$A^p$ is Not an Algebra for $1 \lt p \lt 2$

Abstract

Let $A^p$ be the Banach space of all continuous functions on the torus ${\mathbb T} = \{ z \in {\mathbb C} \vert \vert z \vert = 1 \}$ whose Fourier coefficients are in $\ell ^p$. We show that $A^p$ is not an algebra for all $1 \lt p \lt 2$.