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I would like to show the following isomorphy but not sure how to go about this:

$\mathbb{K}\cong M_{n}(\mathbb{K})$

Also in Blackadar (Operator Algebras, page 171) he states that this isomorphism induces: $B\cong M_{2}(B)$, $M(B)\cong M_{2}(M(B))$ and $Q(B)\cong M_{2}(Q(B))$, which he calls standard isomorphisms.
In the context or the present question, the notation has the following meaning:

  • $B$ is a $C^\ast$-algebra,
  • $Q$ is the outer multiplier algebra of $B$ (hence $Q = M(B)/B$),
  • $M$ is the multiplier algebra,
  • $M_{n}(\mathbb{K})$ are $n\times n$ matrices with entries from $\mathbb{K}$, and finally
  • $\mathbb{K}$ is the $C^\star$-algebra of compact operators on a separable, infinite-dimensional Hilbert space.

Anyway, I don't know how to show this and cannot find a reference on how to do this. Would be very thankful for a reference, how to do it or even some hint.

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    $\begingroup$ Please introduce the notation. Otherwise, the question is senseless. (Please don't assume that we have that book at hand.) $\endgroup$
    – YCor
    Commented Nov 6, 2023 at 14:16
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    $\begingroup$ What is $\mathbb K$? A field? Depending on what sort of isomorphism you mean, $\mathbb K \cong M_n(\mathbb K)$ is usually false. $\endgroup$
    – LSpice
    Commented Nov 6, 2023 at 14:45
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    $\begingroup$ You still haven't introduced $\mathbb{K}$ (of course everybody understands what is $M_n(K)$ when $K$ is given...) $\endgroup$
    – YCor
    Commented Nov 6, 2023 at 14:50
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    $\begingroup$ Hint: Writing $K(H)$ the algebra of compact operators on the Hilbert space $H$, try to identity $K(H^n)$ with $M_n(K(H))$. $\endgroup$
    – YCor
    Commented Nov 6, 2023 at 15:59
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    $\begingroup$ You may want to work with a explicit description of $H$, such as $\ell^2(\mathbb{N})$. In such a case you may want to construct $n$ (explicit) copies of $H$ within itself, take unitaries between $H$ and the copies (so isometries from $H$ onto a proper closed subspace) and then conjugate by these. (Another hint: think of the phenomena behind Hilbert's hotel, or Dedekind's definition of an infinite set). $\endgroup$
    – Diego Martinez
    Commented Nov 6, 2023 at 20:49

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