tailieunhanh - Báo cáo toán học: "Remarks on the singular extension in the C*-algebra of the Heisenberg group "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Nhận xét về phần mở rộng từ trong C *- đại số của nhóm Heisenberg. | Copyright by INCREST 1981 J. OPERATOR THEORY 5 1981 147-170 REMARKS ON THE SINGULAR EXTENSION IN THE C -ALGEBRA OF THE HEISENBERG GROUP DAN VOICULESCU The representation theory of the Heisenberg group shows that its c -algebra C G is an extension of the form 0 -4 C0 R 0 X -4 C G -4 C0 R2 0 where C0 Jf for locally compact X denotes the continuous functions on X which vanish at infinity and X Xf denotes the compact operators on the separable complex infinite-dimensional Hilbert space Xf see for instance 11 . In a certain sense this extension of C0 R 0 is localized at the limit point 0 of R 0 a feature we call singularity in 15 in order to distinguish such extensions from the homogeneous extensions studied in 14 . In this paper we shall exhibit two properties of this extension a rather strong non-splitting property and what we shall call the existence of a limit distribution. The non-splitting property we shall prove is that not only the extension is not split but even the restriction of the extension to a sequence of points in R 0 which converges to 0 is also not split. In particular since C0 R2 is generated by three hermitian elements this implies that there are bounded sequences C4 n i 4 o of compact hermitian operators such that lim II 4 -ổ II lim II 4 c lim II B c 0 -- 00 n- x n- X for which do not exist sequences 7L1 B Ll C LiOĨ commuting compact hermitian operators A n B A C is C i . 0 . such that lim IK - A J lim 114 - B W lim IICn - c ll 0. n-4 x 7 00 - oo This may be of some interest in connection with the open question concerning approximation of two almost commuting compact hermitian operators. We would 148 DAN VOICULESCU like to mention that the rather long ad-hoc argument for the non-splitting result is related in its final part to the non-quasitriangularity of the unilateral shift 9 We mention that the non-splitting of the extension also follows from one of the results announced in Kasparov s short note 10 but the stronger non-splitting result we .