The question is a about the definition of the second topological $K$-group of a Banach algebra $A$.
I was reading a text of Alain Valette (Prop. 3.3.7) where he proves that $$ K_1(SA) \cong \pi_1(\mathrm{GL}_\infty(A),1). $$ Here $A$ is a Banach algebra and $K_1(A) = \mathrm{GL}_\infty(A) / \mathrm{GL}_\infty(A)_0 \cong \pi_0(\mathrm{GL}_\infty(A))$. By $\mathrm{GL}_\infty(A)_0$ one denotes the path component of $1$. The proof goes as follows: $$ \begin{align*} \pi_1(\mathrm{GL}_\infty(A),1) & \cong \pi_0(S\mathrm{GL}_\infty(A)) \\ & \cong \pi_0(\mathrm{GL}_\infty(SA)) \\ & \cong K_1(SA) \end{align*} $$ I am concerned about the following:
- Why is $\pi_0(S\mathrm{GL}_\infty(A))$ a group ?
- What are the first and second isomorphism ?
Remark: This whole question allows us to define $$ K_2^{\mathrm{top}}(A) := \pi_1(\mathrm{GL}_\infty(A),1). $$
Your questions reveal a typo that Valette was not aware of. The third line of the proof of Proposition 3.3.7 should be:$$\pi_n(\text{GL}_\infty(A)) = \pi_{n - 1}(\Omega \text{GL}_\infty(A)) = \pi_{n - 1}(\text{GL}_\infty(SA)),$$where $\Omega X$ is the loop space of $X$. The first equality is a basic property of loop spaces, see e.g. here. The second follows by observing that a loop in the invertible group of $A$ is the same thing as an invertible element in the suspension of $A$ — there are some minor checks regarding units of algebras, but it works.