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. 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 S 1 as the unit circle in the complex plane and let H ⊂ L 2 (S 1) be the Hilbert space H 2 of functions all of whose negative Fourier coefficients vanish. In other words, if we identify functions with their formal Fourier expansions, H = ∞ n=0 c n z n with ∞ n=0 |c n | 2 < ∞. Now let f ∈ C(S 1) and let M f be the operator of multiplication by f on L 2 (S 1). This operator does not necessarily map H into itself, so let P : L 2 (S 1) → H be the orthogonal projection and let T f = P M f , viewed as an operator from H to itself. This is called the Toeplitz operator with continuous symbol f. In terms of the orthonormal basis e 0 (z) = 1, e 1 (z) =