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September  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, 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), 119-124. doi: 10.1017/S0013091500018733.  Google Scholar

[2]

B. Blackadar, "Operator Algebras. Theory of $C^*$-Algebras and von Neumann Algebras," Encyclopaedia of Mathematical Sciences, 122, Operator Algebras and Non-commutative Geometry, III, Springer-Verlag, Berlin, 2006.  Google Scholar

[3]

K. R. Davidson, "$C^*$-Algebras by Example," Fields Institute Monographs, 6, American Mathematical Society, Providence, RI, 1996.  Google Scholar

[4]

E. G. Effros and F. Hahn, "Locally Compact Transformation Groups and $C^*$-Algebras," Memoirs of the American Mathematical Society, No. 75, American Mathematical Society, Providence RI, 1967.  Google Scholar

[5]

G. A. Elliott, Some simple $C^*$-algebras constructed as crossed products with discrete outer automorphism groups, Publ. Res. Inst. Math. Sci., 16 (1980), 299-311. doi: 10.2977/prims/1195187509.  Google Scholar

[6]

R. Exel and A. Vershik, C*-algebras of irreversible dynamical systems, Canad. J. Math., 58 (2006), 39-63. doi: 10.4153/CJM-2006-003-x.  Google Scholar

[7]

S. Kawamura and J. Tomiyama, Properties of topological dynamical systems and corresponding $C^*$-algebras, Tokyo. J. Math., 13 (1990), 251-257. doi: 10.3836/tjm/1270132260.  Google Scholar

[8]

A. Kishimoto, Outer automorphisms and reduced crossed products of simple $C^*$-algebras, Comm. Math. Phys., 81 (1981), 429-435.  Google Scholar

[9]

G. Liu and F. Li, On strongly groupoid graded rings and the corresponding Clifford theorem, Algebra Colloq., 13 (2006), 181-196.  Google Scholar

[10]

P. Lundström, Separable groupoid rings, Comm. Algebra, 34 (2006), 3029-3041. doi: 10.1080/00927870600639906.  Google Scholar

[11]

P. Lundström and J. Öinert, Skew category algebras associated with partially defined dynamical systems, Internat. J. Math., 23 (2012), 1250040, 16 pp. doi: 10.1142/S0129167X12500401.  Google Scholar

[12]

T. Masuda, Groupoid dynamical systems and crossed product. I. The case of W*-systems, Publ. Res. Inst. Math. Sci., 20 (1984), 929-957. doi: 10.2977/prims/1195180873.  Google Scholar

[13]

T. Masuda, Groupoid dynamical systems and crossed product. II. The case of C*-systems, Publ. Res. Inst. Math. Sci., 20 (1984), 959-970. doi: 10.2977/prims/1195180874.  Google Scholar

[14]

J. R. Munkres, "Topology," $2^{nd}$ edition, Prentice-Hall, Inc., Englewood Cliffs, NJ, 2000.  Google Scholar

[15]

F. J. Murray and J. von Neumann, On rings of operators, Ann. of Math. (2), 37 (1936), 116-229. doi: 10.2307/1968693.  Google Scholar

[16]

F. J. Murray and J. von Neumann, On rings of operators. IV, Ann. of Math. (2), 44 (1943), 716-808.  Google Scholar

[17]

J. von Neumann, "Collected Works. Vol. III: Rings of Operators," Pergamon Press, New York-Oxford-London-Paris, 1961.  Google Scholar

[18]

J. Öinert, Simple group graded rings and maximal commutativity, in "Operator Structures and Dynamical Systems," Contemporary Mathematics, 503, American Mathematical Society, Providence, RI, (2009), 159-175. doi: 10.1090/conm/503/09899.  Google Scholar

[19]

J. Öinert and P. Lundström, Commutativity and ideals in category crossed products, Proc. Est. Acad. Sci., 59 (2010), 338-346. doi: 10.3176/proc.2010.4.13.  Google Scholar

[20]

J. Öinert and P. Lundström, The ideal intersection property for groupoid graded rings, Comm. Algebra, 40 (2012), 1860-1871. doi: 10.1080/00927872.2011.559181.  Google Scholar

[21]

J. Öinert and P. Lundström, Miyashita action in strongly groupoid graded rings, Int. Electron. J. Algebra, 11 (2012), 46-63.  Google Scholar

[22]

J. Öinert, J. Richter and S. D. Silvestrov, Maximal commutative subalgebras and simplicity of Ore extensions, J. Algebra Appl., 12 (2013), 1250192, 16 pp. doi: 10.1142/S0219498812501927.  Google Scholar

[23]

J. Öinert and S. D. Silvestrov, Commutativity and ideals in algebraic crossed products, J. Gen. Lie T. Appl., 2 (2008), 287-302. doi: 10.4303/jglta/S070404.  Google Scholar

[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), 216-220.  Google Scholar

[25]

J. Öinert and S. D. Silvestrov, Crossed product-like and pre-crystalline graded rings, in "Generalized Lie Theory in Mathematics, Physics and Beyond" (eds. S. Silvestrov, E. Paal, V. Abramov and A. Stolin), Springer, Berlin, (2009), 281-296. doi: 10.1007/978-3-540-85332-9_24.  Google Scholar

[26]

J. Öinert, S. Silvestrov, T. Theohari-Apostolidi and H. Vavatsoulas, Commutativity and ideals in strongly graded rings, Acta Appl. Math., 108 (2009), 585-602. doi: 10.1007/s10440-009-9435-3.  Google Scholar

[27]

J. Öinert, Simplicity of skew group rings of abelian groups,, to appear in Communications in Algebra, ().   Google Scholar

[28]

A. L. T. Paterson, "Groupoids, Inverse Semigroups, and their Operator Algebras," Progress in Mathematics, 170, Birkhäuser Boston, Inc., Boston, MA, 1999.  Google Scholar

[29]

G. K. Pedersen, "$C^*$-algebras and their Automorphism Groups," London Mathematical Society Monographs, 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979.  Google Scholar

[30]

S. C. Power, Simplicity of $C^*$-algebras of minimal dynamical systems, J. London Math. Soc. (2), 18 (1978), 534-538. doi: 10.1112/jlms/s2-18.3.534.  Google Scholar

[31]

J. C. Quigg and J. S. Spielberg, Regularity and hyporegularity in $C^*$-dynamical system, Houston J. Math., 18 (1992), 139-152.  Google Scholar

[32]

J. S. Spielberg, Free-product groups, Cuntz-Krieger algebras, and covariant maps, Internat. J. Math., 2 (1991), 457-476. doi: 10.1142/S0129167X91000260.  Google Scholar

[33]

C. Svensson, S. Silvestrov and M. de Jeu, Dynamical systems and commutants in crossed products, Internat. J. Math., 18 (2007), 455-471. doi: 10.1142/S0129167X07004217.  Google Scholar

[34]

C. Svensson, S. Silvestrov and M. de Jeu, Connections between dynamical systems and crossed products of Banach algebras by $\mathbbZ$, in "Methods of Spectral Analysis in Mathematical Physics," Operator Theory: Advances and Applications, 186, Birkhäuser Verlag, (2009), 391-401. doi: 10.1007/978-3-7643-8755-6_19.  Google Scholar

[35]

C. Svensson, S. Silvestrov and M. de Jeu, Dynamical systems associated with crossed products, Acta Appl. Math., 108 (2009), 547-559. doi: 10.1007/s10440-009-9506-5.  Google Scholar

[36]

M. Takesaki, "Theory of Operator Algebras. II," Encyclopaedia of Mathematical Sciences, 125, Operator Algebras and Non-Commutative Geometry, 6, Springer-Verlag, Berlin, 2003.  Google Scholar

[37]

J. Tomiyama, "Invitation to $C^*$-Algebras and Topological Dynamics," World Scientific Advanced Series in Dynamical Systems, 3, World Scientific Publishing Co., Singapore, 1987.  Google Scholar

[38]

J. Tomiyama, "The Interplay Between Topological Dynamics and Theory of $C^*$-Algebras," Lecture Notes Series, 2, Seoul National University, Research Institute of Mathematics, Global Anal. Research Center, Seoul, 1992.  Google Scholar

[39]

D. P. Williams, "Crossed Products of $C^*$-Algebras," Mathematical Surveys and Monographs, 134, American Mathematical Society, Providence, RI, 2007.  Google Scholar

[40]

G. Zeller-Meier, Produits croisés d'une $C^*$-algèbre par un groupe d'automorphismes, J. Math. Pures Appl. (9), 47 (1968), 101-239.  Google Scholar

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), 119-124. doi: 10.1017/S0013091500018733.  Google Scholar

[2]

B. Blackadar, "Operator Algebras. Theory of $C^*$-Algebras and von Neumann Algebras," Encyclopaedia of Mathematical Sciences, 122, Operator Algebras and Non-commutative Geometry, III, Springer-Verlag, Berlin, 2006.  Google Scholar

[3]

K. R. Davidson, "$C^*$-Algebras by Example," Fields Institute Monographs, 6, American Mathematical Society, Providence, RI, 1996.  Google Scholar

[4]

E. G. Effros and F. Hahn, "Locally Compact Transformation Groups and $C^*$-Algebras," Memoirs of the American Mathematical Society, No. 75, American Mathematical Society, Providence RI, 1967.  Google Scholar

[5]

G. A. Elliott, Some simple $C^*$-algebras constructed as crossed products with discrete outer automorphism groups, Publ. Res. Inst. Math. Sci., 16 (1980), 299-311. doi: 10.2977/prims/1195187509.  Google Scholar

[6]

R. Exel and A. Vershik, C*-algebras of irreversible dynamical systems, Canad. J. Math., 58 (2006), 39-63. doi: 10.4153/CJM-2006-003-x.  Google Scholar

[7]

S. Kawamura and J. Tomiyama, Properties of topological dynamical systems and corresponding $C^*$-algebras, Tokyo. J. Math., 13 (1990), 251-257. doi: 10.3836/tjm/1270132260.  Google Scholar

[8]

A. Kishimoto, Outer automorphisms and reduced crossed products of simple $C^*$-algebras, Comm. Math. Phys., 81 (1981), 429-435.  Google Scholar

[9]

G. Liu and F. Li, On strongly groupoid graded rings and the corresponding Clifford theorem, Algebra Colloq., 13 (2006), 181-196.  Google Scholar

[10]

P. Lundström, Separable groupoid rings, Comm. Algebra, 34 (2006), 3029-3041. doi: 10.1080/00927870600639906.  Google Scholar

[11]

P. Lundström and J. Öinert, Skew category algebras associated with partially defined dynamical systems, Internat. J. Math., 23 (2012), 1250040, 16 pp. doi: 10.1142/S0129167X12500401.  Google Scholar

[12]

T. Masuda, Groupoid dynamical systems and crossed product. I. The case of W*-systems, Publ. Res. Inst. Math. Sci., 20 (1984), 929-957. doi: 10.2977/prims/1195180873.  Google Scholar

[13]

T. Masuda, Groupoid dynamical systems and crossed product. II. The case of C*-systems, Publ. Res. Inst. Math. Sci., 20 (1984), 959-970. doi: 10.2977/prims/1195180874.  Google Scholar

[14]

J. R. Munkres, "Topology," $2^{nd}$ edition, Prentice-Hall, Inc., Englewood Cliffs, NJ, 2000.  Google Scholar

[15]

F. J. Murray and J. von Neumann, On rings of operators, Ann. of Math. (2), 37 (1936), 116-229. doi: 10.2307/1968693.  Google Scholar

[16]

F. J. Murray and J. von Neumann, On rings of operators. IV, Ann. of Math. (2), 44 (1943), 716-808.  Google Scholar

[17]

J. von Neumann, "Collected Works. Vol. III: Rings of Operators," Pergamon Press, New York-Oxford-London-Paris, 1961.  Google Scholar

[18]

J. Öinert, Simple group graded rings and maximal commutativity, in "Operator Structures and Dynamical Systems," Contemporary Mathematics, 503, American Mathematical Society, Providence, RI, (2009), 159-175. doi: 10.1090/conm/503/09899.  Google Scholar

[19]

J. Öinert and P. Lundström, Commutativity and ideals in category crossed products, Proc. Est. Acad. Sci., 59 (2010), 338-346. doi: 10.3176/proc.2010.4.13.  Google Scholar

[20]

J. Öinert and P. Lundström, The ideal intersection property for groupoid graded rings, Comm. Algebra, 40 (2012), 1860-1871. doi: 10.1080/00927872.2011.559181.  Google Scholar

[21]

J. Öinert and P. Lundström, Miyashita action in strongly groupoid graded rings, Int. Electron. J. Algebra, 11 (2012), 46-63.  Google Scholar

[22]

J. Öinert, J. Richter and S. D. Silvestrov, Maximal commutative subalgebras and simplicity of Ore extensions, J. Algebra Appl., 12 (2013), 1250192, 16 pp. doi: 10.1142/S0219498812501927.  Google Scholar

[23]

J. Öinert and S. D. Silvestrov, Commutativity and ideals in algebraic crossed products, J. Gen. Lie T. Appl., 2 (2008), 287-302. doi: 10.4303/jglta/S070404.  Google Scholar

[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), 216-220.  Google Scholar

[25]

J. Öinert and S. D. Silvestrov, Crossed product-like and pre-crystalline graded rings, in "Generalized Lie Theory in Mathematics, Physics and Beyond" (eds. S. Silvestrov, E. Paal, V. Abramov and A. Stolin), Springer, Berlin, (2009), 281-296. doi: 10.1007/978-3-540-85332-9_24.  Google Scholar

[26]

J. Öinert, S. Silvestrov, T. Theohari-Apostolidi and H. Vavatsoulas, Commutativity and ideals in strongly graded rings, Acta Appl. Math., 108 (2009), 585-602. doi: 10.1007/s10440-009-9435-3.  Google Scholar

[27]

J. Öinert, Simplicity of skew group rings of abelian groups,, to appear in Communications in Algebra, ().   Google Scholar

[28]

A. L. T. Paterson, "Groupoids, Inverse Semigroups, and their Operator Algebras," Progress in Mathematics, 170, Birkhäuser Boston, Inc., Boston, MA, 1999.  Google Scholar

[29]

G. K. Pedersen, "$C^*$-algebras and their Automorphism Groups," London Mathematical Society Monographs, 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979.  Google Scholar

[30]

S. C. Power, Simplicity of $C^*$-algebras of minimal dynamical systems, J. London Math. Soc. (2), 18 (1978), 534-538. doi: 10.1112/jlms/s2-18.3.534.  Google Scholar

[31]

J. C. Quigg and J. S. Spielberg, Regularity and hyporegularity in $C^*$-dynamical system, Houston J. Math., 18 (1992), 139-152.  Google Scholar

[32]

J. S. Spielberg, Free-product groups, Cuntz-Krieger algebras, and covariant maps, Internat. J. Math., 2 (1991), 457-476. doi: 10.1142/S0129167X91000260.  Google Scholar

[33]

C. Svensson, S. Silvestrov and M. de Jeu, Dynamical systems and commutants in crossed products, Internat. J. Math., 18 (2007), 455-471. doi: 10.1142/S0129167X07004217.  Google Scholar

[34]

C. Svensson, S. Silvestrov and M. de Jeu, Connections between dynamical systems and crossed products of Banach algebras by $\mathbbZ$, in "Methods of Spectral Analysis in Mathematical Physics," Operator Theory: Advances and Applications, 186, Birkhäuser Verlag, (2009), 391-401. doi: 10.1007/978-3-7643-8755-6_19.  Google Scholar

[35]

C. Svensson, S. Silvestrov and M. de Jeu, Dynamical systems associated with crossed products, Acta Appl. Math., 108 (2009), 547-559. doi: 10.1007/s10440-009-9506-5.  Google Scholar

[36]

M. Takesaki, "Theory of Operator Algebras. II," Encyclopaedia of Mathematical Sciences, 125, Operator Algebras and Non-Commutative Geometry, 6, Springer-Verlag, Berlin, 2003.  Google Scholar

[37]

J. Tomiyama, "Invitation to $C^*$-Algebras and Topological Dynamics," World Scientific Advanced Series in Dynamical Systems, 3, World Scientific Publishing Co., Singapore, 1987.  Google Scholar

[38]

J. Tomiyama, "The Interplay Between Topological Dynamics and Theory of $C^*$-Algebras," Lecture Notes Series, 2, Seoul National University, Research Institute of Mathematics, Global Anal. Research Center, Seoul, 1992.  Google Scholar

[39]

D. P. Williams, "Crossed Products of $C^*$-Algebras," Mathematical Surveys and Monographs, 134, American Mathematical Society, Providence, RI, 2007.  Google Scholar

[40]

G. Zeller-Meier, Produits croisés d'une $C^*$-algèbre par un groupe d'automorphismes, J. Math. Pures Appl. (9), 47 (1968), 101-239.  Google Scholar

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