Red: If then . Thus the linear functional f is bounded and hence continuous for the sup norm.
Blue: The completion of any normed algebra is a normed algebra for that norm.
I've questions about two steps in the proof of the following theorem:
Red shape: From where we know that this functional is continuous?
Blue shape: From where we know that completion of A is a Sub-norm algebra?
PS. Here is link to the original paper: http://matwbn.icm.edu.pl/ksiazki/sm/sm29/sm29126.pdf