Let be a Banach Algebra without unit and be embeddd into a unital Banach Algebra such that the elements in are of the form and

Define the multiplication in by and the involution by where is the conjugate of .

It is well known that under the norm is a Banach algebra. However, if the norm is defined as , is still a Banach Algebra?

I try to prove that does not hold for a specific element in However, I still do not get a correct one. Can anyone help?