[math] \newcommand{\mathds}{\mathbb}[/math]

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Given a Hilbert space [math]H[/math], prove that we have embeddings of [math]*[/math]-algebras as follows, which are both proper, unless [math]H[/math] is finite dimensional:

[[math]] B(H)\subset\mathcal L(H)\subset M_I(\mathbb C) [[/math]]

Also, prove that in this picture the adjoint operation [math]T\to T^*[/math] takes a very simple form, namely [math](M^*)_{ij}=\overline{M}_{ji}[/math] at the level of the associated matrices.