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Math Help - Subalgebras generated by elements

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    Subalgebras generated by elements

    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 gerated by an element is. Any help will be much appreciated. Thanks!
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    Quote Originally Posted by moesizlac View Post
    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!
    There are two possible answers to this question, depending on whether you want the subalgebra to be unital or not. Since you are proposing to put e into the subalgebra, I assume you want the unital version.

    The set that you have defined is the (unital) linear subspace generated by x. It is finite-dimensional (in fact, two-dimensional, with a basis consisting of x and e) and therefore it is complete. But it need not be closed under multiplication. The Banach subalgebra generated by x must contain all the powers of x. Therefore it must also contain the set of all polynomials \alpha_0e + \alpha_1x + \alpha_2x^2 + \ldots + \alpha_nx^n (for all n\geqslant1). The set of all these polynomials is a subalgebra, but it need not be complete. So you then need to take its closure. That way, you get the Banach subalgebra generated by x.
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