Conference abstracts

Session S05 - Rings and Algebras

July 29, 18:00 ~ 18:25

Partial actions and subshifts

Mikhailo Dokuchaev

Universidade de São Paulo, Brazil   -

An arbitrary (one-sided) subshift $X$ over a finite alphabet $\Lambda $ with $n$ letters can be naturally endowed with a partial action $\theta $ of the free group ${\mathbb F}_n$ with $n$ free generators $g_{\lambda}, (\lambda \in \Lambda ),$ such that $g_{\lambda }$ maps $x$ to $\lambda x,$ where $x$ is an element in $ X$ for which $\lambda x \in X.$ Naturally $g^{-1}_{\lambda }$ removes $\lambda$ from $\lambda x.$ We call $\theta $ the standard partial action, and it is a starting point to constract a $C^*$-algebra ${\mathcal O}^*_X$ associated with $X$, as well as an abstract algbera ${\mathcal O}^K_X$ over an arbitrary field $K$ of characteristic $0.$ Both ${\mathcal O}^K_X$ and ${\mathcal O}^*_X$ are defined in a fairly similar way: using the standard partial action we construct a partial representation $u $ of ${\mathbb F}_n $ into an appropriate algebra (which depends on whether the case is abstract or $C^*$) and then define ${\mathcal O}^K_X$ (or ${\mathcal O}^*_X$) as the subalgebra (respectively, a $C^*$-subalgebra) generated by $u({\mathbb F}_n ).$ Then using a general procedure (see [4,Proposition 10.1] we obtain a partial action $\tau $ of $ {\mathbb F}_n $ on a commutative subalgebra ${\mathcal A}$ and prove that ${\mathcal O}^K_X $ is isomorphic to the crossed product ${\mathcal A}$ $\rtimes _{\tau} {\mathbb F}_n.$ In the $C^*$ case (see [3,Theorem 9.5]), due to an amenability property, ${\mathcal O}^*_X $ is isomorphic to both the full and the reduced crossed product: $ {\mathcal O}^*_X \cong {\mathcal D} \rtimes _{\tau} {\mathbb F}_n \cong {\mathcal D} \rtimes^{\rm red} _{\tau} \, {\mathbb F}_n,$ where $\mathcal D$ is a commutative $C^*$-algebra defined in a similar way as $\mathcal A$. This gives a possibility to study algebras related to subshifts using crossed products by partial actions. It turns out that $ {\mathcal O}^*_X$ is isomorphic to the $C^*$-algebra defined by T. M. Carlsen in [1] in a somewhat different way (see [3,Theorem 10.2]). In particular, if $X$ is a Markov subshift, then $ {\mathcal O}^*_X$ is isomorphic to the Cuntz-Krieger algebra defined in [2]. The $C^*$ version is elaborated in the preprint [3], in which, amongst several related results, a criterion is given for simplicity of $ {\mathcal O}^*_X$ (see [3,Theorem 14.5]).

[1] T. M. Carlsen, Cuntz-Pimsner, $C^*$-algebras associated with subshifts, Internat. J. Math., 19 (2008), 47–70.

[2] J. Cuntz, W. Krieger, A class of $C^*$-algebras and topological Markov chains, Invent. Math., 63 (1981), 25–40.

[3] M. Dokuchaev, R. Exel, Partial actions and subshifts, Preprint, arXiv:1511.00939v1 (2015).

[4] R. Exel, Partial Dynamical Systems, Fell Bundles and Application, to be published in a forthcoming NYJM book series. Available from

Joint work with Ruy Exel (Universidade Federal de Santa Catarina, Brazil).

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