Let $A=\mathbb{C}$ and $E=F=C[0,1]$. Are $E=F$ Hilbert $\mathbb{C}^*$-modules?

Question

Let $A=\mathbb{C}$ and $E=F=C[0,1]$. Are $E=F$ Hilbert $\mathbb{C}^*$-modules?

clearly $E=F$ are $C^*$-algebra

Answer 1

Every C*-algebra $A$ is a Hilbert module over itself with $A$-valued inner product given by

$$\langle a,b\rangle = a^*b.$$

Hilbert modules over complex numbers are just Hilbert spaces, hence a C*-algebra is a $\mathbb C$-Hilbert module if and only if it is one-dimensional.