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Thread: Norm[Wegge-Olsen K theory And C*-algebras]

  1. #1
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    Norm[Wegge-Olsen K theory And C*-algebras]

    Suppose we have a Banach algebra A without unit. We can always embed $\displaystyle A$ into an algebra $\displaystyle A\oplus \mathbb{C} $ with unity such that the elements in $\displaystyle A\oplus \mathbb{C} $ are of the form $\displaystyle (a,\alpha),$ $\displaystyle a\in A ,\alpha \in \mathbb{C}. $

    We know that under the norm $\displaystyle \|(a,\alpha)\| = \|a\|+ |\alpha|$ ,$\displaystyle A\oplus \mathbb{C}$ is a Banach algebra and with the involution $\displaystyle (a,\alpha)^*=(a^*,\bar{\alpha})$, $\displaystyle A\oplus \mathbb{C}$ ia a Banch *-algebra.

    From the book, it is stated there that $\displaystyle A\oplus \mathbb{C}$ can be a Banach *-algebra under many norms. Can anyone give me any example of norms that can make $\displaystyle A\oplus \mathbb{C}$ a Banach *-algebra?
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  2. #2
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    The usual p-norms ($\displaystyle \| (a,b) \|= \left( \| a\|^p+\| b\| ^p\right)^{\frac{1}{p}}$) for the direct sum of two Banach spaces might work (the only thing they might not safisfy would be the C-S type inequality with the product but I haven't really checked it).
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