Extensions for AF algebras and dimension groups

Author:
David Handelman

Journal:
Trans. Amer. Math. Soc. **271** (1982), 537-573

MSC:
Primary 46L05; Secondary 06F20, 16A56, 46M20

DOI:
https://doi.org/10.1090/S0002-9947-1982-0654850-0

MathSciNet review:
654850

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let , be approximately finite dimensional algebras, with nonunital and unital; suppose that either (i) is the algebra of compact operators, or (ii) both , are simple. The classification of extensions of by is studied here, by means of Elliott's dimension groups. In case (i), the weak Ext group of is shown to be , and the strong Ext group is an extension of a cyclic group by the weak Ext group; conditions under which either Ext group is trivial are determined. In case (ii), there is an unnatural and complicated group structure on the classes of extensions when has only finitely many pure finite traces (and somewhat more generally).

**[1]**E. Alfsen,*Compact convex sets and boundary integrals*, Springer-Verlag, Berlin, 1977. MR**0445271 (56:3615)****[2]**H. Bass,*Algebraic*-*theory*, Benjamin, New York, 1968. MR**0249491 (40:2736)****[3]**L. G. Brown, R. G. Douglas and P. A. Fillmore,*Extensions of**algebras and*-*homology*, Ann. of Math.**105**(1977), 265-324. MR**0458196 (56:16399)****[4]**E. G. Effros and J. Rosenberg,*algebras with approximate inner flip*, Pacific J. Math.**77**(1978), 417-443. MR**510932 (80k:46065)****[5]**E. G. Effros, D. E. Handelman and C.-L. Shen,*Affine representations of dimension groups*, Amer. J. Math.**102**(1980), 385-407. MR**564479 (83g:46061)****[6]**G. A. Elliott,*On lifting and extending derivations of**algebras*, J. Funct. Anal.**17**(1974), 395-408. MR**0355616 (50:8090)****[6A]**-,*Automorphisms determined by multipliers on ideals of**algebras*, J. Funct. Anal.**23**(1976), 1-10. MR**0440372 (55:13247)****[7]**-,*On the classification of inductive limits of sequences of semisimple finite dimensional algebras*, J. Algebra**38**(1976), 29-44. MR**0397420 (53:1279)****[8]**K. R. Goodearl,*Von Neumann regular rings*, Pitman, London, 1979. MR**533669 (80e:16011)****[9]**-,*Completions of regular rings*, Math. Ann.**220**(1976), 229-252. MR**0409542 (53:13296)****[10]**K. R. Goodearl and D. E. Handelman,*Rank functions and**of regular rings*, J. Pure Appl. Algebra**7**(1976), 195-216. MR**0389965 (52:10794)****[11]**-,*Metric completions of partially ordered abelian groups*, Indiana Univ. Math. J.**29**(1980), 861-895. MR**589651 (82b:06020)****[12]**D. Handelman,*of von Neumann algebras and**algebras*, Quart. J. Math.**29**(1978), 427-441. MR**517736 (81c:46049)****[13]**-,*Free rank**dense subgroups of*, J. Funct. Anal, (to appear).**[14]**M. Pimsner and S. Popa,*On the Ext group of an**algebra*(to appear).**[15]**-,*On the Ext group of**algebras*(to appear).**[16]**E. G. Effros,*Dimensions and**algebras*, C.B.M.S. Regional Conf. Series in Math., no. 46, Amer. Math. Soc., Providence, R. I., 1981. MR**623762 (84k:46042)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
46L05,
06F20,
16A56,
46M20

Retrieve articles in all journals with MSC: 46L05, 06F20, 16A56, 46M20

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1982-0654850-0

Article copyright:
© Copyright 1982
American Mathematical Society