Extensions for AF algebras and dimension groups
Author:
David Handelman
Journal:
Trans. Amer. Math. Soc. 271 (1982), 537573
MSC:
Primary 46L05; Secondary 06F20, 16A56, 46M20
MathSciNet review:
654850
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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).
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 E. Alfsen, Compact convex sets and boundary integrals, SpringerVerlag, Berlin, 1977. MR 0445271 (56:3615)
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 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), 265324. MR 0458196 (56:16399)
 [4]
 E. G. Effros and J. Rosenberg, algebras with approximate inner flip, Pacific J. Math. 77 (1978), 417443. 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), 385407. MR 564479 (83g:46061)
 [6]
 G. A. Elliott, On lifting and extending derivations of algebras, J. Funct. Anal. 17 (1974), 395408. MR 0355616 (50:8090)
 [6A]
 , Automorphisms determined by multipliers on ideals of algebras, J. Funct. Anal. 23 (1976), 110. MR 0440372 (55:13247)
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 , On the classification of inductive limits of sequences of semisimple finite dimensional algebras, J. Algebra 38 (1976), 2944. MR 0397420 (53:1279)
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 K. R. Goodearl, Von Neumann regular rings, Pitman, London, 1979. MR 533669 (80e:16011)
 [9]
 , Completions of regular rings, Math. Ann. 220 (1976), 229252. MR 0409542 (53:13296)
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 , Metric completions of partially ordered abelian groups, Indiana Univ. Math. J. 29 (1980), 861895. MR 589651 (82b:06020)
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 D. Handelman, of von Neumann algebras and algebras, Quart. J. Math. 29 (1978), 427441. 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)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198206548500
PII:
S 00029947(1982)06548500
Article copyright:
© Copyright 1982
American Mathematical Society
