Metric completions of ordered groups and 𝐾₀ of exchange rings

Pardo, E.

doi:10.1090/S0002-9947-98-01744-9

Metric completions of ordered groups and $K_0$ of exchange rings
HTML articles powered by AMS MathViewer

by E. Pardo PDF

Trans. Amer. Math. Soc. 350 (1998), 913-933 Request permission

Abstract:

We give a description of the closure of the natural affine continuous function representation of $K_0(R)$ for any exchange ring $R$. This goal is achieved by extending the results of Goodearl and Handelman, about metric completions of dimension groups, to a more general class of pre-ordered groups, which includes $K_0$ of exchange rings. As a consequence, the results about $K_0^+$ of regular rings, which the author gave in an earlier paper, can be extended to a wider class of rings, which includes $C^*$-algebras of real rank zero, among others. Also, the framework of pre-ordered groups developed here allows other potential applications.

References

Pere Ara, Strongly $\pi$-regular rings have stable range one, Proc. Amer. Math. Soc. 124 (1996), no. 11, 3293–3298. MR 1343679, DOI 10.1090/S0002-9939-96-03473-9
Pere Ara and K. R. Goodearl, The almost isomorphism relation for simple regular rings, Publ. Mat. 36 (1992), no. 2A, 369–388 (1993). MR 1209808, DOI 10.5565/PUBLMAT_{3}62A92_{0}2
P.Ara, K.R.Goodearl, K.C.O’Meara, E.Pardo, Separative cancellation for projective modules over exchange rings, Israel J. Math. (to appear).
P. Ara, K. R. Goodearl, E. Pardo, and D. V. Tyukavkin, $K$-theoretically simple von Neumann regular rings, J. Algebra 174 (1995), no. 2, 659–677. MR 1334230, DOI 10.1006/jabr.1995.1145
P. Ara and E. Pardo, Refinement monoids with weak comparability and applications to regular rings and $C^*$-algebras, Proc. Amer. Math. Soc. 124 (1996), no. 3, 715–720. MR 1301484, DOI 10.1090/S0002-9939-96-03059-6
Bruce Blackadar, $K$-theory for operator algebras, Mathematical Sciences Research Institute Publications, vol. 5, Springer-Verlag, New York, 1986. MR 859867, DOI 10.1007/978-1-4613-9572-0
Bruce E. Blackadar, Traces on simple AF $C^{\ast }$-algebras, J. Functional Analysis 38 (1980), no. 2, 156–168. MR 587906, DOI 10.1016/0022-1236(80)90062-2
Bruce Blackadar, Comparison theory for simple $C^*$-algebras, Operator algebras and applications, Vol. 1, London Math. Soc. Lecture Note Ser., vol. 135, Cambridge Univ. Press, Cambridge, 1988, pp. 21–54. MR 996438
Bruce Blackadar and David Handelman, Dimension functions and traces on $C^{\ast }$-algebras, J. Functional Analysis 45 (1982), no. 3, 297–340. MR 650185, DOI 10.1016/0022-1236(82)90009-X
Walter D. Burgess and David E. Handelman, The $N^{\ast }$-metric completion of regular rings, Math. Ann. 261 (1982), no. 2, 235–254. MR 675737, DOI 10.1007/BF01456221
Lawrence G. Brown and Gert K. Pedersen, $C^*$-algebras of real rank zero, J. Funct. Anal. 99 (1991), no. 1, 131–149. MR 1120918, DOI 10.1016/0022-1236(91)90056-B
Victor P. Camillo and Hua-Ping Yu, Stable range one for rings with many idempotents, Trans. Amer. Math. Soc. 347 (1995), no. 8, 3141–3147. MR 1277100, DOI 10.1090/S0002-9947-1995-1277100-2
Victor P. Camillo and Hua-Ping Yu, Exchange rings, units and idempotents, Comm. Algebra 22 (1994), no. 12, 4737–4749. MR 1285703, DOI 10.1080/00927879408825098
Peter Crawley and Bjarni Jónsson, Refinements for infinite direct decompositions of algebraic systems, Pacific J. Math. 14 (1964), 797–855. MR 169806, DOI 10.2140/pjm.1964.14.797
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
K. R. Goodearl, von Neumann regular rings, Monographs and Studies in Mathematics, vol. 4, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1979. MR 533669
K. R. Goodearl, Partially ordered abelian groups with interpolation, Mathematical Surveys and Monographs, vol. 20, American Mathematical Society, Providence, RI, 1986. MR 845783, DOI 10.1090/surv/020
K. R. Goodearl, Simple regular rings and rank functions, Math. Ann. 214 (1975), 267–287. MR 409541, DOI 10.1007/BF01352109
B. J. Gardner, Radical classes of regular rings with Artinian primitive images, Pacific J. Math. 99 (1982), no. 2, 337–349. MR 658064, DOI 10.2140/pjm.1982.99.337
K.R.Goodearl, $C^{*}$-algebra’s of real rank zero whose $K_0$ are not Riesz groups, Canad. Math. Bull. 39 (1996), 429–437.
K. R. Goodearl and D. Handelman, Simple self-injective rings, Comm. Algebra 3 (1975), no. 9, 797–834. MR 379593, DOI 10.1080/00927877508822074
K. R. Goodearl and D. E. Handelman, Metric completions of partially ordered abelian groups, Indiana Univ. Math. J. 29 (1980), no. 6, 861–895. MR 589651, DOI 10.1512/iumj.1980.29.29060
K. R. Goodearl and R. B. Warfield Jr., Algebras over zero-dimensional rings, Math. Ann. 223 (1976), no. 2, 157–168. MR 412230, DOI 10.1007/BF01360879
Israel Halperin, Regular rank rings, Canadian J. Math. 17 (1965), 709–719. MR 191926, DOI 10.4153/CJM-1965-071-4
John Hannah and K. C. O’Meara, Depth of idempotent-generated subsemigroups of a regular ring, Proc. London Math. Soc. (3) 59 (1989), no. 3, 464–482. MR 1014867, DOI 10.1112/plms/s3-59.3.464
J. Moncasi, A regular ring whose $K_0$ is not a Riesz group, Comm. Algebra 13 (1985), no. 1, 125–131. MR 768090, DOI 10.1080/00927878508823152
W. K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229 (1977), 269–278. MR 439876, DOI 10.1090/S0002-9947-1977-0439876-2
W. K. Nicholson, $I$-rings, Trans. Amer. Math. Soc. 207 (1975), 361–373. MR 379576, DOI 10.1090/S0002-9947-1975-0379576-9
E. Pardo, On a density condition for $K^+_0$ of von Neumann regular rings, Comm. Algebra 22 (1994), no. 2, 707–719. MR 1255888, DOI 10.1080/00927879408824870
E.Pardo, On the representation of simple Riesz groups, Comm. Algebra (to appear).
Marc A. Rieffel, Dimension and stable rank in the $K$-theory of $C^{\ast }$-algebras, Proc. London Math. Soc. (3) 46 (1983), no. 2, 301–333. MR 693043, DOI 10.1112/plms/s3-46.2.301
Marc A. Rieffel, The cancellation theorem for projective modules over irrational rotation $C^{\ast }$-algebras, Proc. London Math. Soc. (3) 47 (1983), no. 2, 285–302. MR 703981, DOI 10.1112/plms/s3-47.2.285
Josef Stock, On rings whose projective modules have the exchange property, J. Algebra 103 (1986), no. 2, 437–453. MR 864422, DOI 10.1016/0021-8693(86)90145-6
Joan Torrens, On the $N^\ast$-metric completion of regular rings, Arch. Math. (Basel) 47 (1986), no. 6, 529–534. MR 871291, DOI 10.1007/BF01189862
J.Villadsen, Simple $C^*$-algebras with perforation, Preprint, 1995.
R. B. Warfield Jr., Exchange rings and decompositions of modules, Math. Ann. 199 (1972), 31–36. MR 332893, DOI 10.1007/BF01419573
Friedrich Wehrung, Injective positively ordered monoids. I, II, J. Pure Appl. Algebra 83 (1992), no. 1, 43–82, 83–100. MR 1190444, DOI 10.1016/0022-4049(92)90104-N
Friedrich Wehrung, Injective positively ordered monoids. I, II, J. Pure Appl. Algebra 83 (1992), no. 1, 43–82, 83–100. MR 1190444, DOI 10.1016/0022-4049(92)90104-N
F.Wehrung, Tensor products of structures with interpolation, Pacific J.Math. 176 (1996), 267–285.
Shuang Zhang, Matricial structure and homotopy type of simple $C^*$-algebras with real rank zero, J. Operator Theory 26 (1991), no. 2, 283–312. MR 1225518
Shuang Zhang, Diagonalizing projections in multiplier algebras and in matrices over a $C^*$-algebra, Pacific J. Math. 145 (1990), no. 1, 181–200. MR 1066403, DOI 10.2140/pjm.1990.145.181