Isomorphism between compact operators and compact operators tensor matrices ($\mathbb{K}\otimes M_n(\mathbb{C})\cong \mathbb{K}$)

Question

Let $\mathbb{K}$ be the compact operators and $M_n(\mathbb{C})$ the complex valued matrices. I have read the algebra $\mathbb{K}\otimes M_n(\mathbb{C})$ is isomorphic to $\mathbb{K}$. Could you tell me the isomorphism? I thank you in advance for the help.

Answer 1

Of course, we are assuming an infinite-dimensional Hilbert space $H$ here.

Since both $H$ and $H^n$ have the same dimension, there exists a unitary map $V:H\to H^n$. So we can define $\pi:B(H)\to B(H^n)$ by $$ \pi(T)=VTV^*. $$ Since $V$ is a unitary, $\pi(T)$ is an injective $*$-homomorphism.

So it remains to show that $\pi(\mathbb K)=M_n(\mathbb K)$. If $T\in\mathbb K$, then $VTV^*$ is compact because pre and post-composition with a compact is compact. And if $T'\in B(H^n)$ is compact, then so is $V^*T'V\in \mathbb K$, and $\pi(V^*T'V)=T'$.