Topological K-theory is a generalised cohomology theory on the category of compact Hausdorff spaces. It classifies the vector bundles over a space $X$ up to stable equivalences. Equivalently, via the Serre-Swan theorem, it classifies the finitely generated projective modules over the $C^*$ -algebra $C(X)$ .

Let $A$ be a unital $C^*$ -algebra over $\Cset$ and denote by $\Matrix{\infty}{A}$ the algebraic direct limit of matrix algebras $\Matrix{n}{A}$ under the embeddings . Identify the completion of $\Matrix{\infty}{A}$ with the stable algebra $A\otimes\Kset$ (where $\Kset$ is the compact operators on $l_2(\Nset)$ ), which we will continue to denote by $\Matrix{\infty}{A}$ . The $K_0(A)$ group is the Grothendieck group (abelian group of formal differences) of the homotopy classes of the projections in $\Matrix{\infty}{A}$ . Two projections $p$ and $q$ are homotopic if there exists a norm continuous path of projections from $p$ to $q$ . Let $p \in \Matrix{m}{A}$ and $q \in \Matrix{n}{A}$ be two projections. The sum of their homotopy classes $[p]$ and $[q]$ is the homotopy class of their direct sum: $[p]+[q] = [p \oplus q]$ where $p \oplus q = \mathrm{diag}(p,q) \in \Matrix{m+n}{A}$ . Alternatively, one can consider equivalence classes of projections up to unitary transformations. Unitary equivalence coincides with homotopy equivalence in $\Matrix{\infty}{A}$ (or $\Matrix{n}{A}$ for $n$ large enough).

Denote by $\Ugrp_\infty(A)$ the direct limit of unitary groups $\Ugrp_n(A)$ under the embeddings . Give $\Ugrp_\infty(A)$ the direct limit topology, i.e. a subset $U$ of $\Ugrp_\infty(A)$ is open if and only if $U \cap \Ugrp_n(A)$ is an open subset of $\Ugrp_n(A)$ , for all $n$ . The $K_1(A)$ group is the Grothendieck group (abelian group of formal differences) of the homotopy classes of the unitaries in $\Ugrp_\infty(A)$ . Two unitaries $u$ and $v$ are homotopic if there exists a norm continuous path of unitaries from $u$ to $v$ . Let $u \in \Ugrp_m(A)$ and $v \in \Ugrp_n(A)$ be two unitaries. The sum of their homotopy classes $[u]$ and $[v]$ is the homotopy class of their direct sum: $[u]+[v] = [u \oplus v]$ where $u \oplus v = \mathrm{diag}(u,v) \in \Ugrp_{m+n}(A)$ . Equivalently, one can work with invertibles in $\GLgrp_\infty(A)$ (an invertible $g$ is connected to the unitary $u = g|g|^{-1}$ via the homotopy $t \to g|g|^{-t}$ ).

Higher K-groups can be defined through repeated suspensions, \begin{equation} K_n(A) = K_0(S^n A). \end{equation}But, the Bott periodicity theorem means that \begin{equation} K_1(SA) \cong K_0(A). \end{equation} The main properties of $K_i$ are: \begin{eqnarray} K_i(A \oplus B) & = & K_i(A) \oplus K_i(B), \\ K_i(\Matrix{n}{A}) & = & K_i(A) \quad\mbox{(Morita invariance)}, \\ K_i(A \otimes \Kset) & = & K_i(A) \quad\mbox{(stability)}, \\ K_{i+2}(A) & = & K_i(A) \quad\mbox{(Bott periodicity)}. \end{eqnarray} There are three flavours of topological K-theory to handle the cases of $A$ being complex (over $\Cset$ ), real (over $\Rset$ ) or Real (with a given real structure). \begin{eqnarray} K_i(C(X,\Cset)) & = & \mathit{KU}^{-i}(X) \quad\mbox{(complex/unitary)}, \\ K_i(C(X,\Rset)) & = & \mathit{KO}^{-i}(X) \quad\mbox{(real/orthogonal)}, \\ \mathit{KR}_i(C(X),J) & = & \mathit{KR}^{-i}(X,J) \quad\mbox{(Real)}. \end{eqnarray} Real K-theory has a Bott period of 8, rather than 2.

Bibliography

1

N. E. Wegge-Olsen, K-theory and $C^*$ -algebras.
Oxford science publications. Oxford University Press, 1993.

2

B. Blackadar, K-Theory for Operator Algebras.
Cambridge University Press, 2nd ed., 1998.

3

M. Rørdam, F. Larsen and N. J. Laustsen, An Introduction to K-Theory for $C^*$ -Algebras.
Cambridge University Press, 2000.