Illinois Journal of Mathematics

Realizing dimension groups, good measures, and Toeplitz factors

David Handelman

Full-text: Open access

Abstract

Motivated by connections between dimension groups and good measures or minimal actions on Cantor sets (especially Töplitz), we find realizations of classes of dimension groups as limits of primitive matrices all of which have equal column sums, or equal row sums.

Article information

Source
Illinois J. Math., Volume 57, Number 4 (2013), 1057-1109.

Dates
First available in Project Euclid: 1 December 2014

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1417442563

Digital Object Identifier
doi:10.1215/ijm/1417442563

Mathematical Reviews number (MathSciNet)
MR3285868

Zentralblatt MATH identifier
1311.46060

Subjects
Primary: 19K14: $K_0$ as an ordered group, traces 06F20: Ordered abelian groups, Riesz groups, ordered linear spaces [See also 46A40] 46L80: $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22] 15B36: Matrices of integers [See also 11C20] 15B48: Positive matrices and their generalizations; cones of matrices 37A55: Relations with the theory of C-algebras [See mainly 46L55] 20K15: Torsion-free groups, finite rank 20K20: Torsion-free groups, infinite rank

Citation

Handelman, David. Realizing dimension groups, good measures, and Toeplitz factors. Illinois J. Math. 57 (2013), no. 4, 1057--1109. doi:10.1215/ijm/1417442563. https://projecteuclid.org/euclid.ijm/1417442563


Export citation

References

  • E. Akin, Measures on Cantor space, Topology Proc. 24 (1999), 1–34.
  • E. Akin, Good measures on Cantor space, Trans. Amer. Math. Soc. 357 (2005), 2681–2722.
  • S. Bezuglyi and D. Handelman, Measures on Cantor sets: The good, the ugly, the bad, Trans. Amer. Math. Soc. 366 (2014), 6247–6311.
  • M. Boyle and D. Handelman, Algebraic shift equivalence and primitive matrices, Trans. Amer. Math. Soc. 336 (1993), 121–149.
  • E. G. Effros, Dimensions and C*-algebras, CBMS Regional Conference Series in Mathematics, vol. 46, Conference Board of the Mathematical Sciences, Washington, 1981.
  • E. G. Effros, D. Handelman and C.-L. Shen, Dimension groups and their affine representations, Amer. J. Math. 102 (1980), 385–407.
  • G. Elliott, On the classification of inductive limits of sequences of semisimple finite-dimensional algebras, J. Algebra 38 (1976), 29–44.
  • G. Elliott, On totally ordered groups and $K_0$, Proc. conf. ring theory (Waterloo, 1978), Lecture Notes in Math., vol. 734, Springer, Berlin, 1979, pp. 1–50.
  • T. Giordano, I. F. Putnam and C. F. Skau, Topological orbit equivalence and $C^*$-crossed products, J. Reine Angew. Math. 469 (1995), 51–111.
  • T. Giordano, I. F. Putnam and C. F. Skau, Full groups of Cantor minimal systems, Israel J. Math. 111 (1999), 285–320.
  • R. Gjerde and Ø. Johansen, Bratteli–Vershik models for Cantor minimal systems: Applications to Toeplitz flows, Ergodic Theory Dynam. Systems 20 (2000), 1687–1710.
  • D. Handelman, Positive matrices and dimension groups affiliated to C*-algebras and topological Markov chains, J. Operator Theory 6 (1981), 55–74.
  • D. Handelman, Ultrasimplicial dimension groups, Arch. Math. (Basel) 40 (1983), 109–115.
  • D. Handelman, Eventually positive matrices with rational eigenvectors, Ergodic Theory Dynam. Systems 7 (1987), 193–196.
  • D. Handelman, Matrices of positive polynomials, Electron. J. Linear Algebra 19 (2009), 2–89.
  • D. Handelman, Not limits of simple dimension groups with finitely many pure traces, and weakly initial objects, to appear in Indiana Univ. Math. J.
  • D. Handelman, Good measures for non-simple dimension groups, available at \arxivurlarXiv:1309.7424.
  • B. Marcus, Factors and extensions of full shifts, Monatsh. Math. 88 (1979), 239–247.
  • N. Riedel, Classification of dimension groups and iterating systems, Math. Scand. 48 (1981), 226–234.
  • N. Riedel, A counterexample to the unimodular conjecture on finitely generated dimension groups, Proc. Amer. Math. Soc. 83 (1981), 11–15.
  • D. J. S. Robinson, Groups–-St Andrews 1981, London Mathematical Society Lecture Note Series, vol. 71, Cambridge University Press, New York, 1982, pp. 46–81.