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

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

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

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

From the book, it is stated there that $A\oplus \mathbb{C}$ can be a Banach *-algebra under many norms. Can anyone give me any example of norms that can make $A\oplus \mathbb{C}$ a Banach *-algebra?

2. The usual p-norms ( $\| (a,b) \|= \left( \| a\|^p+\| b\| ^p\right)^{\frac{1}{p}}$) 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).