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


where p'\in [p], q'\in [q], and p'\perp q' (see [1] for why this is possible).  This gives an abelian monoid structure on V(A) with identity [0].  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


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.

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

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