1. ## isomorphism

Let $\displaystyle K=K(H)$ with $\displaystyle H$ a infinite dimensional, separable Hilbert space. Let $\displaystyle (e_n)^{\infty}_{n=1}$ be an orthonormal basis for $\displaystyle H$. Let $\displaystyle e_{ij}$ be an operator in $\displaystyle B(H)$ defined by $\displaystyle e_{ij}(x)=\langle x,e_j\rangle e_i$
set $\displaystyle p_n=\sum^{n}_{j=1}e_{jj}$ now show that the following map is a *-isomorphism (bijective *-homomorphism)

$\displaystyle \psi_{n}:M_n(A)\rightarrow p_nKp_n\otimes A, (a_{r,s})\mapsto\sum^{n}_{i,j=1}e_{ij}\otimes a_{r,s}$

2. I managed to show that $\displaystyle \psi_n$ is linear, multiplicative and a bijection (though it took some work)

The only missing part is to show that $\displaystyle \psi(a^*)=\psi(a)^*$
What does $\displaystyle e^*_{ij}$ look like?

3. Let $\displaystyle y\in H$ then consider
$\displaystyle \langle e^{*}_{ij}y,x \rangle=\langle y,e_{ij}x \rangle=\langle y,\langle e_j,x \rangle e_i \rangle=\langle e_j,x \rangle \langle y,e_i \rangle=\langle \langle e_i,y \rangle e_j,x \rangle$

so $\displaystyle e^{*}_{ij}=e_{ji}$

Then everything works out