On the Toeplitz-Jacobson algebra and direct finiteness.(English)Zbl 1370.16016
Bergen, Jeffrey (ed.) et al., Groups, rings, group rings, and Hopf algebras. International conference, Loyola University, Chicago, IL, USA, October 2–4, 2015 and AMS special session, Loyola University, Chicago, IL, USA, October 3–4, 2015. Held in honor of Donald S. Passman’s 75th birthday. Proceedings. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-2805-1/pbk; 978-1-4704-4042-8/ebook). Contemporary Mathematics 688, 113-124 (2017).
Summary: We study the representation theory of the algebraic Toeplitz algebra \(R=\mathbb {K}\langle x,y\rangle /\langle xy-1\rangle \), give a few new structure and homological theorems, completely determine one-sided ideals and survey and re-obtain results from the existing literature as well. We discuss its connection to Kaplansky’s direct finiteness conjecture, and a possible approach to it based on the module theory of \(R\). In addition, we discuss the conjecture’s connections to several central problems in mathematics, including Connes’ embedding conjecture.
For the entire collection see [Zbl 1365.16002].
For the entire collection see [Zbl 1365.16002].
MSC:
| 16R10 | \(T\)-ideals, identities, varieties of associative rings and algebras |
| 16E30 | Homological functors on modules (Tor, Ext, etc.) in associative algebras |
Cite
References:
| [1] | Abrams, Gene, Leavitt path algebras: the first decade, Bull. Math. Sci., 5, 1, 59-120 (2015) ·Zbl 1329.16002 ·doi:10.1007/s13373-014-0061-7 |
| [2] | Abrams, Gene; Mantese, Francesca; Tonolo, Alberto, Extensions of simple modules over Leavitt path algebras, J. Algebra, 431, 78-106 (2015) ·Zbl 1319.16005 ·doi:10.1016/j.jalgebra.2015.01.034 |
| [3] | Alahmedi, Adel; Alsulami, Hamed; Jain, Surender; Zelmanov, Efim I., Structure of Leavitt path algebras of polynomial growth, Proc. Natl. Acad. Sci. USA, 110, 38, 15222-15224 (2013) ·Zbl 1296.16008 ·doi:10.1073/pnas.1311216110 |
| [4] | Ara, Pere; O’Meara, Kevin C.; Perera, Francesc, Stable finiteness of group rings in arbitrary characteristic, Adv. Math., 170, 2, 224-238 (2002) ·Zbl 1018.20006 ·doi:10.1006/aima.2002.2075 |
| [5] | Ara, P.; Moreno, M. A.; Pardo, E., Nonstable \(K\)-theory for graph algebras, Algebr. Represent. Theory, 10, 2, 157-178 (2007) ·Zbl 1123.16006 ·doi:10.1007/s10468-006-9044-z |
| [6] | Ara, Pere; Rangaswamy, Kulumani M., Finitely presented simple modules over Leavitt path algebras, J. Algebra, 417, 333-352 (2014) ·Zbl 1304.16002 ·doi:10.1016/j.jalgebra.2014.06.032 |
| [7] | Ara, Pere; Brustenga, Miquel, Module theory over Leavitt path algebras and \(K\)-theory, J. Pure Appl. Algebra, 214, 7, 1131-1151 (2010) ·Zbl 1189.16013 ·doi:10.1016/j.jpaa.2009.10.001 |
| [8] | Bavula, V. V., The algebra of one-sided inverses of a polynomial algebra, J. Pure Appl. Algebra, 214, 10, 1874-1897 (2010) ·Zbl 1226.16018 ·doi:10.1016/j.jpaa.2009.12.033 |
| [9] | F. Berlai, Groups satisfying Kaplansky’s stable finiteness conjecture, (2015) arXiv/math:1501.02893v1. |
| [10] | Capraro, Valerio; Lupini, Martino, Introduction to sofic and hyperlinear groups and Connes’ embedding conjecture, Lecture Notes in Mathematics 2136, viii+151 pp. (2015), Springer, Cham ·Zbl 1383.20002 ·doi:10.1007/978-3-319-19333-5 |
| [11] | Colak, Pinar, Two-sided ideals in Leavitt path algebras, J. Algebra Appl., 10, 5, 801-809 (2011) ·Zbl 1242.16014 ·doi:10.1142/S0219498811004859 |
| [12] | Chen, Xiao-Wu, Irreducible representations of Leavitt path algebras, Forum Math., 27, 1, 549-574 (2015) ·Zbl 1332.16006 ·doi:10.1515/forum-2012-0020 |
| [13] | Dykema, Ken; Heister, Timo; Juschenko, Kate, Finitely presented groups related to Kaplansky’s direct finiteness conjecture, Exp. Math., 24, 3, 326-338 (2015) ·Zbl 1403.20004 ·doi:10.1080/10586458.2014.993051 |
| [14] | Dykema, Ken; Juschenko, Kate, On stable finiteness of group rings, Algebra Discrete Math., 19, 1, 44-47 (2015) ·Zbl 1333.16027 |
| [15] | Elek, G{\'a}bor; Szab{\'o}, Endre, Sofic groups and direct finiteness, J. Algebra, 280, 2, 426-434 (2004) ·Zbl 1078.16021 ·doi:10.1016/j.jalgebra.2004.06.023 |
| [16] | Gromov, M., Endomorphisms of symbolic algebraic varieties, J. Eur. Math. Soc. (JEMS), 1, 2, 109-197 (1999) ·Zbl 0998.14001 ·doi:10.1007/PL00011162 |
| [17] | Kaplansky, Irving, Fields and rings, ix+198 pp. (1969), The University of Chicago Press, Chicago, Ill.-London ·Zbl 0184.24201 |
| [18] | A. Kwiatkowska, V. Pestov, An introduction to hyperlinear and sofic groups, preprint, 2014. ·Zbl 1367.20052 |
| [19] | Montgomery, M. Susan, Left and right inverses in group algebras, Bull. Amer. Math. Soc., 75, 539-540 (1969) ·Zbl 0174.31204 |
| [20] | Montgomery, Susan, von Neumann finiteness of tensor products of algebras, Comm. Algebra, 11, 6, 595-610 (1983) ·Zbl 0488.16015 ·doi:10.1080/00927878308822867 |
| [21] | Passman, Donald S., Infinite crossed products, Pure and Applied Mathematics 135, xii+468 pp. (1989), Academic Press, Inc., Boston, MA ·Zbl 0662.16001 |
| [22] | Passman, Donald S., The algebraic structure of group rings, Pure and Applied Mathematics, xiv+720 pp. (1977), Wiley-Interscience [John Wiley & Sons], New York-London-Sydney ·Zbl 0368.16003 |
| [23] | Pestov, Vladimir G., Hyperlinear and sofic groups: a brief guide, Bull. Symbolic Logic, 14, 4, 449-480 (2008) ·Zbl 1206.20048 ·doi:10.2178/bsl/1231081461 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
