Recall that two projections in a C*-algebra are equivalent, denoted if there is a partial isometry such that and Let be the set of equivalence classes of projections and We can define addition on by
where , , and (see  for why this is possible). This gives an abelian monoid structure on with identity . If is unital, then we define where is the Grothendieck group of
To proceed with the nonunital case, first consider that is a covariant functor sending since if is a *-homomorphism, then we can define by
Recall the unitization of where with funky structure. We have a *-homomorphism such that Since is a covariant functor, and by composition induces a covariant functor we have a map
In the nonunital case we then define Now define
If is unital, then the condition is always satisfied, so we have Define a norm on by and let be the connected component (in the sense of the norm topology) containing Then we define
Definition 1. Let be a C*-algebra. We define its suspension, denoted as
Hence these are continuous -valued functions that vanish at infinity (so the the suspension is similar to the topological notion).
Theorem 2. There is an isomorphism defined by where defined by and is a path in between and
in the image refers to the constant map We can then define by induction the higher K-groups: All of these are clearly covariant functors since the definition can be reduced to of the -th suspension of
Corollary 3. Let
be a short exact sequence. Then the induced sequence
is exact in the middle.
Let us define a map by
for some where and be a lift of Then since commutes with since
Proposition 4. makes the slightly longer sequence exact at and
Corollary 5. By induction we obtain maps and a long exact sequence
 Blackadar, Bruce. Operator Algebras. Encyclopedia of Mathematical Sciences. Vol. 122. Springer-Verlag. 2006.