Bott projection as $K_1$ class

Question

Consider the Bott projection (described in Exercise 5.I of Wegge-Olsen's book $K$-theory and $C^*$-algebras) given by $b(z)=\frac{1}{1+|z|^2}\begin{pmatrix} 1 & \bar{z} \\ z & |z|^2 \end{pmatrix}$. Regarded as an element in $C_0(\mathbb{R}^2,M_2(\mathbb{C}))$ via $z=x+iy$, $b$ represents a class in $K_0(C_0(\mathbb{R}^2))$. Under the suspension isomorphism $K_1(A)\cong K_0(SA)$ for any $C^*$-algebra $A$, which tells us in particular $K_0(C_0(\mathbb{R}^2))\cong K_1(C(S^1))$, what is the class corresponding to $b$ in $K_1(C(S^1))$? I think it should be the class of the function $z\mapsto\bar{z}$ but I cannot prove it.