Let $A$ and $B$ be $C^{\ast}$-algebras. It is [well known]( https://mathoverflow.net/questions/247428/maximal-tensor-product-of-simple-calgebras-is-non-simple) that maximal tensor product of simple $C^{\ast}$-algebra need not be simple. So basically the ideal structure of $A\otimes_{max}B$  does not really depends on ideal structure of $A$ and of $B$.

> What is known about ideals of $A\otimes_{max}B$ . 

Any references?