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

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- by Nghiem Dang-Ngoc PDF
- Trans. Amer. Math. Soc.
**275**(1983), 143-152 Request permission

## 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.## References

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## Additional Information

- © Copyright 1983 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**275**(1983), 143-152 - MSC: Primary 46L55; Secondary 47A35
- DOI: https://doi.org/10.1090/S0002-9947-1983-0678340-5
- MathSciNet review: 678340