Let me recall the definition of the Bott periodicity isomorphism in the context of $C^*$-algebras. We take a (class of) projection $p \in M_n(A^+)$ and map it to the class of $M_n(A)$ valued loop $f_p$ which is defined by $$f_p(z)=1_n+(z-1)p$$ (here $A^+$ is a unitization of $A$ (even if $A$ already has a unit) and $1_n$ is the identity matrix in $M_n(A)$). Then the Bott isomorphism is defined as $$\beta: K_0(A) \ni [p]-[q] \mapsto [f_pf_q^*] \in K_1(SA).$$ In the classical (meaning: commutative) context Bott isomorphism is defined as multiplication by $[\gamma]-1$ where $\gamma$ is the tautological (complex) line bundle over compex projective space (which is the two sphere $S^2$).
When the (unital) $C^*$-algebra is commutative then it is of the form $C(X)$ for some compact $X$. Then there is a natural isomorphism between $K^0(X)$ and $K_0(C(X))$. Is this isomorphism compatible with the Bott periodicity maps? If the answer is yes, why it is so?
