TAILIEUCHUNG - Báo cáo toán học: "On the topological stable rank of irrational rotation algebras "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Ngày thứ hạng ổn định topo đại số luân chuyển không hợp lý. | J. operator theory 13 1985 143-150 Copyright by INCREST 1985 ON THE TOPOLOGICAL STABLE RANK OF IRRATIONAL ROTATION ALGEBRAS NORBERT RIEDEL INTRODUCTION In this paper we are concerned with the question whether the invertible elements in the irrational rotation algebras are dense or not or to speak in the terminology of Rieffel s theory of the topological stable rank of c -algebras 3 is the topological stable rank of the irrational rotation algebras equal to 1 or 2 cf. 3 . Using an elementary fact in continued fraction theory namely there are irrational numbers a admitting an arbitrarily high order of approximation by rationals in the sense of diophantine approximations we shall show that there are irrational rotation algebras having a dense set of invertible elements. Actually we shall show that for these irrational rotation algebras stfa the invertible polynomials in the two canonical generators uand V of sđa and their adjoints 7 and u are dense in the set of all polynomials. Since the spectra of those polynomials turn out to be very unstable depending on the rotation number a we cannot expect the proof will provide us with explicit formulas for the spectra of polynomials. As an example for this instability we want to mention Hofstadter s empirical investigation of the spectrum of u V u y in 1 Therefore our method of proof will be based on general arguments and very rough estimates. 1 For each n 6 N we define inductively a function ự on N as follows Ml 2 A Ẩ 2n- -W4ỵ11A 0 k ĩ. i l Moreover we define a function Í on N by p n 2 5 27i n4 -1 144 NORBERT RIEDEL Continued fraction theory shows that there are irrational numbers a 0 such that 0-1 a - PM where p qn p and qn are relatively prime integers denotes the convergent of order n of a cf. 2 proof of Theorem 22 . Our main purpose is to prove the following theorem. . Theorem. For any irrational number a 0 satisfying the invertible elements in the corresponding irrational rotation algebra stf are dense in .

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