Let $K\subset(0,+\infty)$ be a non-empty compact set. Consider the $C^*$-algebra $C(K)$ with the standard uniform norm $\|\cdot\|$. Let $\mathcal{P}\subset C(K)$ be a dense set of polynomials. Is it possible to construct another strong Banach norm $\|\cdot\|_1\ge\|\cdot\|$ such that the Banach algebra $H=\overline{\mathcal{P}}^{\|\cdot\|_1}\subset C(K)$ has a different spectrum, e.g. $\sigma_{H}(x)\neq\sigma_{C(K)}(x)$ for the monomial $x$?
(It is obvious that $H$ should be non-closed in $C(K)$ and the norm $\|\cdot\|_1$ should not be equivalent to the norm $\|\cdot\|$.)
It is sufficient to construct $\|\cdot\|_1$ such that $1/x$ can not be approximated by polynomials in this norm, i.e. $1/x\not\in H$. But the norm should be Banach $\|ab\|_1\le\|a\|_1\|b\|_1$.
Thanks.
Yes, the idea you mentioned in your comment works: $K=[1/2,2/3]$ and $||\sum_{n=0}^Na_nx^n||_1=\sum_0^N|a_n|$.
You get $H$ equal to the set of all $f$ of the form $f(x)=\sum_{n=0}^\infty a_nx^n$, with $\sum|a_n|<\infty$. In particular every $f\in H$ extends to a holomorphic function in the unit disk, and hence $1/x\notin H$ by elementary complex analysis.
(In fact $H\cong \ell_1(\mathbb N)$, and hence as is well known $\sigma(H)$ is the closed unit disk.)