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Math Help - Banach Algebra

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    Banach Algebra

    Let A be a Banach Algebra without unit and be embeddd into a unital Banach Algebra A^+ such that the elements in A^+ are of the form (a,\alpha), a \in A and \alpha \in \mathbb{C}.

    Define the multiplication in A^+ by (a,\alpha)(b,\beta)=(ab+\alpha b + \beta a,\alpha \beta) and the involution by (a,\alpha)^*=(a^*,\alpha^-) where \alpha^- is the conjugate of \alpha.

    It is well known that under the norm \|(a,\alpha)\|=\|a\|+|\alpha|, A^+ is a Banach algebra. However, if the norm is defined as \|(a,\alpha)\|=max\{\|a\|,|\alpha|\}, is A^+ still a Banach Algebra?

    I try to prove that  \|(a,\alpha)(b,\beta)\| \le \|(a,\alpha)\|\|(b,\beta)\| does not hold for a specific element in A^+. However, I still do not get a correct one. Can anyone help?
    Last edited by problem; August 3rd 2010 at 10:44 PM.
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