tailieunhanh - Báo cáo toán học: "The Ext groups of the C*-algebras associated with irrational rotations "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Các nhóm máy lẻ của C *- đại số kết hợp với phép quay phi lý. | J OPERATOR THEORY 30980 271 274 Copyright by INCREST 1980 THE EXT GROUPS OF THE C -ALGEBRAS ASSOCIATED WITH IRRATIONAL ROTATIONS SORIN POPA and MARC A. RIEFFEL Let Ớ be an irrational number and let Ag denote the crossed product -algebra 9 for the action of the group of integers z on the algebra of continuous functions on the circle by powers of the rotation of angle 2tzS. It is important for this note that Ag is isomorphic to the c -algebra generated by any two unitaries u and V which satisfy vu ẢUV where Ằ e3Jli0. This can be seen by associating to such u and V the unitaries u and V in A corresponding respectively to rotation by 2tiB and to the identity function which embeds the circle as the unit circle in the complex plane. The purpose of this note is to calculate the Ext groups 3 1 of theXfl and of the n X n matrix algebras over the Ag. For related considerations see 14 It is known that the Ag are simple 7 15 Furthermore they are strongly amenable hence nuclear 17 so that their Ext semigroups are in fact groups 6 1 . In addition the Ag are quasi-diagonal. This follows from results announced by Hadwin and also from the stronger fact obtained in 13 that the Ag can be embedded in AF algebras. This quasi-diagonality is important to US since it enables US to use the homotopy results from 12 Let LiK H denote the Calkin algebra for a separable Hilbert space H and let Exts Ag and Extw Ag denote the strong and weak Ext groups of Ag whose elements consist of equivalence classes of unital -monomorphisms called extensions of Ag into L iK H . Observe that there is a bijection between extensions and the pairs C7 V of unitaries in IJK H satisfying vu 2UV. Observe also that if ind denotes the index function on unitaries in LiK H and if w is a unitary in Ag then ind r w depends only on the class r in Exts a of the extension T. It follows easily that the map p from Exts la to z X z defined by p M ind r w ind r v is a well-defined group homomorphism. Theorem. The map p defined .