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Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Đồng nhất C *- Phần mở rộng $ C (X) \ otimes K (H). phần I. | J. OPERATOR THEORY 1 1979 55-108 Copyright by INCREST 1979 HOMOGENEOUS C -EXTENSIONS OF C X X H . PART I. M.PIMSNER S.POPA and D.VOICULESCU The remarkable work of L. G. Brown R. G. Douglas and p. A. Fillmore 10 12 on extensions of the ideal of compact operators by commutative c -algebras has stimulated further research concerning more general extensions 1 3 4 9 13 16 20 26 34 39 41 47 . This is motivated in part by the desire to extend the Brown-Douglas-Fillmore theory so as to provide a tool for analysing the structure of c -algebras. In particular such a development might lead to a better understanding of the structure of type I c -algebras. Also we should mention the general program for the study of extensions sketched by L. G. Brown in ref. 9 . A class of extensions to be studied as suggested in ref. 26 are those of C X K H . Among these the homogeneous extensions considered here seem to be more tractable. Let US explain what the homogeneity requirement means. Roughly speaking an extension of C X K tT by a c -algebra A separable and with unit gives rise for each X e X to an extension of K H by some quotient A Jx of A. The map which associates to X e X the ideal Jx will be called the ideal symbol of the extension. The extension is called homogeneous if Jx 0 for all xeX. Under a suitable equivalence relation and with some additional conditions on X and A X finite-dimensional and A nuclear the homogeneous extensions yield a group Ext A A which will be the main object of our study. For X reduced to a point this is just the Brown-Douglas-Fillmore group but the consideration of the more general Ext X A will be seen in Part II to be also of some interest for the study of the usual extensions by K H . Passing now to the results of Part I of this paper we should mention a Weyl-von Neumann type theorem for rather general not only homogeneous extensions of C X a short exact sequence for Ext X A in the - variable for general nuclear c -algebras this is new also for the .
