Let *A* be a C*-algebra and let *A _{t}* 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

*A*

_{sup}consisting of the elements

*a*in

*A*for which ||

*a*||

_{sup}:=sup

_{t}||

*a*||

_{t}< $\infty$ is dense in

*A*.

Will the algebra *A*_{sup} be also stable under the holomorphic functional calculus, or some additional considerations are needed?