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 infinite simple C*-algebras, real rank zero simple AT algebras, and AF algebras, ifuandvare almost commuting unitaries whereuhas trivialK1-class,vhas full spectrum, and a certainK0-valued obstruction associated to the pairu,vis trivial, thenucan be deformed to 1 through a path of unitaries in the algebra almost commuting withv, and the length of the path can be estimated by a universal constant. This result is used to identify the obstruction with Loringʹs Bott element associated to the pairu,vand also to prove the more universal statement that if (ui, vi),i=1, 2, are two pairs of almost commuting unitaries with [ui]1, [vi]1, and Bott (ui, vi) each independent ofi, then one pair can be deformed into the other along a path of pairs of almost commuting unitaries in the algebra, the length of the path being bounded by a universal constant.