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2013, 33(9): 4157-4171. doi: 10.3934/dcds.2013.33.4157

## Simple skew category algebras associated with minimal partially defined dynamical systems

 1 University West, Department of Engineering Science, SE-46186 Trollhättan, Sweden 2 Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark

Received  June 2012 Revised  February 2013 Published  March 2013

In this article, we continue our study of category dynamical systems, that is functors $s$ from a category $G$ to $Top^{op}$, and their corresponding skew category algebras. Suppose that the spaces $s(e)$, for $e ∈ ob(G)$, are compact Hausdorff. We show that if (i) the skew category algebra is simple, then (ii) $G$ is inverse connected, (iii) $s$ is minimal and (iv) $s$ is faithful. We also show that if $G$ is a locally abelian groupoid, then (i) is equivalent to (ii), (iii) and (iv). Thereby, we generalize results by Öinert for skew group algebras to a large class of skew category algebras.
Citation: Patrik Nystedt, Johan Öinert. Simple skew category algebras associated with minimal partially defined dynamical systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4157-4171. doi: 10.3934/dcds.2013.33.4157
##### References:
 [1] R. J. Archbold and J. S. Spielberg, Topologically free actions and ideals in discrete $C^*$-dynamical systems,, Proc. of Edinburgh Math. Soc. (2), 37 (1993), 119. doi: 10.1017/S0013091500018733. [2] B. Blackadar, "Operator Algebras. Theory of $C^*$-Algebras and von Neumann Algebras,", Encyclopaedia of Mathematical Sciences, 122 (2006). [3] K. R. Davidson, "$C^*$-Algebras by Example,", Fields Institute Monographs, 6 (1996). [4] E. G. Effros and F. Hahn, "Locally Compact Transformation Groups and $C^*$-Algebras,", Memoirs of the American Mathematical Society, (1967). [5] G. A. Elliott, Some simple $C^*$-algebras constructed as crossed products with discrete outer automorphism groups,, Publ. Res. Inst. Math. Sci., 16 (1980), 299. doi: 10.2977/prims/1195187509. [6] R. Exel and A. Vershik, C*-algebras of irreversible dynamical systems,, Canad. J. Math., 58 (2006), 39. doi: 10.4153/CJM-2006-003-x. [7] S. Kawamura and J. Tomiyama, Properties of topological dynamical systems and corresponding $C^*$-algebras,, Tokyo. J. Math., 13 (1990), 251. doi: 10.3836/tjm/1270132260. [8] A. Kishimoto, Outer automorphisms and reduced crossed products of simple $C^*$-algebras,, Comm. Math. Phys., 81 (1981), 429. [9] G. Liu and F. Li, On strongly groupoid graded rings and the corresponding Clifford theorem,, Algebra Colloq., 13 (2006), 181. [10] P. Lundström, Separable groupoid rings,, Comm. Algebra, 34 (2006), 3029. doi: 10.1080/00927870600639906. [11] P. Lundström and J. Öinert, Skew category algebras associated with partially defined dynamical systems,, Internat. J. Math., 23 (2012). doi: 10.1142/S0129167X12500401. [12] T. Masuda, Groupoid dynamical systems and crossed product. I. The case of W*-systems,, Publ. Res. Inst. Math. Sci., 20 (1984), 929. doi: 10.2977/prims/1195180873. [13] T. Masuda, Groupoid dynamical systems and crossed product. II. The case of C*-systems,, Publ. Res. Inst. Math. Sci., 20 (1984), 959. doi: 10.2977/prims/1195180874. [14] J. R. Munkres, "Topology,", $2^{nd}$ edition, (2000). [15] F. J. Murray and J. von Neumann, On rings of operators,, Ann. of Math. (2), 37 (1936), 116. doi: 10.2307/1968693. [16] F. J. Murray and J. von Neumann, On rings of operators. IV,, Ann. of Math. (2), 44 (1943), 716. [17] J. von Neumann, "Collected Works. Vol. III: Rings of Operators,", Pergamon Press, (1961). [18] J. Öinert, Simple group graded rings and maximal commutativity,, in, 503 (2009), 159. doi: 10.1090/conm/503/09899. [19] J. Öinert and P. Lundström, Commutativity and ideals in category crossed products,, Proc. Est. Acad. Sci., 59 (2010), 338. doi: 10.3176/proc.2010.4.13. [20] J. Öinert and P. Lundström, The ideal intersection property for groupoid graded rings,, Comm. Algebra, 40 (2012), 1860. doi: 10.1080/00927872.2011.559181. [21] J. Öinert and P. Lundström, Miyashita action in strongly groupoid graded rings,, Int. Electron. J. Algebra, 11 (2012), 46. [22] J. Öinert, J. Richter and S. D. Silvestrov, Maximal commutative subalgebras and simplicity of Ore extensions,, J. Algebra Appl., 12 (2013). doi: 10.1142/S0219498812501927. [23] J. Öinert and S. D. Silvestrov, Commutativity and ideals in algebraic crossed products,, J. Gen. Lie T. Appl., 2 (2008), 287. doi: 10.4303/jglta/S070404. [24] J. Öinert and S. D. Silvestrov, On a correspondence between ideals and commutativity in algebraic crossed products,, J. Gen. Lie T. Appl., 2 (2008), 216. [25] J. Öinert and S. D. Silvestrov, Crossed product-like and pre-crystalline graded rings,, in, (2009), 281. doi: 10.1007/978-3-540-85332-9_24. [26] J. Öinert, S. Silvestrov, T. Theohari-Apostolidi and H. Vavatsoulas, Commutativity and ideals in strongly graded rings,, Acta Appl. Math., 108 (2009), 585. doi: 10.1007/s10440-009-9435-3. [27] J. Öinert, Simplicity of skew group rings of abelian groups,, to appear in Communications in Algebra, (). [28] A. L. T. Paterson, "Groupoids, Inverse Semigroups, and their Operator Algebras,", Progress in Mathematics, 170 (1999). [29] G. K. Pedersen, "$C^*$-algebras and their Automorphism Groups,", London Mathematical Society Monographs, 14 (1979). [30] S. C. Power, Simplicity of $C^*$-algebras of minimal dynamical systems,, J. London Math. Soc. (2), 18 (1978), 534. doi: 10.1112/jlms/s2-18.3.534. [31] J. C. Quigg and J. S. Spielberg, Regularity and hyporegularity in $C^*$-dynamical system,, Houston J. Math., 18 (1992), 139. [32] J. S. Spielberg, Free-product groups, Cuntz-Krieger algebras, and covariant maps,, Internat. J. Math., 2 (1991), 457. doi: 10.1142/S0129167X91000260. [33] C. Svensson, S. Silvestrov and M. de Jeu, Dynamical systems and commutants in crossed products,, Internat. J. Math., 18 (2007), 455. doi: 10.1142/S0129167X07004217. [34] C. Svensson, S. Silvestrov and M. de Jeu, Connections between dynamical systems and crossed products of Banach algebras by $\mathbbZ$,, in, 186 (2009), 391. doi: 10.1007/978-3-7643-8755-6_19. [35] C. Svensson, S. Silvestrov and M. de Jeu, Dynamical systems associated with crossed products,, Acta Appl. Math., 108 (2009), 547. doi: 10.1007/s10440-009-9506-5. [36] M. Takesaki, "Theory of Operator Algebras. II,", Encyclopaedia of Mathematical Sciences, 125 (2003). [37] J. Tomiyama, "Invitation to $C^*$-Algebras and Topological Dynamics,", World Scientific Advanced Series in Dynamical Systems, 3 (1987). [38] J. Tomiyama, "The Interplay Between Topological Dynamics and Theory of $C^*$-Algebras,", Lecture Notes Series, 2 (1992). [39] D. P. Williams, "Crossed Products of $C^*$-Algebras,", Mathematical Surveys and Monographs, 134 (2007). [40] G. Zeller-Meier, Produits croisés d'une $C^*$-algèbre par un groupe d'automorphismes,, J. Math. Pures Appl. (9), 47 (1968), 101.

show all references

##### References:
 [1] R. J. Archbold and J. S. Spielberg, Topologically free actions and ideals in discrete $C^*$-dynamical systems,, Proc. of Edinburgh Math. Soc. (2), 37 (1993), 119. doi: 10.1017/S0013091500018733. [2] B. Blackadar, "Operator Algebras. Theory of $C^*$-Algebras and von Neumann Algebras,", Encyclopaedia of Mathematical Sciences, 122 (2006). [3] K. R. Davidson, "$C^*$-Algebras by Example,", Fields Institute Monographs, 6 (1996). [4] E. G. Effros and F. Hahn, "Locally Compact Transformation Groups and $C^*$-Algebras,", Memoirs of the American Mathematical Society, (1967). [5] G. A. Elliott, Some simple $C^*$-algebras constructed as crossed products with discrete outer automorphism groups,, Publ. Res. Inst. Math. Sci., 16 (1980), 299. doi: 10.2977/prims/1195187509. [6] R. Exel and A. Vershik, C*-algebras of irreversible dynamical systems,, Canad. J. Math., 58 (2006), 39. doi: 10.4153/CJM-2006-003-x. [7] S. Kawamura and J. Tomiyama, Properties of topological dynamical systems and corresponding $C^*$-algebras,, Tokyo. J. Math., 13 (1990), 251. doi: 10.3836/tjm/1270132260. [8] A. Kishimoto, Outer automorphisms and reduced crossed products of simple $C^*$-algebras,, Comm. Math. Phys., 81 (1981), 429. [9] G. Liu and F. Li, On strongly groupoid graded rings and the corresponding Clifford theorem,, Algebra Colloq., 13 (2006), 181. [10] P. Lundström, Separable groupoid rings,, Comm. Algebra, 34 (2006), 3029. doi: 10.1080/00927870600639906. [11] P. Lundström and J. Öinert, Skew category algebras associated with partially defined dynamical systems,, Internat. J. Math., 23 (2012). doi: 10.1142/S0129167X12500401. [12] T. Masuda, Groupoid dynamical systems and crossed product. I. The case of W*-systems,, Publ. Res. Inst. Math. Sci., 20 (1984), 929. doi: 10.2977/prims/1195180873. [13] T. Masuda, Groupoid dynamical systems and crossed product. II. The case of C*-systems,, Publ. Res. Inst. Math. Sci., 20 (1984), 959. doi: 10.2977/prims/1195180874. [14] J. R. Munkres, "Topology,", $2^{nd}$ edition, (2000). [15] F. J. Murray and J. von Neumann, On rings of operators,, Ann. of Math. (2), 37 (1936), 116. doi: 10.2307/1968693. [16] F. J. Murray and J. von Neumann, On rings of operators. IV,, Ann. of Math. (2), 44 (1943), 716. [17] J. von Neumann, "Collected Works. Vol. III: Rings of Operators,", Pergamon Press, (1961). [18] J. Öinert, Simple group graded rings and maximal commutativity,, in, 503 (2009), 159. doi: 10.1090/conm/503/09899. [19] J. Öinert and P. Lundström, Commutativity and ideals in category crossed products,, Proc. Est. Acad. Sci., 59 (2010), 338. doi: 10.3176/proc.2010.4.13. [20] J. Öinert and P. Lundström, The ideal intersection property for groupoid graded rings,, Comm. Algebra, 40 (2012), 1860. doi: 10.1080/00927872.2011.559181. [21] J. Öinert and P. Lundström, Miyashita action in strongly groupoid graded rings,, Int. Electron. J. Algebra, 11 (2012), 46. [22] J. Öinert, J. Richter and S. D. Silvestrov, Maximal commutative subalgebras and simplicity of Ore extensions,, J. Algebra Appl., 12 (2013). doi: 10.1142/S0219498812501927. [23] J. Öinert and S. D. Silvestrov, Commutativity and ideals in algebraic crossed products,, J. Gen. Lie T. Appl., 2 (2008), 287. doi: 10.4303/jglta/S070404. [24] J. Öinert and S. D. Silvestrov, On a correspondence between ideals and commutativity in algebraic crossed products,, J. Gen. Lie T. Appl., 2 (2008), 216. [25] J. Öinert and S. D. Silvestrov, Crossed product-like and pre-crystalline graded rings,, in, (2009), 281. doi: 10.1007/978-3-540-85332-9_24. [26] J. Öinert, S. Silvestrov, T. Theohari-Apostolidi and H. Vavatsoulas, Commutativity and ideals in strongly graded rings,, Acta Appl. Math., 108 (2009), 585. doi: 10.1007/s10440-009-9435-3. [27] J. Öinert, Simplicity of skew group rings of abelian groups,, to appear in Communications in Algebra, (). [28] A. L. T. Paterson, "Groupoids, Inverse Semigroups, and their Operator Algebras,", Progress in Mathematics, 170 (1999). [29] G. K. Pedersen, "$C^*$-algebras and their Automorphism Groups,", London Mathematical Society Monographs, 14 (1979). [30] S. C. Power, Simplicity of $C^*$-algebras of minimal dynamical systems,, J. London Math. Soc. (2), 18 (1978), 534. doi: 10.1112/jlms/s2-18.3.534. [31] J. C. Quigg and J. S. Spielberg, Regularity and hyporegularity in $C^*$-dynamical system,, Houston J. Math., 18 (1992), 139. [32] J. S. Spielberg, Free-product groups, Cuntz-Krieger algebras, and covariant maps,, Internat. J. Math., 2 (1991), 457. doi: 10.1142/S0129167X91000260. [33] C. Svensson, S. Silvestrov and M. de Jeu, Dynamical systems and commutants in crossed products,, Internat. J. Math., 18 (2007), 455. doi: 10.1142/S0129167X07004217. [34] C. Svensson, S. Silvestrov and M. de Jeu, Connections between dynamical systems and crossed products of Banach algebras by $\mathbbZ$,, in, 186 (2009), 391. doi: 10.1007/978-3-7643-8755-6_19. [35] C. Svensson, S. Silvestrov and M. de Jeu, Dynamical systems associated with crossed products,, Acta Appl. Math., 108 (2009), 547. doi: 10.1007/s10440-009-9506-5. [36] M. Takesaki, "Theory of Operator Algebras. II,", Encyclopaedia of Mathematical Sciences, 125 (2003). [37] J. Tomiyama, "Invitation to $C^*$-Algebras and Topological Dynamics,", World Scientific Advanced Series in Dynamical Systems, 3 (1987). [38] J. Tomiyama, "The Interplay Between Topological Dynamics and Theory of $C^*$-Algebras,", Lecture Notes Series, 2 (1992). [39] D. P. Williams, "Crossed Products of $C^*$-Algebras,", Mathematical Surveys and Monographs, 134 (2007). [40] G. Zeller-Meier, Produits croisés d'une $C^*$-algèbre par un groupe d'automorphismes,, J. Math. Pures Appl. (9), 47 (1968), 101.
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