Direct Sum of $C^*$-algebras.

Question

If a have two $C^*$-algebras, I can do the external direct sum of then. My question is when you have two sub-$*$-algebras $A,B$ of a $C^*$-algebra $C$, what is exactly the concept of direct sum? To be more precise, I have $C = A + B$ and $A \cap B = \{0\}$? I know that if $AB = 0$ then $A \cap B = \{0\}$ but, the reverse is true? I'm asking this question because in a exemple that i'm trying to do this sum, i have two sub-*-algebras that $AB \neq 0$. Is this condition necessary?

Answer 1

You are mixing the concept of direct sum as vector spaces (which only requires $A\cap B=\{0\}$) with direct sum as C$^*$-algebras (which requires $AB=\{0\}$).