THE FUNDAMENTAL GROUP OF THE AUTOMORPHISM GROUP OF A NONCOMMUTATIVE TORUS THE FUNDAMENTAL GROUP OF THE AUTOMORPHISM GROUP OF A NONCOMMUTATIVE TORUS

THE FUNDAMENTAL GROUP OFTHE AUTOMORPHISM GROUP OFA NONCOMMUTATIVE TORUSTHE FUNDAMENTAL GROUP OFTHE AUTOMORPHISM GROUP OFA NONCOMMUTATIVE TORU

Assume that each completely irrational noncommutative torus is realized as an inductive limit of circle algebras, and that for a completely irrational noncommutative torus Aω of rank m there are a completely irrational noncommutative torus Aρ of rank m and a positive integer d such that tr（Aω） = tr（Aρ）. It is proved that the set of all C*-algebras of sections of locally trivial C*-algebra bundles over S2 with fibres Aω. has a group structure, denoted by π1（Aut（Aω.））, which is isomorphic to Z if d > 1 and {0} if d > 1. Let Bcd be a cd-homogeneous C*-algebra over S2 x T2 of which no non-trivial matrix algebra can be factored out. The spherical noncommutative torns Sρcd is defined by twisting C*（T2 x Zm-2） in Bcd C* （Z（m-2）） by a totally skew multiplier ρ on T2 x Z（m-2）. It is shown that Sρcd Mp∞ is isomorphic to C（S2） C* （T2 x Zm-2, ρ） Mcd（C） Mp∞ if and only if the set of prime factors of cd is a subset of the set of prime factors of p.