Homotopy of a Pair of Approximately Commuting Unitaries in a Simple C*-Algebra

✍ Scribed by Ola Bratteli; George A Elliott; David E Evans; Akitaka Kishimoto

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 603 KB
• Volume: 160
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1998.3261

✦ Synopsis

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, if u and v are almost commuting unitaries where u has trivial K 1 -class, v has full spectrum, and a certain K 0 -valued obstruction associated to the pair u, v is trivial, then u can be deformed to 1 through a path of unitaries 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 obstruction with Loring's Bott element 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 other along a path of pairs of almost commuting unitaries in the algebra, the length of the path being bounded by a universal constant. 1998 Academic Press 1. INTRODUCTION This paper originated in the course of the classification of purely infinite simple C*-algebras obtained as inductive limits of direct sums of algebras article no.