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**moesizlac** If I have a Banach algebra X which contains the element x, the "sub-banach algebra generated by x" is the smallest set containing 'x' which is a Banach algebra, right? But is there a more explicit way to write it? For example I thought maybe

[tex]\{\alpha x+\beta e: \alpha,\beta\in\mathbb{C}\}[\math] with 'e' denoting the unit.

That set is linear... But I'm not sure about some of the other properties it needs to be Banach algebra... for example is it complete?

... Or am I on the wrong track completely here? I haven't been able to find any concrete material on what exactly a sub-algebra generated by an element is. Any help will be much appreciated. Thanks!