Endomorphisms of certain irrational rotation $C^{\ast}$-algebras

Abstract

KAZUNORI KODAKA1. Preliminaries for quadratic irrational numbers First we will give definitions and well known facts on quadratic irrational numbers.Let GL(2, Z) be the group of all 2 2-matrices over Z with determinant _+ 1.Let k l] GL(2, Z) g= rn n and 0 be an irrational number.We define m+nO gO k + lO and we call g a fractional transformation.Furthermore if g4: 0 1 0 -1 then we say that g is non-trivial.Let Q be the ring of rational numbers.We suppose that 0 is a quadratic irrational number.If 0=x+yv/-d where x,yQ and dN, then we define 0'= x-yv and we call 0' the conjugate of 0. We say that 0 is reduced if 0 > 1 and -1 < 0' < 0 where 0' is the conjugate of 0.For any quadratic irrational number 0 there are a fractional transforma- tion k GL(2, Z) 177 rt