Let $A$ be a semisimple commutative Banach algebra with the maximal ideal space $X$. Further, assume that $A$ is regular i.e. for every closed set $E\subseteq X$ and $x\in X\setminus E$, there is some $a\in A$ such that $\widehat{a}|_E\equiv 0$ and $\widehat{a}(x)=1$ (where $\widehat{a}$ is the Gelfand transform of $a$).

I was wondering if one knows such a Banach algebra $A$ which is not closed under complex conjugate. In other words, there is some $a\in A$ such that for every other $b\in A$, $$\widehat{a}\neq \overline{\widehat{b}}.$$