About X-theory

K-theory was developed by Atiyah and Hirzebruch in the 1960s based on work of Grothendieck in algebraic geometry. It was introduced as a tool in C*-algebra theory in the early 1970s through some specific applications described below. Very briefly, K-theory (for C*-algebras) is a pair of functors, called K0 and K1 that to each C*-algebra A associate two Abelian groups K0(A) and K1(A). The group K0(A) is given an ordering that (in special cases) makes it an ordered Abelian group. There are powerful machines, some of which are described in this book, making it possible to calculate the K-theory of a great many C*-algebras. K-theory contains much information about the individual C*-algebras — one can learn about the structure of a given C*-algebra by knowing its K-theory, and one can distinguish two C*-algebras from each other by distinguishing their K-theories. For certain classes of C*-algebras, K-theory is actually a complete invariant, K-theory is also a natural home for index theory.

Two applications demonstrated the importance of K-theory to C*-algebras. George Elliott showed in the early 1970s (in a work published in 1976, [18]) that AF-algebras (the so-called “approximately finite dimensional” C*-algebras; see Chapter 7 for a precise definition) are classified by their ordered K0-groups. (The K1-group of an AF-algebra is always zero.) As a consequence, all information about an AF-algebra is contained in its ordered K0-group. This result indicated the possibility of classifying a more general class of C*-algebras by their K-theory.