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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Author: Nghiem Dang-Ngoc
Journal: Trans. Amer. Math. Soc. 275 (1983), 143-152
MSC: Primary 46L55; Secondary 47A35
MathSciNet review: 678340
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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.

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Article copyright: © Copyright 1983 American Mathematical Society

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