalently, whether the cohomology group \(H^1(L^1(G), M(G))\) is trivial.
B. E. Johnson, in the above mentioned Memoirs, solved the problem for a wide variety of groups, including amenable groups and [SIN]-groups. He also solved it for the case of connected groups in [J. Lond. Math. Soc., II. Ser. 63, No. 2, 441–452 (2001; Zbl 1012.43001)]. F. Ghahramani, V. Runde and G. Willis in [Proc. Lond. Math. Soc., III. Ser. 80, No. 2, 360–390 (2000; Zbl 1029.22007)] also gave a number of affirmative answers to the problem.
In the paper under review the author presents a deep functional analytic proof and solves the problem for its full general case by showing that: Every continuous derivation from \(L^1(G)\) into \(M(G)\) is inner, in which \(G\) is an arbitrary locally compact group.

MSC:

43A20 \(L^1\)-algebras on groups, semigroups, etc.
47B47 Commutators, derivations, elementary operators, etc.
46H99 Topological algebras, normed rings and algebras, Banach algebras
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
22F05 General theory of group and pseudogroup actions
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