Foliations formed by orbits of maximal dimension in the co-adjoint representation of a class of solvable Lie groups
The method of studying group \(C^*\)-algebras in finding the topological invariants in \(KK\)-theory was initialized by the reviewer in 1975 for the example of the group of affine transformations of the straight line. The method was then largely extended and developed. It should be very interesting to find more classes of locally compact groups, the \(C^*\)- algebras of which should be studied by the same idea. Under the supervision of the reviewer, \textit{V. M. Son} and \textit{H. H. Viet} [J. Oper. Theory 11, 77-90 (1984; Zbl 0537.46058)] completely described the structure of group \(C^*\)-algebras of the class \(\overline {MD}\) of solvable Lie groups, the coadjoint orbits of which are open sets or points. Under the supervision of the reviewer, in the paper under review the author classifies all the 4-dimensional solvable Lie groups in the class \(MD4\), the coadjoint orbits of which are points of maximal dimension (Theorem 1). Taking together all the generic orbits of maximal dimension (Theorem 2), the author classifies all the available topological types of the foliations (Theorems 3, 4) and finds the \(KK\)- theory invariants of the arized Connes type measurable foliations (Theorems 5, 6). The main case of the real diamond group was previously published by the author [J. Oper. Theory 24, 227-238 (1990; Zbl 0785.46054)].
scientific article; zbMATH DE number 797919