The following is an exercise in the book "$K$-theory and $C^\ast$-algebras" by Wegge-Olsen:

Let $S$ denote the suspension functor. Suppose that we have natural transformations $\Phi^0$ from the functor $K_0S$ to $K_1$ and $\Phi^1$ from $K_1S$ to $K_0$. Also suppose that for $A=\mathbb{C}$, $\Phi^0_\mathbb{C}:K_0(C_0(\mathbb{R}))\rightarrow K_1(\mathbb{C})$ and $\Phi^1_\mathbb{C}:K_1(C_0(\mathbb{R}))\rightarrow K_0(\mathbb{C})$ are isomorphisms. Then we get Bott periodicity (i.e., $\Phi^0_A$ and $\Phi^1_A$ are isomorphisms for an arbitrary $C^\ast$-algebra $A$).

I check surjectivity as follows:

Given $[p]\in K_0(A)$, consider the homomorphism $\alpha:\mathbb{C}\rightarrow A\otimes\mathbb{K}$ given by $1\mapsto p\otimes e_{11}$, where $\mathbb{K}$ denotes the algebra of compact operators on Hilbert space, and $e_{11}$ is a rank 1 projection. Naturality gives a commutative diagram, which gives $\Phi^1_A\circ(S\alpha)_*\circ(\Phi^1_\mathbb{C})^{-1}([1])=[p]$, i.e., $\Phi^1_A$ is surjective. Similarly, $\Phi^0_A$ is surjective.

However, I need help in figuring out injectivity.


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