Homotopy of a pair of approximately commuting unitaries in a simple C*-algebra

It is shown that, for a class of unital C*-algebras including purely infini te simple C*-algebras, real rank zero simple AB algebras, and AT algebras, if u and v are almost commuting unitaries where u has trivial K-1-class, v has full spectrum, and a certain K-o-valued obstruction associated to the p air u, v is trivial, then u can be deformed to 1 through a path of unitarie s in the algebra almost commuting with v, and the length of the path can be estimated by a universal constant. This result is used to identify the obs truction with Luring's Bott clement associated to the pair u, v and also to prove the more universal statement that if (u(i), v(i)), i = 1, 2, are two pairs of almost commuting unitaries with [u(i)](1),[v(i)](1), and Bott (u( i), v(i)) each independent of i. then one pair can be deformed into the oth er along a path of pairs of almost commuting unitaries in the algebra, the length of the path being bounded by a universal constant. (C) 1998 Academic Press.