I can see that it might be possible to identify $\displaystyle B(\mathbb{C}^{n})$ with $\displaystyle M_{n}(\mathbb{C})$

However, I am finding some trouble showing that

$\displaystyle \varphi:B(\mathbb{C}^{n})\rightarrow M_{n}(\mathbb{C})$

is a *-isomorphism...?

Can you think of a specific *-isomorphism $\displaystyle \varphi$ which will identify$\displaystyle B(\mathbb{C}^{n})$ with $\displaystyle M_{n}(\mathbb{C})$?

$\displaystyle B(\mathbb{C}^{n})$ is the set of all bounded linear operators form the Hilbert space $\displaystyle \mathbb{C}^{n}$ to itself.

$\displaystyle M_{n}(\mathbb{C})$ is the space of all $\displaystyle n\times n$ matrices with complex entries.