Let A be a C*-algebra and let At be a set of dense *-subalgebras of A, stable under holomorphic functional calculus on A, which are also Banach algebras complete with respect to the norms ||$\cdot$||t. Suppose also that the algebra Asup consisting of the elements a in A for which ||a||sup:=supt ||a||t < $\infty$ is dense in A.
Will the algebra Asup be also stable under the holomorphic functional calculus, or some additional considerations are needed?