Finite-dimensional left ideals in some algebras associated with a locally compact group.

*(English)*Zbl 0918.43003Let \(G\) be a locally compact group. There are many Banach algebras which are associated with \(G\), such as \(L^1(G)\), \(M(G)\), the bidual \(L^1(G)^{**}\) of \(L^1(G)\), and the duals of spaces of continuous bounded functions on \(G\) such as \(WAP(G)^*\) and \(LUC(G)^*\). For many of these algebras (except \(L^1(G)\)) there tends to be a rather unmanageable family of maximal ideals, and also the algebras may not be semisimple. It is therefore of interest to study their minimal ideals. The author studies the minimal ideals of finite dimension. He describes such ideals for each of the algebras listed above – they are obtained from (finite-dimensional) representations of \(G\). In particular, he identifies those groups for which such ideals exist.

Reviewer: John W.Baker (Sheffield)

##### MSC:

43A10 | Measure algebras on groups, semigroups, etc. |

22D15 | Group algebras of locally compact groups |

22A15 | Structure of topological semigroups |