Comparison Between Algebraic and Topological K-Theory for Banach Algebras and C*-Algebras

Abstract

For a Banach algebra, one can define two kinds of K-theory: topological K-theory, which satisfies Bott periodicity, and algebraic K-theory, which usually does not. It was discovered, starting in the early 80’s, that the “comparison map” from algebraic to topological K-theory is a surprisingly rich object. About the same time, it was also found that the algebraic (as opposed to topological) K-theory of operator algebras does have some direct applications in operator theory. This article will summarize what is known about these applications and the comparison map.