Compact group automorphisms, addition formulas and Fuglede-Kadison determinants

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 $f$.

  • [Arveson] W. Arveson, An Invitation to $C^*$-Algebras, New York: Springer-Verlag, 1976, vol. 39.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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} $-Algebras and their Automorphism Groups, London: Academic Press [Harcourt Brace Jovanovich Publishers], 1979, vol. 14.
    @book {Ped, MRKEY = {0548006},
      AUTHOR = {Pedersen, Gert K.},
      TITLE = {{$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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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},
     }

Authors

Hanfeng Li

Department of Mathematics
Chongqing University
Chongqing
China

and

Department of Mathematics
SUNY at Buffalo
Buffalo, NY 14260-2900