isomorphism

**Mauritzvdworm** · July 19th 2010, 01:46 AM

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

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

**Mauritzvdworm** · July 23rd 2010, 06:31 AM

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

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

**Mauritzvdworm** · August 1st 2010, 12:23 AM

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

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

Then everything works out

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