Operator K-theory

Definition

Let $A$ be a unital C*-algebra, and let $M_n(A)$ be the C*-algebra of $n \times n$ matrices over $A$. Embed $M_n(A)$ in $M_{n+1}(A)$ by $a \mapsto \begin{pmatrix} a & 0 \\ 0 & 0 \end{pmatrix}$ and let $M_\infty(A) = \bigcup_n M_n(A)$.

1. A projection is an element $p$ with $p = p^2 = p^*$. Two projections $p, q \in M_\infty(A)$ are Murray–von Neumann equivalent if $p = v^* v$ and $q = v v^*$ for some $v \in M_\infty(A)$. The classes form a commutative monoid under $[p] + [q] = [\operatorname{diag}(p, q)]$, and $K_0(A)$ is its Grothendieck group.
2. Let $U_n(A)$ be the group of unitaries of $M_n(A)$, embedded in $U_{n+1}(A)$ by $u \mapsto \operatorname{diag}(u, 1)$, and let $U_\infty(A) = \bigcup_n U_n(A)$. Then $K_1(A)$ is the group of classes of $U_\infty(A)$. Here $u$ and $w$ in $U_n(A)$ are identified when they are joined by a continuous path in $U_m(A)$ for some $m \ge n$.
3. If $A$ is not unital, let $\tilde{A}$ be its unitization; then $K_0(A)$ is the kernel of the map $K_0(\tilde{A}) \to K_0(\C)$ induced by $\tilde{A} \to \C$, and $K_1(A) = K_1(\tilde{A})$.

Operator K-theory is the pair of functors $A \mapsto K_0(A)$ and $A \mapsto K_1(A)$.