Suppose $A,B$ are $C^*$-algebras,the multiplication on $A\oplus B$ is defined as following:$(a_1,b_1)(a_2,b_2)=(a_1a_2,b_1b_2),$my question is:how to define the norm on $A\oplus B$ such that $A\oplus B$ is a $C^*$ algebra.
I am still confused about the unitization of a $C^*$– algebra .When $A$ is non-unital ,$\tilde{A}$ is not isomporhic to $A\oplus \Bbb C$.The reason is: according to the multiplication mentioned above,if $\tilde{A}$ is isomporhic to $A\oplus \Bbb C$,then $A$ contains a unit,a contradiction.But if the mulpilication defined above ,how to define the norm of $A\oplus C$ such that it becomes a $C^*$-algebra.
Best Answer
For the product structure in your first paragraph, the norm on $A\oplus B$ given by $$\|(a,b)\|=\max\{\|a\|,\|b\|\},$$ is a $C^*$-norm.
The reason that this does not work for the unitization $\tilde A$ of a $C^*$-algebra $A$ is because of the following: although we define $\tilde A$ to be $A\oplus\mathbb C$ as a vector space (moreover as a vector space with involution), the algebra structure on $\tilde A$ is given by $$(a,\lambda)(b,\mu)=(ab+\mu a+\lambda b,\lambda\mu).$$ Hence the above norm may not (and does not) produce a $C^*$-norm on $\tilde A$. Thus we need to be more clever when trying to norm $\tilde A$.