Sous-groupes maximaux de groupes classiques associés à certaines C *- algèbres. (Maximal classical subgroups associated to some C * algebras) (Q1106429)

This paper studies maximal subgroups of the connected component G resp. U of the identity in the general linear group GL(A) or in the unitary group U(A) of some C *-algebras A with identity, e.g. simple C *-algebras and factors. We describe the results in some detail. For simple C *-algebras with identity the following results are established: If p,q are non-zero projections from A with sum 1 then the subgroup consisting of elements x in G with \(qxp=qx^{-1}p=0\) is a maximal subgroup of G. Also, if \(G^{\alpha}\) denotes the set of elements in G commuting with p-q then the normalizer of \(G^{\alpha}\) in G is a maximal subgroup of G. If A is a factor then if p,q are non-zero non-equivalent projections with sum 1 then the set of unitaries commuting with p-q is a maximal subgroup of U. If p and q are equivalent and K is the maximal proper ideal of A then the normalizer in U of the set of unitaries which commute with p-q modulo K is a maximal subgroup of U. If A is a factor then the normalizer in G of the set of elements in G which is a sum of a unitary and an element from K is a maximal subgroup of G.