a question on the definition of direct sum of $C^*$ algebras

Question

According to the definition in the Olsen's book,if $A=B\bigoplus C$,the intersection of $B$ and $C$ should be zero.But in other reference books ,when talking about direct sum of matrix algebras,there are many examples such as $\Bbb C\bigoplus\Bbb C$,$M_2(\Bbb C)\oplus M_2(\Bbb C)$,the intersection is not zero. What are the differences between two definitions?

Answer 1

The same term "direct sum" is used for two different things (that, in the end, are basically the same in spirit).

If you have two subspaces $B,C\subset A$, you say that $A$ is the (internal) direct sum of $B$ and $C$ if $A=B+C$ and $B\cap C=\{0\}$.

If, on the other hand, you have two spaces (not necessarily living in the same environment) you may construct the (external) direct sum $A=B\oplus C=\{(b,c):\ b\in B,\ c\in C\}$. Now $A$ is an internal direct sum $A=(A\oplus 0) + (0\oplus B)$.

Answer 2

By definition, $\mathbb C \oplus \mathbb C$ is the set of ordered pairs $(x,y)$ with $x$ and $y$ both in $\mathbb C$.

Inside $\mathbb C \oplus \mathbb C$, there are two copies of $\mathbb C$. One of them is $\{ (x,0) : x \in \mathbb C \}$, the other one is $\{ (0,x) : x \in \mathbb C \}$. These have intersection $(0,0)$ as required.