Book Review
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MathSciNet review:
1567798
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Book Information:
Author:
K. R. Goodearl
Title:
Partially ordered abelian groups with interpolation
Additional book information:
Mathematical Surveys and Monographs, number 20, American Mathematical Society, Providence, R.I., 1986, xxii + 336 pp., ISBN 0-8218-1520-2.
Garrett Birkhoff, Lattice, ordered groups, Ann. of Math. (2) 43 (1942), 298–331. MR 6550, DOI 10.2307/1968871
Ola Bratteli, George A. Elliott, and Akitaka Kishimoto, The temperature state space of a $C^\ast$-dynamical system. II, Ann. of Math. (2) 123 (1986), no. 2, 205–263. MR 835762, DOI 10.2307/1971271
3. A. Connes, Non commutative differential geometry, Proceedings of Arbeitstagung 1987 (F. Hirzebruch, ed. ), Max Planck Institute, Univ. of Bonn.
Joachim Cuntz and Wolfgang Krieger, Topological Markov chains with dicyclic dimension groups, J. Reine Angew. Math. 320 (1980), 44–51. MR 592141, DOI 10.1515/crll.1980.320.44
Edward G. Effros, Dimensions and $C^{\ast }$-algebras, CBMS Regional Conference Series in Mathematics, vol. 46, Conference Board of the Mathematical Sciences, Washington, D.C., 1981. MR 623762
Edward G. Effros, David E. Handelman, and Chao Liang Shen, Dimension groups and their affine representations, Amer. J. Math. 102 (1980), no. 2, 385–407. MR 564479, DOI 10.2307/2374244
Edward G. Effros and Chao Liang Shen, Approximately finite $C^{\ast }$-algebras and continued fractions, Indiana Univ. Math. J. 29 (1980), no. 2, 191–204. MR 563206, DOI 10.1512/iumj.1980.29.29013
George A. Elliott, On the classification of inductive limits of sequences of semisimple finite-dimensional algebras, J. Algebra 38 (1976), no. 1, 29–44. MR 397420, DOI 10.1016/0021-8693(76)90242-8
George A. Elliott, A property of totally ordered abelian groups, C. R. Math. Rep. Acad. Sci. Canada 1 (1978/79), no. 2, 63–66. MR 519524
George A. Elliott, On totally ordered groups, and $K_{0}$, Ring theory (Proc. Conf., Univ. Waterloo, Waterloo, 1978) Lecture Notes in Math., vol. 734, Springer, Berlin, 1979, pp. 1–49. MR 548122
L. Fuchs, Riesz groups, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 19 (1965), 1–34. MR 180609
David E. Handelman, Positive polynomials, convex integral polytopes, and a random walk problem, Lecture Notes in Mathematics, vol. 1282, Springer-Verlag, Berlin, 1987. MR 914972, DOI 10.1007/BFb0078909
V. F. R. Jones, Braid groups, Hecke algebras and type $\textrm {II}_1$ factors, Geometric methods in operator algebras (Kyoto, 1983) Pitman Res. Notes Math. Ser., vol. 123, Longman Sci. Tech., Harlow, 1986, pp. 242–273. MR 866500
A. J. Lazar and J. Lindenstrauss, Banach spaces whose duals are $L_{1}$ spaces and their representing matrices, Acta Math. 126 (1971), 165–193. MR 291771, DOI 10.1007/BF02392030
Daniele Mundici, Farey stellar subdivisions, ultrasimplicial groups, and $K_0$ of AF $C^*$-algebras, Adv. in Math. 68 (1988), no. 1, 23–39. MR 931170, DOI 10.1016/0001-8708(88)90006-0
Adrian Ocneanu, Quantized groups, string algebras and Galois theory for algebras, Operator algebras and applications, Vol. 2, London Math. Soc. Lecture Note Ser., vol. 136, Cambridge Univ. Press, Cambridge, 1988, pp. 119–172. MR 996454
M. Pimsner and D. Voiculescu, Imbedding the irrational rotation $C^{\ast }$-algebra into an AF-algebra, J. Operator Theory 4 (1980), no. 2, 201–210. MR 595412
Jean Renault, A groupoid approach to $C^{\ast }$-algebras, Lecture Notes in Mathematics, vol. 793, Springer, Berlin, 1980. MR 584266
Frédéric Riesz, Sur quelques notions fondamentales dans la théorie générale des opérations linéaires, Ann. of Math. (2) 41 (1940), 174–206 (French). MR 902, DOI 10.2307/1968825
Chao Liang Shen, On the classification of the ordered groups associated with the approximately finite-dimensional $C^{\ast }$-algebras, Duke Math. J. 46 (1979), no. 3, 613–633. MR 544249
21. A. M. Vershik and S. V. Kerov, Locally semisimple algebras, combinatorial theory, and the K, J. Soviet Math. 38 (1987), 1701- 1733.
- 1.
- G. Birkhoff, Lattice-ordered groups, Ann. of Math. (2) 43 (1942), 298-331. MR 0006550
- 2.
- O. Bratteli, G. A. Elliott, and A. Kishimoto, The temperature state space of a C*-dynamical system. II, Ann. of Math. (2) 123 (1986), 205-263. MR 835762
- 3.
- A. Connes, Non commutative differential geometry, Proceedings of Arbeitstagung 1987 (F. Hirzebruch, ed. ), Max Planck Institute, Univ. of Bonn.
- 4.
- J. Cuntz and W. Krieger, Topological Markov chains with dicyclic dimension groups, J. Reine Angew. Math. 320 (1980), 44-51. MR 592141
- 5.
- E. G. Effros, Dimensions and C*-algebras, CBMS Regional Conf. Ser. in Math., No. 46, Amer. Math. Soc., Providence, R. I., 1981. MR 623762
- 6.
- E. G. Effros, D. E. Handelman, and C.-L. Shen, Dimension groups and their affine representations, Amer. J. Math. 102 (1980), 385-407. MR 564479
- 7.
- E. G. Effros and C.-L. Shen, Approximately finite C*-algebras and continued fractions, Indiana Univ. Math. J. 29 (1980), 191-204. MR 563206
- 8.
- G. A. Elliott, On the classification of inductive limits of sequences of semisimple finite-dimensional algebras, J. Algebra 38 (1976), 29-44. MR 397420
- 9.
- G. A. Elliott, A property of totally ordered groups, C. R. Math. Rep. Acad. Sci. Canada 1 (1979), 63-66. MR 519524
- 10.
- G. A. Elliott, On totally ordered groups, and K0, Ring Theory, Waterloo, 1978 (D. Handelman and J. Lawrence, eds. ), Lecture Notes in Math., vol. 734, Springer-Verlag, Berlin and New York, 1979, pp. 1-49. MR 548122
- 11.
- L. Fuchs, Riesz groups, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 19 (1965), 1-34. MR 180609
- 12.
- D. E. Handelman, Positive polynomials, convex integral polytopes, and a random walk problem, Lecture Notes in Math., vol. 1282, Springer-Verlag, Berlin and New York, 1987. MR 914972
- 13.
- V. F. R. Jones, Braid groups, Hecke algebras and type II, Geometric Methods in Operator Algebras (H. Araki and E. G. Effros, eds. ), Longman, London, 1986, pp. 242-273. MR 866500
- 14.
- A. Lazar and J. Lindenstrauss, Banach spaces whose duals are L, Acta Math. 126 (1971), 165-193. MR 291771
- 15.
- D. Mundici, Farey stellar subdivisions, ultrasimplicial groups, and K, Adv. in Math. 68 (1988), 23-39. MR 931170
- 16.
- A. Ocneanu, Quantized groups, string algebras, and Galois theory for algebras, preprint. MR 996454
- 17.
- M. Pimsner and D. Voiculescu, Imbedding the irrational rotation C*-algebra into an AF-algebra, J. Operator Theory 4 (1980), 201-210. MR 595412
- 18.
- J. Renault, A groupoid approach to C*-algebras, Lecture Notes in Math., vol. 793, Springer-Verlag, Berlin and New York, 1980. MR 584266
- 19.
- F. Riesz, Sur quelques notions fondamentales dans la théorie générale des opérations linéaires, Ann. of Math. (2) 41 (1940), 174-206. MR 902
- 20.
- C.-L. Shen, On the classification of the ordered groups associated with the approximately finite-dimensional C*-algebras, Duke Math. J. 46 (1979), 613-633. MR 544249
- 21.
- A. M. Vershik and S. V. Kerov, Locally semisimple algebras, combinatorial theory, and the K, J. Soviet Math. 38 (1987), 1701- 1733.
Review Information:
Reviewer:
George A. Elliott
Journal:
Bull. Amer. Math. Soc.
21 (1989), 200-204
DOI:
https://doi.org/10.1090/S0273-0979-1989-15822-9