Rokhlin actions and self-absorbing C*-algebras
Let A be a unital separable C∗-algebra, and D a K1-injective strongly selfabsorbing C∗-algebra. We show that if A is D-absorbing, then the crossed product of A by a compact second countable group or by Z or by R is D-absorbing as well, assuming the action satisfying a Rokhlin property. In the case of a compact Rokhlin action we prove a similar statement about approximate divisibility.
Finite group actions on C∗-algebras with the Rohlin property—II
Abstract We classify finite group actions on some classes of C ∗ -algebras with the Rohlin property in terms of the K-groups. A group cohomological obstruction is obtained for the K-groups of a C ∗ -algebra allowing such an action, for example, the Tate cohomology groups with the K-groups as the coefficient modules must vanish. Model actions with the Rohlin property are constructed for given K-theoretical invariants. Quasi-free actions of finite cyclic groups on the Cuntz algebras and their dual actions are investigated from the view point of the Rohlin property.
On Hilbert's thirteenth problem and related questions
Hilbert's thirteenth problem involves the study of solutions of algebraic equations. The object is to obtain a complexity estimate for an algebraic function. As of now, the problem remains open. There are only a few partial algebraic results in this connection, but at the same time the problem has stimulated a series of studies in the theory of functions with their subsequent applications. The most brilliant result in this cycle is Kolmogorov's theorem on superpositions of continuous functions.