The usual p-norms ( ) 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).
Suppose we have a Banach algebra A without unit. We can always embed into an algebra with unity such that the elements in are of the form
We know that under the norm , is a Banach algebra and with the involution , ia a Banch *-algebra.
From the book, it is stated there that can be a Banach *-algebra under many norms. Can anyone give me any example of norms that can make a Banach *-algebra?