# K-Theory of C*-algebras

Recall that two projections $p,q$ in a C*-algebra $A$ are equivalent, denoted $p\sim q,$ if there is a partial isometry $u$ such that $p=u^*u$ and $q=uu^*.$  Let $V_0(A)$ be the set of equivalence classes of projections and $V(A)=V_0(M_\infty(A)).$  We can define addition on $V(A)$ by $[p]+[q]=[p'+q']$

where $p'\in [p]$, $q'\in [q]$, and $p'\perp q'$ (see  for why this is possible).  This gives an abelian monoid structure on $V(A)$ with identity $$.  If $A$ is unital, then we define $K_0(A)=G(V(A))$ where $G(V(A))$ is the Grothendieck group of $V(A).$

To proceed with the nonunital case, first consider that $V:C^*algebras\to Mon$ is a covariant functor sending $A\mapsto V(A),$ since if $\varphi:A\to B$ is a *-homomorphism, then we can define $V(\varphi):V(A)\to V(B)$ by $V(\varphi)([p])=V(\varphi\circ V^{-1}[p]).$

Recall the unitization of $A$ where $A^\dagger=A\oplus\mathbb{C}$ with funky structure.  We have a *-homomorphism $\mu:A^\dagger\to\mathbb{C}$ such that $\ker\mu=0.$  Since $V$ is a covariant functor, and by composition induces a covariant functor $K_0:C^*algebras\to Grp,$ we have a map $\mu_*=K_0(\mu):K_0(A^\dagger)\to K_0(\mathbb{C})=\mathbb{Z}.$

In the nonunital case we then define $K_0(A)=\ker\mu_*.$  Now define $U_n(A)=\{x\in U(M_n(A^\dagger)):x=1_n\mbox{~mod~}M_n(A)\}.$

If $A$ is unital, then the condition is always satisfied, so we have $U_n(A)=U(M_n(A)).$  Define a norm on $M_n(A)$ by $\|x\|=\max_{ij}\|x_{ij}\|$ and let $U_n(A)_0$ be the connected component (in the sense of the norm topology) containing $1_n.$  Then we define $K_1(A)=\lim U_n(A)/U_n(A)_0.$

Definition 1.  Let $A$ be a C*-algebra.  We define its suspension, denoted $SA,$ as $C_0((0,1),A)=C_0(\mathbb{R},A).$

Hence these are continuous $A$-valued functions that vanish at infinity (so the the suspension is similar to the topological notion).

Theorem 2.  There is an isomorphism $\varphi:K_1(A)\to K_0(SA)$ defined by $\varphi([v])=[p]-[q_n]$ where $[p]=[p_t:(0,1)\to A]$ defined by $p_t=w_tq_nw_t^*$ and $w_t$ is a path in $U_{2n}(A)$ between $1_{2n}$ and $diag(v,v^*).$ $[q_n]$ in the image refers to the constant map $q_n:(0,1)\to\{q_n\}.$  We can then define by induction the higher K-groups: $K_{n+1}(A)=K_n(SA).$  All of these are clearly covariant functors since the definition can be reduced to $K_0$ of the $(n+1)$-th suspension of $A.$

Corollary 3.  Let $0\longrightarrow J\stackrel{i}{\longrightarrow} A\stackrel{\pi}{\longrightarrow}A/J\to 0$

be a short exact sequence.  Then the induced sequence $K_1(J)\stackrel{i_*}{\longrightarrow} K_1(A)\stackrel{\pi_*}{\longrightarrow}K_1(A/J)$

is exact in the middle.

Let us define a map $\partial_1:K_1(A/J)\to K_0(J)$ by $\partial_1([u])=[wq_nw^*]-[q_n]$

for some $q_n\in J$ where $u\in U_n(A/J)$ and $w\in U_{2n}(A)$ be a lift of $diag(u,u^{-1}).$  Then $\partial_1([u])\in K_0(J)$ since $diag(u,u^{-1})$ commutes with $q_n$ since $u\notin J.$

Proposition 4 $\partial_1$ makes the slightly longer sequence exact at $K_1(A/J)$ and $K_0(J).$

Corollary 5.  By induction we obtain maps $\partial=\{\partial_n\}$ and a long exact sequence $\cdots K_n(J)\stackrel{i_*}{\to}K_n(A)\stackrel{\pi_*}{\to}K_n(A/J)\stackrel{\partial}{\to}K_{n-1}(J)\stackrel{i_*}{\to}\cdots\stackrel{\pi_*}{\to}K_0(A/J)\to 0.$

  Blackadar, Bruce.  Operator Algebras.  Encyclopedia of Mathematical Sciences.  Vol. 122.  Springer-Verlag.  2006.