arXiv subjects
Publications and source records attributed to E. Troitsky.
A C*-algebra $A$ is C*-reflexive if any countably generated Hilbert C*-module $M$ over $A$ is C*-reflexive, i.e. the second dual module $M''$ coincides with $M$. We show that a commutative C*-algebra $A$ is C*-reflexive if and only if for any sequence $I_k$ of disjoint non-zero C*-subalgebras, the canonical inclusion $\oplus_k I_k\subset A$ doesn't extend to an inclusion of $\prod_k I_k$.
Continuous actions of topological groups on compact Hausdorff spaces $X$ are investigated which induce almost periodic functions in the corresponding commutative C*-algebra. The unique invariant mean on the group resulting from averaging allows to derive a C*-valued inner product and a Hilbert C*-module which serve as an environment to describe characteristics of the group action. For uniformly continuous, Lyapunov stable actions the derived invariant mean $M(ϕ_x)$ is continuous on $X$ for any element $ϕ\in C(X)$, and the induced C*-valued inner product corresponds to a conditional expectation from $C(X)$ onto the fixed point algebra of the action defined by averaging on orbits. In the case of selfduality of the Hilbert C*-module all orbits are shown to have the same cardinality. Stable actions on compact metric spaces give rise to C*-reflexive Hilbert C*-modules. The same is true if the cardinality of finite orbits is uniformly bounded and the number of closures of infinite orbits is finite. A number of examples illustrate typical situations appearing beyond the classified cases.