Dynamical systems and extensions of states on $C^{\ast }$-algebras

Abstract:

Let $(A,G,\tau )$ be a noncommutative dynamical system, i.e. $A$ is a ${C^{\ast } }$-algebra, $G$ a topological group and $\tau$ an action of $G$ on $A$ by $^{\ast }$-automorphisms, and let $({M_\alpha })$ be an $M$-net on $G$. We characterize the set of $a$ in $A$ such that ${M_\alpha }a$ converges in norm. We show that this set is intimately related to the problem of extensions of pure states of R. V. Kadison and I. M. Singer: if $B$ is a maximal abelian subalgebra of $A$, we can associate a dynamical system $(A,G,\tau )$ such that ${M_\alpha }a$ converges in norm if and only if all extensions to $A$, of a homomorphism of $B$, coincide on $a$. This result allows us to construct different examples of a ${C^{\ast } }$-algebra $A$ with maximal abelian subalgebra $B$ (isomorphic to $C({\mathbf {R}}/{\mathbf {Z}})$ or ${L^\infty }[0,1])$ for which the property of unique pure state extension of homomorphisms is or is not verified.