Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: K-lý thuyết về hành động của nhóm vòng tròn trên C *- đại số. | J. OPERATOR THEORY 6 1981 135-141 Copyright by INCREST 1981 EXT OF CERTAIN FREE PRODUCT C -ALGEBRAS LAWRENCE G. BROWN 1. In this note we prove an exact sequence which makes it possible to compute Ext l ổ up to a group extension where A B is the amalgamated free product c is finite dimensional and Ext X and Ext fi are groups. Since A B is not generally nuclear it is not possible to deduce this result from the general theory and our proof is quite elementary. The Calkin algebra Q is where is the c -algebra of bounded operators on the separable infinite dimensional Hilbert space and is the ideal of compact operators. The quotient map from to Q is denoted by n. Since all separable infinite dimensional Hilbert spaces are isomorphic we are free to change when convenient. For A a separable c -algebra Ext X is defined as a set of equivalence classes r . Here t a - Q is a -monomorphism and Tj T2 if and only if there is a partial isometry u e Q such that M uTj Tj ứ and T2 a MTi ri M for all a e A. An operation is defined on Ext 4 by means of the direct sum operation for Hilbert spaces and Ext A becomes an abelian semigroup T or r is called trivial if there is a -homomorphism a A - such that T 7TƠ. Voiculescu s theorem 11 shows that there is only one trivial element of Ext T and that it is a unit for Ext 4 . Ext is a contravariant functor. If A has a unit an object ExtsC4 containing equivalence classes ĩ s can be defined similarly. The only differences are that Ĩ and ff are required to be unital and T T2 if and only if there is a unitary Ĩ7e such that Ta 7i C Ti7r Z7 _1. Further details on Ext can be found in 2 3 5 8 9 and other works. Voiculescu s theorem is the only major result we will use. If Pỵ and p2 are projections in such that 7IÍP1 ir p2 then an integer Px p2 called the essential codimension can be defined 4 has the following properties 1. If V Vi 1 vy Pị i 1 2 then Pi p2 index Ị 36 LAWRENCE G. BROWN 2. If p2 A Pj p2 Py P2Ji usual codimension 3. - P 4. Pj .
