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#### Dynamical SystemsFunctional Analysismathscidoc:1908.12003

J. Operator Theory, 78, (1), 159-172, 2017
Motivated by reformulating Furstenberg's $\times p,\times q$ conjecture via representations of a crossed product $C^*$-algebra, we show that in a discrete $C^*$-dynamical system $(A,\Gamma)$, the space of (ergodic) $\Gamma$-invariant states on $A$ is homeomorphic to a subspace of (pure) state space of $A\rtimes\Gamma$. Various applications of this in topological dynamical systems and representation theory are obtained. In particular, we prove that the classification of ergodic $\Gamma$-invariant regular Borel probability measures on a compact Hausdorff space $X$ is equivalent to the classification a special type of irreducible representations of $C(X)\rtimes \Gamma$.
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@inproceedings{huichi2017ergodic,
title={Ergodic invariant states and irreducible representations of crossed product $C^∗$-algebras.},
author={Huichi Huang, and Jianchao Wu},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190822114503812551446},
booktitle={J. Operator Theory},
volume={78},
number={1},
pages={159-172},
year={2017},
}

Huichi Huang, and Jianchao Wu. Ergodic invariant states and irreducible representations of crossed product $C^∗$-algebras.. 2017. Vol. 78. In J. Operator Theory. pp.159-172. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190822114503812551446.
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