Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

Tychonoff's theorem for locally compact spaces and an elementary approach to the topology of path spaces

Author(s): Alan L. T. Paterson; Amy E. Welch
Journal: Proc. Amer. Math. Soc. 133 (2005), 2761-2770.
MSC (2000): Primary 54B10, 46L05; Secondary 22A22, 46L85, 54B15
Posted: April 20, 2005
MathSciNet review: 2146225
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Abstract: The path spaces of a directed graph play an important role in the study of graph -algebras. These are topological spaces that were originally constructed using groupoid and inverse semigroup techniques. In this paper, we develop a simple, purely topological, approach to this construction, based on Tychonoff's theorem. In fact, the approach is shown to work even for higher dimensional graphs satisfying the finitely aligned condition, and we construct the groupoid of the graph. Motivated by these path space results, we prove a Tychonoff theorem for an infinite, countable product of locally compact spaces. The main idea is to include certain finite products of the spaces along with the infinite product. We show that the topology is, in a reasonable sense, a pointwise topology.

References:

1.: T. Bates, D. Pask, I. Raeburn and W. Szymanski, The -algebras of row-finite graphs, New York J. Math. 6(2000), 307-324. MR 1777234 (2001k:46084)
2.: N. Bourbaki, Elements of Mathematics, General Topology, Chapters 1-4, Springer-Verlag, New York, 1989. MR 0979294 (90a:54001a)
3.: N. Bourbaki, Elements of Mathematics, General Topology, Chapters 5-10, Springer-Verlag, New York, 1989. MR 0979295 (90a:54001b)
4.: J. Cuntz, Simple -algebras generated by isometries, Comm. Math. Phys. 57(1977), 173-185. MR 0467330 (57:7189)
5.: J. Cuntz and W. Krieger, A class of -algebras and topological Markov chains, Invent. Math. 56(1980), 251-268. MR 0561974 (82f:46073a)
6.: R. Exel and M. Laca, Cuntz-Krieger algebras for infinite matrices, J. Reine Angew. Math. 512 (1999), 119-172. MR 1703078 (2000i:46064)
7.: A. an Huef and I. Raeburn, The ideal structure of Cuntz-Krieger algebras, Ergodic Theory and Dynamical Systems 17(1997), 611-624. MR 1452183 (98k:46098)
8.: O. Kallenberg, Foundations of Modern Probability, Springer, New York, 1997. MR 1464694 (99e:60001)
9.: M. Khoshkam and G. Skandalis, Regular Representation of Groupoids and Applications to Inverse Semigroups, J. Reine Angew. Math. 546(2002), 47-72. MR 1900993 (2003f:46084)
10.: A. Kumjian, Notes on -algebras of graphs, Contemporary Mathematics 228(1998), 189-200. MR 1667662 (99m:46137)
11.: A. Kumjian, D. Pask and I. Raeburn, Cuntz-Krieger algebras of directed graphs, Pacific J. Math. 184(1998), 161-174. MR 1626528 (99i:46049)
12.: A. Kumjian, D. Pask, I. Raeburn and J. Renault, Graphs, Groupoids and Cuntz-Krieger algebras, J. Funct. Anal. 144(1997), 505-541. MR 1432596 (98g:46083)
13.: A. Kumjian and D. Pask, Higher Rank Graph -algebras, New York Journal of Mathematics 6(2000), 1-20. MR 1745529 (2001b:46102)
14.: J. Nagata, Modern General Topology, North Holland, Amsterdam, 1968. MR 0264579 (41:9171)
15.: D.A. Pask and C.E. Sutherland, Filtered inclusions of path algebras: a combinatorial approach to Doplicher-Roberts duality, J. Operator Theory 31 (1994) 99-121. MR 1316986 (96i:46073)
16.: A.L.T. Paterson Groupoids, inverse semigroups and their operator algebras, Progress in Mathematics, v. 170, Birkäuser Boston, New York, 1999. MR 1724106 (2001a:22003)
17.: A.L.T. Paterson Graph inverse semigroups, groupoids, and their -algebras, J. Operator Theory 48(2002), 645-662. MR 1962477 (2004h:46066)
18.: I. Raeburn and A. Sims, Product systems of graphs and the Toeplitz algebras of higher-rank graphs, preprint, 2001.
19.: I. Raeburn, A. Sims and T. Yeend, The -algebras of finitely aligned higher-rank graphs, J. Funct. Anal. 213(2004), 206-240. MR 2069786
20.: I. Raeburn, A. Sims and T. Yeend, Higher-rank graphs and their -algebras, Proc. Edinburgh Math. Soc. 46(2003), 99-115. MR 1961175 (2004f:46068)
21.: J. N. Renault, A groupoid approach to $C^{*}$ -algebras, Lecture Notes in Mathematics, Vol. 793, Springer-Verlag, New York, 1980. MR 0584266 (82h:46075)
22.: W. Szymanski, Simplicity of Cuntz-Krieger algebras of infinite matrices, Pacific J. Math. 199(2001) 249-256. MR 1847155 (2002i:46056)
23.: W. Szymanski, Bimodules for Cuntz-Krieger algebras of infinite matrices, Bull. Austral. Math. Soc. 62(2000), 87-94. MR 1775890 (2001g:46151)

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