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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.

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.

Reviewer: Hamid Vishki (Mashhad)

### 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 |