Holomorphic stability of inverse limit of pre-$C^*$-algebras

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?

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You probably already know this, but I think this is if and only if $A_{\sup}$ has spectral permanence-- i.e. for each $a\in A_{\sup}$, the spectrum of $a$ in either $A$ or $A_{\sup}$ is the same. But that doesn't (to me) seem easy to check. –  Matthew Daws Mar 17 '11 at 10:51
Yep, that's the point. I've tried several techniques, in particular considering the MacLaurin series of the function in the point $\zeta \in \mathrm{C}$\Sp(a), and then making an estimation, but apparently this doesn't work. In fact I'm free to use the notion of holomorphic stability by Blackadar and Cuntz since I have the families of $A_\sup$ norms. But I just wanted to know, whether this information is redundant. –  Kolya Ivankov Mar 17 '11 at 16:13