The letter $\Bbb K$ in Bruce Blackadar, on operator algebra denotes the algebra of compact operators on a separable infinite dimensional hilbert space, $H$.
In my other post, it is shown that $M_\infty(\Bbb C)$ is isomoprhic to $\Bbb K$ set-wise.
My questions are as follows:
(i) We can give $M_\infty(\Bbb C)$ a $*$-algebra structure. If each embedding is an isometric isomorphism, in the colimit diagram, we can also give it the norm. Does this also make $M_\infty(\Bbb C)$ a $C^*$ algebra?
(ii) How is $\Bbb K$ independent of choice of $H$? Where is separablility used?
Partial replies or references are appreciated.