Comparison between algebraic and topological K-theory for Banach algebras and C ∗ -algebras

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. 1 Some Problems in Operator Theory 1.1 Toeplitz operators and K-Theory The connection between operator theory and K-theory has very old roots, although it took a long time for the connection to be understood. We begin with an example. Think of S1 as the unit circle in the complex plane and let H ⊂ L2(S1) be the Hilbert space H2 of functions all of whose negative Fourier coefficients vanish. In other words, if we identify functions with their formal Fourier expansions, H = n=0 cnz n with n=