Suppose that we have two Banach algebras A and B and subsets S_1 of A and S_2 such that there exists a "natural" bijection between S_1 and S_2. We are interested in A_1 and B_1 which are smallest closed subalgebras of A (B) generated by S_1 (S_2). In my particular case I would like to show that A_1 and A_2 are (isometrically) isomorphic. Problem is that in my situation I don't have complete description of neither A_1 or A_2. Do you have any suggestions on how to tackle problem of this form?

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general; so you're going to have to tell us more background I think... – Matthew Daws May 23 '11 at 19:46