Motivation for studying Operator $K_1$

Question

I'm struggling to find motivation for studying $K_1$ for C*-algebras (Here I am talking about $K_1$ as the abelian group built from unitaries in $M_\infty(A)$ up to homotopy). Operator $K_0$ is the study of projective modules over C*-algebras up to an appropiate notion of equivalence, why is $K_1$ built the way it is?

Answer 1

The "better" definition for operator $K_1$ is that it's $K_0$ of the suspension of your algebra. In this way you can define $K_n$ as $K_0$ of an $n$-fold suspension. Defining the higher $K$-groups in this way is motivated by what happens for (co)homology theories in algebraic topology and get's you nice long exact sequences. It is then a very nice coincidence that $K_1$ happens to be describable in terms of unitaries. Also, the Bott periodicity Theorem is a non-trivial result which states that there are essentially only two $K$-groups in complex $K$-theory.

As for actual applications of $K_1$, well for one, you get further invariants of your algbera. As an example $K_1(C(\mathbb{T})) \cong \mathbb{Z}$, with the isomorphism here taking the class of a function in $C(\mathbb{T})$ to the winding number of the function: a well-known invariant. In this case $K_1$ is essentially detecting the hole in the circle. In the noncommuatative case this can be a bit whackier. I could go on, but hopefully this gives you some idea of "Why $K_1$?".