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 [1] 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

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