Compact group automorphisms, addition formulas and Fuglede-Kadison determinants

Hanfeng Li

doi:https://doi.org/10.4007/annals.2012.176.1.5

Abstract

For a countable amenable group $\Gamma$ and an element $f$ in the integral group ring $\mathbb{Z}\Gamma$ being invertible in the group von Neumann algebra of $\Gamma$, we show that the entropy of the shift action of $\Gamma$ on the Pontryagin dual of the quotient of $\mathbb{Z}\Gamma$ by its left ideal generated by $f$ is the logarithm of the Fuglede-Kadison determinant of $f$. For the proof, we establish an $\ell^p$-version of Rufus Bowen’s definition of topological entropy, addition formulas for group extensions of countable amenable group actions, and an approximation formula for the Fuglede-Kadison determinant of $f$ in terms of the determinants of perturbations of the compressions of $f$.

Show/hide bibliography for this article

[Arveson] W. Arveson, An Invitation to $C^*$-Algebras, New York: Springer-Verlag, 1976, vol. 39.

Show bibtex

@book {Arveson, MRKEY = {0512360}, AUTHOR = {Arveson, William}, TITLE = {An Invitation to {$C^*$}-Algebras}, SERIES = {Grad. Texts in Math.}, VOLUME={39}, PUBLISHER = {Springer-Verlag}, ADDRESS = {New York}, YEAR = {1976}, PAGES = {x+106}, MRCLASS = {46L05}, MRNUMBER = {0512360}, MRREVIEWER = {V. Ptak}, ZBLNUMBER = {0344.46123}, }
[Berg] $Go to document$ K. R. Berg, "Convolution of invariant measures, maximal entropy," Math. Systems Theory, vol. 3, pp. 146-150, 1969.

Show bibtex

@article {Berg, MRKEY = {0248330}, AUTHOR = {Berg, Kenneth R.}, TITLE = {Convolution of invariant measures, maximal entropy}, JOURNAL = {Math. Systems Theory}, FJOURNAL = {Mathematical Systems Theory. An International Journal on Mathematical Computing Theory}, VOLUME = {3}, YEAR = {1969}, PAGES = {146--150}, ISSN = {0025-5661}, MRCLASS = {28.75}, MRNUMBER = {0248330}, MRREVIEWER = {R. L. Adler}, DOI = {10.1007/BF01746521}, ZBLNUMBER = {0179.08301}, }
[BG] $Go to document$ V. Bergelson and A. Gorodnik, "Ergodicity and mixing of non-commuting epimorphisms," Proc. Lond. Math. Soc., vol. 95, iss. 2, pp. 329-359, 2007.

Show bibtex

@article {BG, MRKEY = {2352564}, AUTHOR = {Bergelson, Vitaly and Gorodnik, Alexander}, TITLE = {Ergodicity and mixing of non-commuting epimorphisms}, JOURNAL = {Proc. Lond. Math. Soc.}, FJOURNAL = {Proceedings of the London Mathematical Society. Third Series}, VOLUME = {95}, YEAR = {2007}, NUMBER = {2}, PAGES = {329--359}, ISSN = {0024-6115}, MRCLASS = {37A25 (22F10 28D15 37A15 37A30)}, MRNUMBER = {2352564}, MRREVIEWER = {Daniel Berend}, DOI = {10.1112/plms/pdm007}, ZBLNUMBER = {1127.37007}, }
[Bow2] $Go to document$ L. Bowen, "Measure conjugacy invariants for actions of countable sofic groups," J. Amer. Math. Soc., vol. 23, iss. 1, pp. 217-245, 2010.

Show bibtex

@article {Bow2, MRKEY = {2552252}, AUTHOR = {Bowen, Lewis}, TITLE = {Measure conjugacy invariants for actions of countable sofic groups}, JOURNAL = {J. Amer. Math. Soc.}, FJOURNAL = {Journal of the American Mathematical Society}, VOLUME = {23}, YEAR = {2010}, NUMBER = {1}, PAGES = {217--245}, ISSN = {0894-0347}, MRCLASS = {37A35 (28D05 37A15 37A20)}, MRNUMBER = {2552252}, MRREVIEWER = {Konstantin Medynets}, DOI = {10.1090/S0894-0347-09-00637-7}, ZBLNUMBER = {1201.37005}, }
[Bow4] $Go to document$ L. Bowen, "Entropy for expansive algebraic actions of residually finite groups," Ergodic Theory Dynam. Systems, vol. 31, iss. 3, pp. 703-718, 2011.

Show bibtex

@article {Bow4, MRKEY = {2794944}, AUTHOR = {Bowen, Lewis}, TITLE = {Entropy for expansive algebraic actions of residually finite groups}, JOURNAL = {Ergodic Theory Dynam. Systems}, FJOURNAL = {Ergodic Theory and Dynamical Systems}, VOLUME = {31}, YEAR = {2011}, NUMBER = {3}, PAGES = {703--718}, ISSN = {0143-3857}, MRCLASS = {37Axx (20E26)}, MRNUMBER = {2794944}, DOI = {10.1017/S0143385710000179}, ZBLNUMBER = {05918090}, }
[Bow5] $Go to document$ L. Bowen, "Sofic entropy and amenable groups," Ergodic Theory Dynam. Systems, vol. 32, pp. 427-466, 2011.

Show bibtex

@article{Bow5, author={Bowen, Lewis}, TITLE = {Sofic entropy and amenable groups}, JOURNAL={Ergodic Theory Dynam. Systems} , VOLUME = {32}, PAGES = {427--466}, YEAR = {2011}, DOI = {10.1017/S0143385711000253}, }
[BowenLi] L. Bowen and H. Li, Harmonic models and spanning forests of residually finite groups, 2011.

Show bibtex

@misc{BowenLi, author={Bowen, Lewis and Li, H.}, TITLE = {Harmonic models and spanning forests of residually finite groups}, NOTE={preprint}, YEAR={2011}, ARXIV = {1108.4192}, }
[Bow] $Go to document$ R. Bowen, "Entropy for group endomorphisms and homogeneous spaces," Trans. Amer. Math. Soc., vol. 153, pp. 401-414, 1971.

Show bibtex

@article {Bow, MRKEY = {0274707}, AUTHOR = {Bowen, Rufus}, TITLE = {Entropy for group endomorphisms and homogeneous spaces}, JOURNAL = {Trans. Amer. Math. Soc.}, FJOURNAL = {Transactions of the American Mathematical Society}, VOLUME = {153}, YEAR = {1971}, PAGES = {401--414}, ISSN = {0002-9947}, MRCLASS = {28.70 (22.00)}, MRNUMBER = {0274707}, MRREVIEWER = {G. Della Riccia}, DOI = {10.2307/1995565}, ZBLNUMBER = {0212.29201}, }
[Chou] $Go to document$ C. Chou, "Elementary amenable groups," Illinois J. Math., vol. 24, iss. 3, pp. 396-407, 1980.

Show bibtex

@article {Chou, MRKEY = {0573475}, AUTHOR = {Chou, Ching}, TITLE = {Elementary amenable groups}, JOURNAL = {Illinois J. Math.}, FJOURNAL = {Illinois Journal of Mathematics}, VOLUME = {24}, YEAR = {1980}, NUMBER = {3}, PAGES = {396--407}, ISSN = {0019-2082}, CODEN = {IJMTAW}, MRCLASS = {43A07 (20E99)}, MRNUMBER = {0573475}, MRREVIEWER = {J. Wiegold}, URL = {http://projecteuclid.org/euclid.ijm/1256047608}, ZBLNUMBER = {0439.20017}, }
[Conway] J. B. Conway, A Course in Functional Analysis, Second ed., New York: Springer-Verlag, 1990, vol. 96.

Show bibtex

@book {Conway, MRKEY = {1070713}, AUTHOR = {Conway, J. B.}, TITLE = {A Course in Functional Analysis}, SERIES = {Grad. Texts in Math.}, VOLUME = {96}, EDITION = {Second}, PUBLISHER = {Springer-Verlag}, ADDRESS = {New York}, YEAR = {1990}, PAGES = {xvi+399}, ISBN = {0-387-97245-5}, MRCLASS = {46-01 (47-01)}, MRNUMBER = {1070713}, ZBLNUMBER = {0706.46003}, }
[CS] J. H. Conway and N. A. Sloane, Sphere Packings, Lattices and Groups, Third ed., New York: Springer-Verlag, 1999, vol. 290.

Show bibtex

@book {CS, MRKEY = {1662447}, AUTHOR = {Conway, J. H. and Sloane, NJ A.}, TITLE = {Sphere Packings, Lattices and Groups}, SERIES = {Grundl. Math. Wissen.}, VOLUME = {290}, EDITION = {Third}, PUBLISHER = {Springer-Verlag}, ADDRESS = {New York}, YEAR = {1999}, PAGES = {lxxiv+703}, ISBN = {0-387-98585-9}, MRCLASS = {11H31 (05B40 11H06 20D08 52C07 52C17 94B75 94C30)}, MRNUMBER = {1662447}, MRREVIEWER = {Renaud Coulangeon}, ZBLNUMBER = {0915.52003}, }
[Dan] $Go to document$ A. I. Danilenko, "Entropy theory from the orbital point of view," Monatsh. Math., vol. 134, iss. 2, pp. 121-141, 2001.

Show bibtex

@article {Dan, MRKEY = {1878075}, AUTHOR = {Danilenko, Alexandre I.}, TITLE = {Entropy theory from the orbital point of view}, JOURNAL = {Monatsh. Math.}, FJOURNAL = {Monatshefte für Mathematik}, VOLUME = {134}, YEAR = {2001}, NUMBER = {2}, PAGES = {121--141}, ISSN = {0026-9255}, CODEN = {MNMTA2}, MRCLASS = {37A35 (28D20 37A15 37A20)}, MRNUMBER = {1878075}, MRREVIEWER = {Thomas Ward}, DOI = {10.1007/s006050170003}, ZBLNUMBER = {0996.37007}, }
[Den] $Go to document$ C. Deninger, "Fuglede-Kadison determinants and entropy for actions of discrete amenable groups," J. Amer. Math. Soc., vol. 19, iss. 3, pp. 737-758, 2006.

Show bibtex

@article {Den, MRKEY = {2220105}, AUTHOR = {Deninger, Christopher}, TITLE = {Fuglede-{K}adison determinants and entropy for actions of discrete amenable groups}, JOURNAL = {J. Amer. Math. Soc.}, FJOURNAL = {Journal of the American Mathematical Society}, VOLUME = {19}, YEAR = {2006}, NUMBER = {3}, PAGES = {737--758}, ISSN = {0894-0347}, MRCLASS = {37A35 (22D25 37B40 43A07 46L55)}, MRNUMBER = {2220105}, MRREVIEWER = {Thomas Ward}, DOI = {10.1090/S0894-0347-06-00519-4}, ZBLNUMBER = {1104.22010}, }
[Den09] C. Deninger, "Mahler measures and Fuglede-Kadison determinants," Münster J. Math., vol. 2, pp. 45-63, 2009.

Show bibtex

@article {Den09, MRKEY = {2545607}, AUTHOR = {Deninger, Christopher}, TITLE = {Mahler measures and {F}uglede-{K}adison determinants}, JOURNAL = {Münster J. Math.}, FJOURNAL = {Münster Journal of Mathematics}, VOLUME = {2}, YEAR = {2009}, PAGES = {45--63}, ISSN = {1867-5778}, MRCLASS = {37A15 (11R06 28D20 37A35 37A45 46L10)}, MRNUMBER = {2545607}, MRREVIEWER = {Thomas Ward}, ZBLNUMBER = {05657123}, }
[Den06] $Go to document$ C. Deninger, "$p$-adic entropy and a $p$-adic Fuglede-Kadison determinant," in Algebra, Arithmetic, and Geometry: in Honor of Yu. I. Manin. Vol. I, Boston, MA: Birkhäuser, 2009, vol. 269, pp. 423-442.

Show bibtex

@incollection {Den06, MRKEY = {2641178}, AUTHOR = {Deninger, Christopher}, TITLE = {{$p$}-adic entropy and a {$p$}-adic {F}uglede-{K}adison determinant}, BOOKTITLE = {Algebra, Arithmetic, and Geometry: in Honor of {Y}u. {I}. {M}anin. {V}ol. {I}}, SERIES = {Progr. Math.}, VOLUME = {269}, PAGES = {423--442}, PUBLISHER = {Birkhäuser}, ADDRESS = {Boston, MA}, YEAR = {2009}, MRCLASS = {37P35 (11R06 11S82 37A35 37C85 37P20)}, MRNUMBER = {2641178}, MRREVIEWER = {Thomas Ward}, DOI = {10.1007/978-0-8176-4745-2_10}, ZBLNUMBER = {1208.37004}, }
[Den07] $Go to document$ C. Deninger, "Determinants on von Neumann algebras, Mahler measures and Ljapunov exponents," J. Reine Angew. Math., vol. 651, pp. 165-185, 2011.

Show bibtex

@article {Den07, MRKEY = {2774314}, AUTHOR = {Deninger, Christopher}, TITLE = {Determinants on von {N}eumann algebras, {M}ahler measures and {L}japunov exponents}, JOURNAL = {J. Reine Angew. Math.}, FJOURNAL = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]}, VOLUME = {651}, YEAR = {2011}, PAGES = {165--185}, ISSN = {0075-4102}, CODEN = {JRMAA8}, MRCLASS = {46L10 (11R06 37A35 47C15)}, MRNUMBER = {2774314}, MRREVIEWER = {Fyodor A. Sukochev}, DOI = {10.1515/CRELLE.2011.012}, ZBLNUMBER = {1220.46038}, }
[DSch] $Go to document$ C. Deninger and K. Schmidt, "Expansive algebraic actions of discrete residually finite amenable groups and their entropy," Ergodic Theory Dynam. Systems, vol. 27, iss. 3, pp. 769-786, 2007.

Show bibtex

@article {DSch, MRKEY = {2322178}, AUTHOR = {Deninger, Christopher and Schmidt, Klaus}, TITLE = {Expansive algebraic actions of discrete residually finite amenable groups and their entropy}, JOURNAL = {Ergodic Theory Dynam. Systems}, FJOURNAL = {Ergodic Theory and Dynamical Systems}, VOLUME = {27}, YEAR = {2007}, NUMBER = {3}, PAGES = {769--786}, ISSN = {0143-3857}, MRCLASS = {37A35 (37B40)}, MRNUMBER = {2322178}, MRREVIEWER = {A. I. Danilenko}, DOI = {10.1017/S0143385706000939}, ZBLNUMBER = {1128.22003}, }
[DLMSY] $Go to document$ J. Dodziuk, P. Linnell, V. Mathai, T. Schick, and S. Yates, "Approximating $L^2$-invariants and the Atiyah conjecture," Comm. Pure Appl. Math., vol. 56, iss. 7, pp. 839-873, 2003.

Show bibtex

@article {DLMSY, MRKEY = {1990479}, AUTHOR = {Dodziuk, J{ó}zef and Linnell, Peter and Mathai, Varghese and Schick, Thomas and Yates, Stuart}, TITLE = {Approximating {$L\sp 2$}-invariants and the {A}tiyah conjecture}, JOURNAL = {Comm. Pure Appl. Math.}, FJOURNAL = {Communications on Pure and Applied Mathematics}, VOLUME = {56}, YEAR = {2003}, NUMBER = {7}, PAGES = {839--873}, ISSN = {0010-3640}, CODEN = {CPAMA}, MRCLASS = {58J50 (11J81)}, MRNUMBER = {1990479}, MRREVIEWER = {Alan L. Carey}, DOI = {10.1002/cpa.10076}, ZBLNUMBER = {1036.58017}, }
[DS] $Go to document$ T. Downarowicz and J. Serafin, "Fiber entropy and conditional variational principles in compact non-metrizable spaces," Fund. Math., vol. 172, iss. 3, pp. 217-247, 2002.

Show bibtex

@article {DS, MRKEY = {1898686}, AUTHOR = {Downarowicz, Tomasz and Serafin, Jacek}, TITLE = {Fiber entropy and conditional variational principles in compact non-metrizable spaces}, JOURNAL = {Fund. Math.}, FJOURNAL = {Fundamenta Mathematicae}, VOLUME = {172}, YEAR = {2002}, NUMBER = {3}, PAGES = {217--247}, ISSN = {0016-2736}, MRCLASS = {37B40 (37A35)}, MRNUMBER = {1898686}, MRREVIEWER = {Doris Fiebig}, DOI = {10.4064/fm172-3-2}, ZBLNUMBER = {1115.37308}, }
[ER] $Go to document$ M. Einsiedler and H. Rindler, "Algebraic actions of the discrete Heisenberg group and other non-abelian groups," Aequationes Math., vol. 62, iss. 1-2, pp. 117-135, 2001.

Show bibtex

@article {ER, MRKEY = {1849144}, AUTHOR = {Einsiedler, Manfred and Rindler, Harald}, TITLE = {Algebraic actions of the discrete {H}eisenberg group and other non-abelian groups}, JOURNAL = {Aequationes Math.}, FJOURNAL = {Aequationes Mathematicae}, VOLUME = {62}, YEAR = {2001}, NUMBER = {1-2}, PAGES = {117--135}, ISSN = {0001-9054}, CODEN = {AEMABN}, MRCLASS = {37A15 (22D99 28A15 37A35)}, MRNUMBER = {1849144}, MRREVIEWER = {Mahesh Nerurkar}, DOI = {10.1007/PL00000133}, ZBLNUMBER = {0987.37003}, }
[EW] $Go to document$ M. Einsiedler and T. Ward, "Entropy geometry and disjointness for zero-dimensional algebraic actions," J. Reine Angew. Math., vol. 584, pp. 195-214, 2005.

Show bibtex

@article {EW, MRKEY = {2155090}, AUTHOR = {Einsiedler, Manfred and Ward, Thomas}, TITLE = {Entropy geometry and disjointness for zero-dimensional algebraic actions}, JOURNAL = {J. Reine Angew. Math.}, FJOURNAL = {Journal für die Reine und Angewandte Mathematik}, VOLUME = {584}, YEAR = {2005}, PAGES = {195--214}, ISSN = {0075-4102}, CODEN = {JRMAA8}, MRCLASS = {37A15 (28D20 37A35)}, MRNUMBER = {2155090}, MRREVIEWER = {Sergey L. Gefter}, DOI = {10.1515/crll.2005.2005.584.195}, ZBLNUMBER = {1198.37010}, }
[FGLLM] $Go to document$ G. Fendler, K. Gröchenig, M. Leinert, J. Ludwig, and C. Molitor-Braun, "Weighted group algebras on groups of polynomial growth," Math. Z., vol. 245, iss. 4, pp. 791-821, 2003.

Show bibtex

@article {FGLLM, MRKEY = {2020712}, AUTHOR = {Fendler, G. and Gr{ö}chenig, K. and Leinert, M. and Ludwig, J. and Molitor-Braun, C.}, TITLE = {Weighted group algebras on groups of polynomial growth}, JOURNAL = {Math. Z.}, FJOURNAL = {Mathematische Zeitschrift}, VOLUME = {245}, YEAR = {2003}, NUMBER = {4}, PAGES = {791--821}, ISSN = {0025-5874}, CODEN = {MAZEAX}, MRCLASS = {43A20 (22D12 22D15)}, MRNUMBER = {2020712}, MRREVIEWER = {Adolf R. Mirotin}, DOI = {10.1007/s00209-003-0571-6}, ZBLNUMBER = {1050.43003}, }
[Frey] A. H. Frey, Studies in amenable semigroups, 1960.

Show bibtex

@misc{Frey, author={Frey, A. H.}, TITLE={Studies in amenable semigroups}, NOTE={Ph.D. thesis, University of Washington, Seattle, Washington}, YEAR={1960}, }
[FK] $Go to document$ B. Fuglede and R. V. Kadison, "Determinant theory in finite factors," Ann. of Math., vol. 55, pp. 520-530, 1952.

Show bibtex

@article {FK, MRKEY = {0052696}, AUTHOR = {Fuglede, Bent and Kadison, Richard V.}, TITLE = {Determinant theory in finite factors}, JOURNAL = {Ann. of Math.}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {55}, YEAR = {1952}, PAGES = {520--530}, ISSN = {0003-486X}, MRCLASS = {46.3X}, MRNUMBER = {0052696}, MRREVIEWER = {F. I. Mautner}, DOI = {10.2307/1969645}, ZBLNUMBER = {0046.33604}, }
[Glasner] E. Glasner, Ergodic Theory via Joinings, Providence, RI: Amer. Math. Soc., 2003, vol. 101.

Show bibtex

@book {Glasner, MRKEY = {1958753}, AUTHOR = {Glasner, Eli}, TITLE = {Ergodic Theory via Joinings}, SERIES = {Math. Surveys Monogr.}, VOLUME = {101}, PUBLISHER = {Amer. Math. Soc.}, ADDRESS = {Providence, RI}, YEAR = {2003}, PAGES = {xii+384}, ISBN = {0-8218-3372-3}, MRCLASS = {37A15 (28Dxx 37A25 37A35 37A45 37B99 54H20)}, MRNUMBER = {1958753}, MRREVIEWER = {Andr{é}s del Junco}, ZBLNUMBER = {1038.37002}, }
[Hochster] $Go to document$ M. Hochster, "Subsemigroups of amenable groups," Proc. Amer. Math. Soc., vol. 21, pp. 363-364, 1969.

Show bibtex

@article {Hochster, MRKEY = {0240223}, AUTHOR = {Hochster, M.}, TITLE = {Subsemigroups of amenable groups}, JOURNAL = {Proc. Amer. Math. Soc.}, FJOURNAL = {Proceedings of the American Mathematical Society}, VOLUME = {21}, YEAR = {1969}, PAGES = {363--364}, ISSN = {0002-9939}, MRCLASS = {20.92 (46.00)}, MRNUMBER = {0240223}, MRREVIEWER = {E. Granirer}, DOI = {10.2307/2037004}, ZBLNUMBER = {0174.30801}, }
[Hulanicki72] $Go to document$ A. Hulanicki, "On the spectrum of convolution operators on groups with polynomial growth," Invent. Math., vol. 17, pp. 135-142, 1972.

Show bibtex

@article {Hulanicki72, MRKEY = {0323951}, AUTHOR = {Hulanicki, A.}, TITLE = {On the spectrum of convolution operators on groups with polynomial growth}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, VOLUME = {17}, YEAR = {1972}, PAGES = {135--142}, ISSN = {0020-9910}, MRCLASS = {22D12 (43A20)}, MRNUMBER = {0323951}, MRREVIEWER = {J. M. G. Fell}, DOI = {10.1007/BF01418936}, ZBLNUMBER = {0264.43007}, }
[Jenkins69] $Go to document$ J. W. Jenkins, "An amenable group with a nonsymmetric group algebra," Bull. Amer. Math. Soc., vol. 75, pp. 357-360, 1969.

Show bibtex

@article {Jenkins69, MRKEY = {0238090}, AUTHOR = {Jenkins, Joe W.}, TITLE = {An amenable group with a nonsymmetric group algebra}, JOURNAL = {Bull. Amer. Math. Soc.}, FJOURNAL = {Bulletin of the American Mathematical Society}, VOLUME = {75}, YEAR = {1969}, PAGES = {357--360}, ISSN = {0002-9904}, MRCLASS = {46.80}, MRNUMBER = {0238090}, MRREVIEWER = {T. Hannula}, DOI = {10.1090/S0002-9904-1969-12170-1}, ZBLNUMBER = {0186.46201}, }
[Jenkins70] $Go to document$ J. W. Jenkins, "Symmetry and nonsymmetry in the group algebras of discrete groups," Pacific J. Math., vol. 32, pp. 131-145, 1970.

Show bibtex

@article {Jenkins70, MRKEY = {0256185}, AUTHOR = {Jenkins, Joe W.}, TITLE = {Symmetry and nonsymmetry in the group algebras of discrete groups}, JOURNAL = {Pacific J. Math.}, FJOURNAL = {Pacific Journal of Mathematics}, VOLUME = {32}, YEAR = {1970}, PAGES = {131--145}, ISSN = {0030-8730}, MRCLASS = {46.80}, MRNUMBER = {0256185}, MRREVIEWER = {A. Hulanicki}, URL = {http://projecteuclid.org/euclid.pjm/1102977529}, ZBLNUMBER = {0172.03603}, }
[Jenkins71] $Go to document$ J. W. Jenkins, "On the spectral radius of elements in a group algebra," Illinois J. Math., vol. 15, pp. 551-554, 1971.

Show bibtex

@article {Jenkins71, MRKEY = {0287332}, AUTHOR = {Jenkins, Joe W.}, TITLE = {On the spectral radius of elements in a group algebra}, JOURNAL = {Illinois J. Math.}, FJOURNAL = {Illinois Journal of Mathematics}, VOLUME = {15}, YEAR = {1971}, PAGES = {551--554}, ISSN = {0019-2082}, MRCLASS = {46.80}, MRNUMBER = {0287332}, MRREVIEWER = {L. J. Lardy}, URL = {http://projecteuclid.org/euclid.ijm/1256052510}, ZBLNUMBER = {0222.43002}, }
[KR] R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras. Vol. I. Elementary Theory, New York: Academic Press [Harcourt Brace Jovanovich Publishers], 1983, vol. 100.

Show bibtex

@book {KR, MRKEY = {0719020}, AUTHOR = {Kadison, Richard V. and Ringrose, John R.}, TITLE = {Fundamentals of the Theory of Operator Algebras. {V}ol. {I}. Elementary Theory}, SERIES = {Pure Appl. Math.}, VOLUME = {100}, PUBLISHER = {Academic Press [Harcourt Brace Jovanovich Publishers]}, ADDRESS = {New York}, YEAR = {1983}, PAGES = {xv+398}, ISBN = {0-12-393301-3}, MRCLASS = {46Lxx (46-01 46-02 47-01)}, MRNUMBER = {0719020}, MRREVIEWER = {Robert S. Doran}, ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER}, ZBLNUMBER = {0888.46039}, }
[K1] A. S. Kechris, Classical Descriptive Set Theory, New York: Springer-Verlag, 1995, vol. 156.

Show bibtex

@book {K1, MRKEY = {1321597}, AUTHOR = {Kechris, Alexander S.}, TITLE = {Classical Descriptive Set Theory}, SERIES = {Grad. Texts in Math.}, VOLUME = {156}, PUBLISHER = {Springer-Verlag}, ADDRESS = {New York}, YEAR = {1995}, PAGES = {xviii+402}, ISBN = {0-387-94374-9}, MRCLASS = {03E15 (03-01 03-02 04A15 28A05 54H05 90D44)}, MRNUMBER = {1321597}, MRREVIEWER = {Jakub Jasi{ń}ski}, ZBLNUMBER = {0819.04002}, }
[Kelley] J. L. Kelley, General Topology, New York: Springer-Verlag, 1975, vol. 27.

Show bibtex

@book {Kelley, MRKEY = {0370454}, AUTHOR = {Kelley, John L.}, TITLE = {General Topology}, NOTE = {reprint of the 1955 edition [Van Nostrand, Toronto, Ont.]}, SERIES={Grad. Texts in Math.}, VOLUME={27}, PUBLISHER = {Springer-Verlag}, ADDRESS = {New York}, YEAR = {1975}, PAGES = {xiv+298}, MRCLASS = {54-XX}, MRNUMBER = {0370454}, ZBLNUMBER = {0306.54002}, }
[KL11a] $Go to document$ D. Kerr and H. Li, "Entropy and the variational principle for actions of sofic groups," Invent. Math., vol. 186, pp. 501-558, 2011.

Show bibtex

@article{KL11a, author={Kerr, D. and Li, H.}, TITLE={Entropy and the variational principle for actions of sofic groups}, JOURNAL={Invent. Math.}, VOLUME={186}, YEAR={2011}, PAGES={501--558}, MRNUMBER = {2854085}, ZBLNUMBER = {06004405}, DOI = {10.1007/s00222-011-0324-9}, }
[KL11b] D. Kerr and H. Li, Soficity, amenability and dynamical entropy.

Show bibtex

@misc{KL11b, author={Kerr, D. and Li, H.}, TITLE={Soficity, amenability and dynamical entropy}, SORTYEAR={2012}, NOTE={{\it Amer. J. Math.}, to appear}, }
[KS00] $Go to document$ B. Kitchens and K. Schmidt, "Isomorphism rigidity of irreducible algebraic ${\bf Z}^d$-actions," Invent. Math., vol. 142, iss. 3, pp. 559-577, 2000.

Show bibtex

@article {KS00, MRKEY = {1804161}, AUTHOR = {Kitchens, Bruce and Schmidt, Klaus}, TITLE = {Isomorphism rigidity of irreducible algebraic {${\bf Z}\sp d$}-actions}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, VOLUME = {142}, YEAR = {2000}, NUMBER = {3}, PAGES = {559--577}, ISSN = {0020-9910}, CODEN = {INVMBH}, MRCLASS = {37A15 (28D15 28D20 37A35)}, MRNUMBER = {1804161}, MRREVIEWER = {Thomas Ward}, DOI = {10.1007/PL00005793}, ZBLNUMBER = {0970.22006}, }
[Lang] S. Lang, Algebra, third ed., New York: Springer-Verlag, 2002, vol. 211.

Show bibtex

@book {Lang, MRKEY = {1878556}, AUTHOR = {Lang, Serge}, TITLE = {Algebra}, SERIES = {Grad. Texts in Math.}, VOLUME = {211}, EDITION = {third}, PUBLISHER = {Springer-Verlag}, ADDRESS = {New York}, YEAR = {2002}, PAGES = {xvi+914}, ISBN = {0-387-95385-X}, MRCLASS = {00A05 (15-02)}, MRNUMBER = {1878556}, ZBLNUMBER = {0984.00001}, }
[LP] $Go to document$ H. Leptin and D. Poguntke, "Symmetry and nonsymmetry for locally compact groups," J. Funct. Anal., vol. 33, iss. 2, pp. 119-134, 1979.

Show bibtex

@article {LP, MRKEY = {0546502}, AUTHOR = {Leptin, Horst and Poguntke, Detlev}, TITLE = {Symmetry and nonsymmetry for locally compact groups}, JOURNAL = {J. Funct. Anal.}, FJOURNAL = {Journal of Functional Analysis}, VOLUME = {33}, YEAR = {1979}, NUMBER = {2}, PAGES = {119--134}, ISSN = {0022-1236}, CODEN = {JFUAAW}, MRCLASS = {43A20}, MRNUMBER = {0546502}, MRREVIEWER = {A. Hulanicki}, DOI = {10.1016/0022-1236(79)90107-1}, ZBLNUMBER = {0414.43004}, }
[Lind] $Go to document$ D. A. Lind, "The structure of skew products with ergodic group automorphisms," Israel J. Math., vol. 28, iss. 3, pp. 205-248, 1977.

Show bibtex

@article {Lind, MRKEY = {0460593}, AUTHOR = {Lind, Douglas A. }, TITLE = {The structure of skew products with ergodic group automorphisms}, JOURNAL = {Israel J. Math.}, FJOURNAL = {Israel Journal of Mathematics}, VOLUME = {28}, YEAR = {1977}, NUMBER = {3}, PAGES = {205--248}, ISSN = {0021-2172}, MRCLASS = {28A65 (22D40 58F15)}, MRNUMBER = {0460593}, MRREVIEWER = {R. K. Thomas}, DOI = {10.1007/BF02759810}, ZBLNUMBER = {0365.28015}, }
[LS99] $Go to document$ D. A. Lind and K. Schmidt, "Homoclinic points of algebraic ${\bf Z}^d$-actions," J. Amer. Math. Soc., vol. 12, iss. 4, pp. 953-980, 1999.

Show bibtex

@article {LS99, MRKEY = {1678035}, AUTHOR = {Lind, Douglas A. and Schmidt, Klaus}, TITLE = {Homoclinic points of algebraic {${\bf Z}\sp d$}-actions}, JOURNAL = {J. Amer. Math. Soc.}, FJOURNAL = {Journal of the American Mathematical Society}, VOLUME = {12}, YEAR = {1999}, NUMBER = {4}, PAGES = {953--980}, ISSN = {0894-0347}, MRCLASS = {37A15 (22D40 37C85 37D20)}, MRNUMBER = {1678035}, MRREVIEWER = {Thomas Ward}, DOI = {10.1090/S0894-0347-99-00306-9}, ZBLNUMBER = {0940.22004}, }
[LS02] $Go to document$ D. A. Lind and K. Schmidt, "Symbolic and algebraic dynamical systems," in Handbook of Dynamical Aystems, Vol. 1A, Amsterdam: North-Holland, 2002, pp. 765-812.

Show bibtex

@incollection {LS02, MRKEY = {1928527}, AUTHOR = {Lind, Douglas A. and Schmidt, Klaus}, TITLE = {Symbolic and algebraic dynamical systems}, BOOKTITLE = {Handbook of Dynamical Aystems, {V}ol. 1{A}}, PAGES = {765--812}, PUBLISHER = {North-Holland}, ADDRESS = {Amsterdam}, YEAR = {2002}, MRCLASS = {37B50 (37A15 37B10)}, MRNUMBER = {1928527}, MRREVIEWER = {Anthony Quas}, DOI = {10.1016/S1874-575X(02)80012-1}, ZBLNUMBER = {1044.37010}, }
[LSW] $Go to document$ D. A. Lind, K. Schmidt, and T. Ward, "Mahler measure and entropy for commuting automorphisms of compact groups," Invent. Math., vol. 101, iss. 3, pp. 593-629, 1990.

Show bibtex

@article {LSW, MRKEY = {1062797}, AUTHOR = {Lind, Douglas A. and Schmidt, Klaus and Ward, Tom}, TITLE = {Mahler measure and entropy for commuting automorphisms of compact groups}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, VOLUME = {101}, YEAR = {1990}, NUMBER = {3}, PAGES = {593--629}, ISSN = {0020-9910}, CODEN = {INVMBH}, MRCLASS = {22D40 (28D20)}, MRNUMBER = {1062797}, MRREVIEWER = {Meir Smorodinsky}, DOI = {10.1007/BF01231517}, ZBLNUMBER = {0774.22002}, }
[LW88] $Go to document$ D. A. Lind and T. Ward, "Automorphisms of solenoids and $p$-adic entropy," Ergodic Theory Dynam. Systems, vol. 8, iss. 3, pp. 411-419, 1988.

Show bibtex

@article {LW88, MRKEY = {0961739}, AUTHOR = {Lind, Douglas A. and Ward, T.}, TITLE = {Automorphisms of solenoids and {$p$}-adic entropy}, JOURNAL = {Ergodic Theory Dynam. Systems}, FJOURNAL = {Ergodic Theory and Dynamical Systems}, VOLUME = {8}, YEAR = {1988}, NUMBER = {3}, PAGES = {411--419}, ISSN = {0143-3857}, MRCLASS = {28D20 (58F11)}, MRNUMBER = {0961739}, MRREVIEWER = {Kenneth R. Berg}, DOI = {10.1017/S0143385700004545}, ZBLNUMBER = {0634.22005}, }
[LW] $Go to document$ E. Lindenstrauss and B. Weiss, "Mean topological dimension," Israel J. Math., vol. 115, pp. 1-24, 2000.

Show bibtex

@article {LW, MRKEY = {1749670}, AUTHOR = {Lindenstrauss, Elon and Weiss, Benjamin}, TITLE = {Mean topological dimension}, JOURNAL = {Israel J. Math.}, FJOURNAL = {Israel Journal of Mathematics}, VOLUME = {115}, YEAR = {2000}, PAGES = {1--24}, ISSN = {0021-2172}, CODEN = {ISJMAP}, MRCLASS = {37B99 (37B05 37B40)}, MRNUMBER = {1749670}, MRREVIEWER = {Kathleen M. Madden}, DOI = {10.1007/BF02810577}, ZBLNUMBER = {0978.54026}, }
[Linnell] $Go to document$ P. A. Linnell, "Division rings and group von Neumann algebras," Forum Math., vol. 5, iss. 6, pp. 561-576, 1993.

Show bibtex

@article {Linnell, MRKEY = {1242889}, AUTHOR = {Linnell, Peter A.}, TITLE = {Division rings and group von {N}eumann algebras}, JOURNAL = {Forum Math.}, FJOURNAL = {Forum Mathematicum}, VOLUME = {5}, YEAR = {1993}, NUMBER = {6}, PAGES = {561--576}, ISSN = {0933-7741}, CODEN = {FOMAEF}, MRCLASS = {20C07 (16K40 16S35 22D25 46L10)}, MRNUMBER = {1242889}, MRREVIEWER = {Alain Valette}, DOI = {10.1515/form.1993.5.561}, ZBLNUMBER = {0794.22008}, }
[Lowner] $Go to document$ K. Löwner, "Über monotone Matrixfunktionen," Math. Z., vol. 38, iss. 1, pp. 177-216, 1934.

Show bibtex

@article {Lowner, MRKEY = {1545446}, AUTHOR = {L{ö}wner, Karl}, TITLE = {Über monotone {M}atrixfunktionen}, JOURNAL = {Math. Z.}, FJOURNAL = {Mathematische Zeitschrift}, VOLUME = {38}, YEAR = {1934}, NUMBER = {1}, PAGES = {177--216}, ISSN = {0025-5874}, CODEN = {MAZEAX}, MRCLASS = {Contributed Item}, MRNUMBER = {1545446}, DOI = {10.1007/BF01170633}, ZBLNUMBER = {0008.11301}, }
[Luck94] $Go to document$ W. Lück, "Approximating $L^2$-invariants by their finite-dimensional analogues," Geom. Funct. Anal., vol. 4, iss. 4, pp. 455-481, 1994.

Show bibtex

@article {Luck94, MRKEY = {1280122}, AUTHOR = {L{ü}ck, W.}, TITLE = {Approximating {$L\sp 2$}-invariants by their finite-dimensional analogues}, JOURNAL = {Geom. Funct. Anal.}, FJOURNAL = {Geometric and Functional Analysis}, VOLUME = {4}, YEAR = {1994}, NUMBER = {4}, PAGES = {455--481}, ISSN = {1016-443X}, CODEN = {GFANFB}, MRCLASS = {58G12 (46L85 57Q10)}, MRNUMBER = {1280122}, MRREVIEWER = {Peter Haskell}, DOI = {10.1007/BF01896404}, ZBLNUMBER = {0853.57021}, }
[Luck] W. Lück, $L^2$-Invariants: Theory and Applications to Geometry and $K$-theory, New York: Springer-Verlag, 2002, vol. 44.

Show bibtex

@book{Luck, author={L{ü}ck, W.}, TITLE = {$L^2$-Invariants: Theory and Applications to Geometry and $K$-theory}, SERIES = {Ergeb. Math. Grenzgeb.}, VOLUME = {44}, PUBLISHER = {Springer-Verlag}, ADDRESS = {New York}, YEAR = {2002}, PAGES = {xvi+595}, ISBN = {3-540-43566-2}, MRCLASS = {58J22 (19K56 46L80 57Q10 57R20 58J52)}, MRNUMBER = {1926649}, MRREVIEWER = {Thomas Schick}, ZBLNUMBER = {1009.55001}, }
[Ludwig] $Go to document$ J. Ludwig, "A class of symmetric and a class of Wiener group algebras," J. Funct. Anal., vol. 31, iss. 2, pp. 187-194, 1979.

Show bibtex

@article {Ludwig, MRKEY = {0525950}, AUTHOR = {Ludwig, Jean}, TITLE = {A class of symmetric and a class of {W}iener group algebras}, JOURNAL = {J. Funct. Anal.}, FJOURNAL = {Journal of Functional Analysis}, VOLUME = {31}, YEAR = {1979}, NUMBER = {2}, PAGES = {187--194}, ISSN = {0022-1236}, CODEN = {JFUAAW}, MRCLASS = {43A20 (22D15)}, MRNUMBER = {0525950}, MRREVIEWER = {A. Hulanicki}, DOI = {10.1016/0022-1236(79)90060-0}, ZBLNUMBER = {0402.22003}, }
[Mahler1] $Go to document$ K. Mahler, "An application of Jensen’s formula to polynomials," Mathematika, vol. 7, pp. 98-100, 1960.

Show bibtex

@article {Mahler1, MRKEY = {0124467}, AUTHOR = {Mahler, K.}, TITLE = {An application of {J}ensen's formula to polynomials}, JOURNAL = {Mathematika}, FJOURNAL = {Mathematika. A Journal of Pure and Applied Mathematics}, VOLUME = {7}, YEAR = {1960}, PAGES = {98--100}, ISSN = {0025-5793}, MRCLASS = {30.10}, MRNUMBER = {0124467}, MRREVIEWER = {L. Mirsky}, DOI = {10.1112/S0025579300001637}, ZBLNUMBER = {0099.25003}, }
[Mahler2] $Go to document$ K. Mahler, "On some inequalities for polynomials in several variables," J. London Math. Soc., vol. 37, pp. 341-344, 1962.

Show bibtex

@article {Mahler2, MRKEY = {0138593}, AUTHOR = {Mahler, K.}, TITLE = {On some inequalities for polynomials in several variables}, JOURNAL = {J. London Math. Soc.}, FJOURNAL = {Journal of the London Mathematical Society. Second Series}, VOLUME = {37}, YEAR = {1962}, PAGES = {341--344}, ISSN = {0024-6107}, MRCLASS = {10.30 (32.05)}, MRNUMBER = {0138593}, MRREVIEWER = {L. Mirsky}, DOI = {10.1112/jlms/s1-37.1.341}, ZBLNUMBER = {0105.06301}, }
[MT] G. Miles and R. K. Thomas, "Generalized torus automorphisms are Bernoullian," in Studies in Probability and Ergodic Theory, New York: Academic Press, 1978, vol. 2, pp. 231-249.

Show bibtex

@incollection {MT, MRKEY = {0517264}, AUTHOR = {Miles, G. and Thomas, R. K.}, TITLE = {Generalized torus automorphisms are {B}ernoullian}, BOOKTITLE = {Studies in Probability and Ergodic Theory}, SERIES = {Adv. in Math. Suppl. Stud.}, VOLUME = {2}, PAGES = {231--249}, PUBLISHER = {Academic Press}, ADDRESS = {New York}, YEAR = {1978}, MRCLASS = {22D40 (28D05)}, MRNUMBER = {0517264}, MRREVIEWER = {S. G. Dani}, ZBLNUMBER = {0499.22006}, }
[Miles] $Go to document$ R. Miles, "The entropy of algebraic actions of countable torsion-free abelian groups," Fund. Math., vol. 201, iss. 3, pp. 261-282, 2008.

Show bibtex

@article {Miles, MRKEY = {2457481}, AUTHOR = {Miles, Richard}, TITLE = {The entropy of algebraic actions of countable torsion-free abelian groups}, JOURNAL = {Fund. Math.}, FJOURNAL = {Fundamenta Mathematicae}, VOLUME = {201}, YEAR = {2008}, NUMBER = {3}, PAGES = {261--282}, ISSN = {0016-2736}, MRCLASS = {37A35 (11C08 28D15 28D20 37A15 37B40)}, MRNUMBER = {2457481}, MRREVIEWER = {Thomas Ward}, DOI = {10.4064/fm201-3-4}, ZBLNUMBER = {1154.37006}, }
[MB] $Go to document$ R. Miles and M. Björklund, "Entropy range problems and actions of locally normal groups," Discrete Contin. Dyn. Syst., vol. 25, iss. 3, pp. 981-989, 2009.

Show bibtex

@article {MB, MRKEY = {2533986}, AUTHOR = {Miles, Richard and Bj{ö}rklund, Michael}, TITLE = {Entropy range problems and actions of locally normal groups}, JOURNAL = {Discrete Contin. Dyn. Syst.}, FJOURNAL = {Discrete and Continuous Dynamical Systems. Series A}, VOLUME = {25}, YEAR = {2009}, NUMBER = {3}, PAGES = {981--989}, ISSN = {1078-0947}, MRCLASS = {37A35 (37B40)}, MRNUMBER = {2533986}, MRREVIEWER = {Thomas Ward}, DOI = {10.3934/dcds.2009.25.981}, ZBLNUMBER = {1179.37012}, }
[MW] $Go to document$ R. Miles and T. Ward, "Orbit-counting for nilpotent group shifts," Proc. Amer. Math. Soc., vol. 137, iss. 4, pp. 1499-1507, 2009.

Show bibtex

@article {MW, MRKEY = {2465676}, AUTHOR = {Miles, Richard and Ward, Thomas}, TITLE = {Orbit-counting for nilpotent group shifts}, JOURNAL = {Proc. Amer. Math. Soc.}, FJOURNAL = {Proceedings of the American Mathematical Society}, VOLUME = {137}, YEAR = {2009}, NUMBER = {4}, PAGES = {1499--1507}, ISSN = {0002-9939}, CODEN = {PAMYAR}, MRCLASS = {37A45 (22D40 37A15 37A35)}, MRNUMBER = {2465676}, MRREVIEWER = {Douglas P. Dokken}, DOI = {10.1090/S0002-9939-08-09649-4}, ZBLNUMBER = {1160.22005}, }
[MV] G. Mislin and A. Valette, Proper Group Actions and the Baum-Connes Conjecture, Basel: Birkhäuser, 2003.

Show bibtex

@book {MV, MRKEY = {2027168}, AUTHOR = {Mislin, Guido and Valette, Alain}, TITLE = {Proper Group Actions and the {B}aum-{C}onnes Conjecture}, SERIES = {Adv. Courses in Math. CRM Barcelona}, PUBLISHER = {Birkhäuser}, ADDRESS = {Basel}, YEAR = {2003}, PAGES = {viii+131}, ISBN = {3-7643-0408-1}, MRCLASS = {19K35 (46L80 55N20 58J22)}, MRNUMBER = {2027168}, ZBLNUMBER = {1028.46001}, }
[JMO] $Go to document$ J. Moulin Ollagnier, Ergodic Theory and Statistical Mechanics, New York: Springer-Verlag, 1985, vol. 1115.

Show bibtex

@book {JMO, MRKEY = {0781932}, AUTHOR = {Moulin Ollagnier, Jean}, TITLE = {Ergodic Theory and Statistical Mechanics}, SERIES = {Lecture Notes in Math.}, VOLUME = {1115}, PUBLISHER = {Springer-Verlag}, ADDRESS = {New York}, YEAR = {1985}, PAGES = {vi+147}, ISBN = {3-540-15192-3}, MRCLASS = {28Dxx (43A07 54H20 82A05)}, MRNUMBER = {0781932}, MRREVIEWER = {Manfred Denker}, ZBLNUMBER = {0558.28010}, DOI = {10.1007/BFb0101575}, }
[Naimark] M. A. Nauimark, Normed Algebras, Third ed., Wolters-Noordhoff Publishing, Groningen, 1972.

Show bibtex

@book {Naimark, MRKEY = {0438123}, AUTHOR = {Na{\u\i}mark, M. A.}, TITLE = {Normed Algebras}, EDITION = {Third}, NOTE = {translated from the second Russian edition by Leo F. Boron, Wolters-Noordhoff Series of Monographs and Textbooks on Pure and Applied Mathematics}, PUBLISHER = {Wolters-Noordhoff Publishing, Groningen}, YEAR = {1972}, PAGES = {xvi+598}, MRCLASS = {46H05 (43-02 46J05)}, MRNUMBER = {0438123}, ZBLNUMBER = {0254.46025}, }
[Nica] $Go to document$ B. Nica, "Relatively spectral morphisms and applications to $K$-theory," J. Funct. Anal., vol. 255, iss. 12, pp. 3303-3328, 2008.

Show bibtex

@article {Nica, MRKEY = {2469024}, AUTHOR = {Nica, Bogdan}, TITLE = {Relatively spectral morphisms and applications to {$K$}-theory}, JOURNAL = {J. Funct. Anal.}, FJOURNAL = {Journal of Functional Analysis}, VOLUME = {255}, YEAR = {2008}, NUMBER = {12}, PAGES = {3303--3328}, ISSN = {0022-1236}, CODEN = {JFUAAW}, MRCLASS = {19K99 (46H05 46L80)}, MRNUMBER = {2469024}, MRREVIEWER = {Vladimir Manuilov}, DOI = {10.1016/j.jfa.2008.04.018}, ZBLNUMBER = {1170.46042}, }
[OW] $Go to document$ D. S. Ornstein and B. Weiss, "Entropy and isomorphism theorems for actions of amenable groups," J. Analyse Math., vol. 48, pp. 1-141, 1987.

Show bibtex

@article {OW, MRKEY = {0910005}, AUTHOR = {Ornstein, Donald S. and Weiss, Benjamin}, TITLE = {Entropy and isomorphism theorems for actions of amenable groups}, JOURNAL = {J. Analyse Math.}, FJOURNAL = {Journal d'Analyse Mathématique}, VOLUME = {48}, YEAR = {1987}, PAGES = {1--141}, ISSN = {0021-7670}, CODEN = {JOAMAV}, MRCLASS = {28D20 (22D40)}, MRNUMBER = {0910005}, MRREVIEWER = {John C. Kieffer}, DOI = {10.1007/BF02790325}, ZBLNUMBER = {0637.28015}, }
[Parry] W. Parry, Topics in Ergodic Theory, Cambridge: Cambridge Univ. Press, 2004, vol. 75.

Show bibtex

@book {Parry, MRKEY = {2140546}, AUTHOR = {Parry, William}, TITLE = {Topics in Ergodic Theory}, SERIES = {Cambridge Tracts in Math.}, VOLUME = {75}, NOTE = {Reprint of the 1981 original}, PUBLISHER = {Cambridge Univ. Press}, ADDRESS = {Cambridge}, YEAR = {2004}, PAGES = {x+110}, ISBN = {0-521-60490-7}, MRCLASS = {37Axx (28D05)}, MRNUMBER = {2140546}, ZBLNUMBER = {1096.37001}, }
[Passman] D. S. Passman, The Algebraic Structure of Group Rings, New York: Wiley-Interscience [John Wiley & Sons], 1977.

Show bibtex

@book {Passman, MRKEY = {0470211}, AUTHOR = {Passman, Donald S.}, TITLE = {The Algebraic Structure of Group Rings}, SERIES = {Pure Appl. Math.}, PUBLISHER = {Wiley-Interscience [John Wiley \& Sons]}, ADDRESS = {New York}, YEAR = {1977}, PAGES = {xiv+720}, ISBN = {0-471-02272-1}, MRCLASS = {16-02 (16A27 20C07)}, MRNUMBER = {0470211}, MRREVIEWER = {Uno Kaljulaid}, ZBLNUMBER = {0368.16003}, }
[Pat] A. L. T. Paterson, Amenability, Providence, RI: Amer. Math. Soc., 1988, vol. 29.

Show bibtex

@book {Pat, MRKEY = {0961261}, AUTHOR = {Paterson, Alan L. T.}, TITLE = {Amenability}, SERIES = {Math. Surveys Monogr.}, VOLUME = {29}, PUBLISHER = {Amer. Math. Soc.}, ADDRESS = {Providence, RI}, YEAR = {1988}, PAGES = {xx+452}, ISBN = {0-8218-1529-6}, MRCLASS = {43-02 (22-02 43A07 46Lxx)}, MRNUMBER = {0961261}, MRREVIEWER = {C. Chou}, ZBLNUMBER = {0648.43001}, }
[Ped] G. K. Pedersen, $C^{\ast} $-Algebras and their Automorphism Groups, London: Academic Press [Harcourt Brace Jovanovich Publishers], 1979, vol. 14.

Show bibtex

@book {Ped, MRKEY = {0548006}, AUTHOR = {Pedersen, Gert K.}, TITLE = {{$C\sp{\ast} $}-Algebras and their Automorphism Groups}, SERIES = {London Mat. Soc. Monogr.}, VOLUME = {14}, PUBLISHER = {Academic Press [Harcourt Brace Jovanovich Publishers]}, ADDRESS = {London}, YEAR = {1979}, PAGES = {ix+416}, ISBN = {0-12-549450-5}, MRCLASS = {46Lxx}, MRNUMBER = {0548006}, MRREVIEWER = {J. W. Bunce}, ZBLNUMBER = {0416.46043}, }
[Raikov] D. A. Raikov, "To the theory of normed rings with involution," C. R. $($Doklady$)$ Acad. Sci. URSS, vol. 54, pp. 387-390, 1946.

Show bibtex

@article {Raikov, MRKEY = {0019845}, AUTHOR = {Raikov, D. A.}, TITLE = {To the theory of normed rings with involution}, JOURNAL = {C. R. $($Doklady$)$ Acad. Sci. URSS}, VOLUME = {54}, YEAR = {1946}, PAGES = {387--390}, MRCLASS = {46.0X}, MRNUMBER = {0019845}, MRREVIEWER = {C. E. Rickart}, ZBLNUMBER = {0060.27102}, }
[Rudin] W. Rudin, Real and Complex Analysis, Third ed., New York: McGraw-Hill Book Co., 1987.

Show bibtex

@book {Rudin, MRKEY = {0924157}, AUTHOR = {Rudin, Walter}, TITLE = {Real and Complex Analysis}, EDITION = {Third}, PUBLISHER = {McGraw-Hill Book Co.}, ADDRESS = {New York}, YEAR = {1987}, PAGES = {xiv+416}, ISBN = {0-07-054234-1}, MRCLASS = {00A05 (26-01 30-01 46-01)}, MRNUMBER = {0924157}, ZBLNUMBER = {0925.00005}, }
[RS] $Go to document$ D. J. Rudolph and K. Schmidt, "Almost block independence and Bernoullicity of ${\bf Z}^d$-actions by automorphisms of compact abelian groups," Invent. Math., vol. 120, iss. 3, pp. 455-488, 1995.

Show bibtex

@article {RS, MRKEY = {1334481}, AUTHOR = {Rudolph, Daniel J. and Schmidt, Klaus}, TITLE = {Almost block independence and {B}ernoullicity of {${\bf Z}\sp d$}-actions by automorphisms of compact abelian groups}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, VOLUME = {120}, YEAR = {1995}, NUMBER = {3}, PAGES = {455--488}, ISSN = {0020-9910}, CODEN = {INVMBH}, MRCLASS = {22D40 (28D15)}, MRNUMBER = {1334481}, MRREVIEWER = {Thomas Ward}, DOI = {10.1007/BF01241139}, ZBLNUMBER = {0835.28007}, }
[RW] $Go to document$ D. J. Rudolph and B. Weiss, "Entropy and mixing for amenable group actions," Ann. of Math., vol. 151, iss. 3, pp. 1119-1150, 2000.

Show bibtex

@article {RW, MRKEY = {1779565}, AUTHOR = {Rudolph, Daniel J. and Weiss, Benjamin}, TITLE = {Entropy and mixing for amenable group actions}, JOURNAL = {Ann. of Math.}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {151}, YEAR = {2000}, NUMBER = {3}, PAGES = {1119--1150}, ISSN = {0003-486X}, CODEN = {ANMAAH}, MRCLASS = {37A15 (37A25 37A35)}, MRNUMBER = {1779565}, MRREVIEWER = {Thomas Ward}, DOI = {10.2307/121130}, ZBLNUMBER = {0957.37003}, }
[Schick1] $Go to document$ T. Schick, "Erratum: “Integrality of $L^2$-Betti numbers”," Math. Ann., vol. 322, iss. 2, pp. 421-422, 2002.

Show bibtex

@article {Schick1, MRKEY = {1894160}, AUTHOR = {Schick, Thomas}, TITLE = {Erratum: ``{I}ntegrality of {$L\sp 2$}-{B}etti numbers''}, JOURNAL = {Math. Ann.}, FJOURNAL = {Mathematische Annalen}, VOLUME = {322}, YEAR = {2002}, NUMBER = {2}, PAGES = {421--422}, ISSN = {0025-5831}, CODEN = {MAANA}, MRCLASS = {55N25 (16S34 46L99 58J22)}, MRNUMBER = {1894160}, MRREVIEWER = {Warren Dicks}, DOI = {10.1007/s002080100282}, }
[Schick01] $Go to document$ T. Schick, "$L^2$-determinant class and approximation of $L^2$-Betti numbers," Trans. Amer. Math. Soc., vol. 353, iss. 8, pp. 3247-3265, 2001.

Show bibtex

@article {Schick01, MRKEY = {1828605}, AUTHOR = {Schick, Thomas}, TITLE = {{$L\sp 2$}-determinant class and approximation of {$L\sp 2$}-{B}etti numbers}, JOURNAL = {Trans. Amer. Math. Soc.}, FJOURNAL = {Transactions of the American Mathematical Society}, VOLUME = {353}, YEAR = {2001}, NUMBER = {8}, PAGES = {3247--3265}, ISSN = {0002-9947}, CODEN = {TAMTAM}, MRCLASS = {58J52 (55N25 55P25 57Q10)}, MRNUMBER = {1828605}, MRREVIEWER = {Patrick T. McDonald}, DOI = {10.1090/S0002-9947-01-02699-X}, ZBLNUMBER = {0979.55004}, }
[Sch] K. Schmidt, Dynamical Systems of Algebraic Origin, Basel: Birkhäuser, 1995, vol. 128.

Show bibtex

@book {Sch, MRKEY = {1345152}, AUTHOR = {Schmidt, Klaus}, TITLE = {Dynamical Systems of Algebraic Origin}, SERIES = {Progr. Math.}, VOLUME = {128}, PUBLISHER = {Birkhäuser}, ADDRESS = {Basel}, YEAR = {1995}, PAGES = {xviii+310}, ISBN = {3-7643-5174-8}, MRCLASS = {28D15 (11K55 22D40 28D20)}, MRNUMBER = {1345152}, MRREVIEWER = {Mahesh Nerurkar}, ZBLNUMBER = {0833.28001}, }
[Schmidt01] K. Schmidt, "The dynamics of algebraic $\Bbb Z^d$-actions," in European Congress of Mathematics, Vol. I, Basel: Birkhäuser, 2001, vol. 201, pp. 543-553.

Show bibtex

@incollection {Schmidt01, MRKEY = {1905342}, AUTHOR = {Schmidt, Klaus}, TITLE = {The dynamics of algebraic {$\Bbb Z\sp d$}-actions}, BOOKTITLE = {European {C}ongress of {M}athematics, {V}ol. {I}}, VENUE={{B}arcelona, 2000}, SERIES = {Progr. Math.}, VOLUME = {201}, PAGES = {543--553}, PUBLISHER = {Birkhäuser}, ADDRESS = {Basel}, YEAR = {2001}, MRCLASS = {37A15 (28D20)}, MRNUMBER = {1905342}, MRREVIEWER = {Thomas Ward}, ZBLNUMBER = {1071.28011}, }
[SW] $Go to document$ K. Schmidt and T. Ward, "Mixing automorphisms of compact groups and a theorem of Schlickewei," Invent. Math., vol. 111, iss. 1, pp. 69-76, 1993.

Show bibtex

@article {SW, MRKEY = {1193598}, AUTHOR = {Schmidt, Klaus and Ward, Tom}, TITLE = {Mixing automorphisms of compact groups and a theorem of {S}chlickewei}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, VOLUME = {111}, YEAR = {1993}, NUMBER = {1}, PAGES = {69--76}, ISSN = {0020-9910}, CODEN = {INVMBH}, MRCLASS = {22D40}, MRNUMBER = {1193598}, DOI = {10.1007/BF01231280}, ZBLNUMBER = {0824.28012}, }
[Solomyak] $Go to document$ R. Solomyak, "On coincidence of entropies for two classes of dynamical systems," Ergodic Theory Dynam. Systems, vol. 18, iss. 3, pp. 731-738, 1998.

Show bibtex

@article {Solomyak, MRKEY = {1631756}, AUTHOR = {Solomyak, Rita}, TITLE = {On coincidence of entropies for two classes of dynamical systems}, JOURNAL = {Ergodic Theory Dynam. Systems}, FJOURNAL = {Ergodic Theory and Dynamical Systems}, VOLUME = {18}, YEAR = {1998}, NUMBER = {3}, PAGES = {731--738}, ISSN = {0143-3857}, MRCLASS = {58F11 (22D40)}, MRNUMBER = {1631756}, MRREVIEWER = {Thomas Ward}, DOI = {10.1017/S0143385798108313}, ZBLNUMBER = {0924.58047}, }
[Takesaki] M. Takesaki, Theory of Operator Algebras. I, New York: Springer-Verlag, 2002, vol. 124.

Show bibtex

@book {Takesaki, MRKEY = {1873025}, AUTHOR = {Takesaki, M.}, TITLE = {Theory of Operator Algebras. {I}}, SERIES = {Encycl. Math. Sci.}, VOLUME = {124}, NOTE = {reprint of the first (1979) edition, {\it Operator Algebras and Non-commutative Geometry} {\bf 5}}, PUBLISHER = {Springer-Verlag}, ADDRESS = {New York}, YEAR = {2002}, PAGES = {xx+415}, ISBN = {3-540-42248-X}, MRCLASS = {46Lxx (46-01)}, MRNUMBER = {1873025}, ZBLNUMBER = {0990.46034}, }
[Thomas2] $Go to document$ R. K. Thomas, "The addition theorem for the entropy of transformations of $G$-spaces," Trans. Amer. Math. Soc., vol. 160, pp. 119-130, 1971.

Show bibtex

@article {Thomas2, MRKEY = {0293064}, AUTHOR = {Thomas, R. K.}, TITLE = {The addition theorem for the entropy of transformations of {$G$}-spaces}, JOURNAL = {Trans. Amer. Math. Soc.}, FJOURNAL = {Transactions of the American Mathematical Society}, VOLUME = {160}, YEAR = {1971}, PAGES = {119--130}, ISSN = {0002-9947}, MRCLASS = {28A65}, MRNUMBER = {0293064}, MRREVIEWER = {Michael Weiss}, DOI = {10.2307/1995794}, ZBLNUMBER = {0232.28015}, }
[Wal] P. Walters, An Introduction to Ergodic Theory, New York: Springer-Verlag, 1982, vol. 79.

Show bibtex

@book {Wal, MRKEY = {0648108}, AUTHOR = {Walters, Peter}, TITLE = {An Introduction to Ergodic Theory}, SERIES = {Grad. Texts in Math.}, VOLUME = {79}, PUBLISHER = {Springer-Verlag}, ADDRESS = {New York}, YEAR = {1982}, PAGES = {ix+250}, ISBN = {0-387-90599-5}, MRCLASS = {28Dxx (54H20 58F11)}, MRNUMBER = {0648108}, MRREVIEWER = {M. A. Akcoglu}, ZBLNUMBER = {0475.28009}, }
[Ward92] $Go to document$ T. B. Ward, "The Bernoulli property for expansive $\Bbb Z^2$ actions on compact groups," Israel J. Math., vol. 79, iss. 2-3, pp. 225-249, 1992.

Show bibtex

@article {Ward92, MRKEY = {1248915}, AUTHOR = {Ward, T. B.}, TITLE = {The {B}ernoulli property for expansive {$\bold Z\sp 2$} actions on compact groups}, JOURNAL = {Israel J. Math.}, FJOURNAL = {Israel Journal of Mathematics}, VOLUME = {79}, YEAR = {1992}, NUMBER = {2-3}, PAGES = {225--249}, ISSN = {0021-2172}, CODEN = {ISJMAP}, MRCLASS = {28D15}, MRNUMBER = {1248915}, MRREVIEWER = {B. Kami{ń}ski}, DOI = {10.1007/BF02808217}, ZBLNUMBER = {0789.28014}, }
[WZ] $Go to document$ T. Ward and Q. Zhang, "The Abramov-Rokhlin entropy addition formula for amenable group actions," Monatsh. Math., vol. 114, iss. 3-4, pp. 317-329, 1992.

Show bibtex

@article {WZ, MRKEY = {1203977}, AUTHOR = {Ward, Thomas and Zhang, Qing}, TITLE = {The {A}bramov-{R}okhlin entropy addition formula for amenable group actions}, JOURNAL = {Monatsh. Math.}, FJOURNAL = {Monatshefte für Mathematik}, VOLUME = {114}, YEAR = {1992}, NUMBER = {3-4}, PAGES = {317--329}, ISSN = {0026-9255}, CODEN = {MNMTA2}, MRCLASS = {28D20 (28D15)}, MRNUMBER = {1203977}, MRREVIEWER = {Mahesh Nerurkar}, DOI = {10.1007/BF01299386}, ZBLNUMBER = {0764.28014}, }
[Weiss01] B. Weiss, "Monotileable amenable groups," in Topology, Ergodic Theory, Real Algebraic Geometry, Providence, RI: Amer. Math. Soc., 2001, vol. 202, pp. 257-262.

Show bibtex

@incollection {Weiss01, MRKEY = {1819193}, AUTHOR = {Weiss, Benjamin}, TITLE = {Monotileable amenable groups}, BOOKTITLE = {Topology, Ergodic Theory, Real Algebraic Geometry}, SERIES = {Amer. Math. Soc. Transl. Ser. 2}, VOLUME = {202}, PAGES = {257--262}, PUBLISHER = {Amer. Math. Soc.}, ADDRESS = {Providence, RI}, YEAR = {2001}, MRCLASS = {22D40}, MRNUMBER = {1819193}, ZBLNUMBER = {0982.22004}, }
[Yuz1] S. A. Yuzvinskiui, "Metric properties of the endomorphisms of compact groups," Izv. Akad. Nauk SSSR Ser. Mat., vol. 29, pp. 1295-1328, 1965.

Show bibtex

@article {Yuz1, MRKEY = {0194588}, AUTHOR = {Yuzvinski{\u\i}, S. A.}, TITLE = {Metric properties of the endomorphisms of compact groups}, JOURNAL = {Izv. Akad. Nauk SSSR Ser. Mat.}, FJOURNAL = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya}, VOLUME = {29}, YEAR = {1965}, PAGES = {1295--1328}, ISSN = {0373-2436}, MRCLASS = {28.75 (10.33)}, NOTE={(Russian); translated in {\it Amer. Math. Soc. Transl.} {\bf 66} (1968), 63--98}, MRNUMBER = {0194588}, MRREVIEWER = {K. R. Parthasarathy}, ZBLNUMBER = {0206.03602}, }
[Yuz2] S. A. Yuzvinskiui, "Computing the entropy of a group of endomorphisms," Sibirsk. Mat. \^Z., vol. 8, pp. 230-239, 1967.

Show bibtex

@article{Yuz2, author={Yuzvinski{\u\i}, S. A.}, TITLE={Computing the entropy of a group of endomorphisms}, JOURNAL={Sibirsk. Mat. \^{Z}.}, VOLUME={8}, YEAR={1967}, PAGES={230--239}, NOTE={(Russian); translated in {\it Siberian Math. J.} {\bf 8} (1967), 172--178}, ZBLNUMBER = {0211.34901}, }

Compact group automorphisms, addition formulas and Fuglede-Kadison determinants

Abstract

Authors