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Two embedding of non commutative algebra of operators into C*-algebra of bounded operators

Is there a well-established correspondence between C*-algebras (containing identity) and von Neumann algebras?
If it exists, does it hold for non commutative algebras?
If it exists, do we know which if any von Neumann algebra can be “embedded” inside a C*-algebra of operators?
Thanks
P.S. I am not sure the word embedding is right here

There is such a correspondence, but it is not equivalent to the notion of “von Neumann algebra”.
Let $A$ be a C*-algebra, and $\pi$ a non-degenerate representation of $A$ on a Hilbert space $H$. Let $B$ be the C*-subalgebra of $B(H)$ consisting of the elements $x$ satisfying $\pi(a)x=x\pi(a)$ for all $a\in A$. One can check that $B$ is a von Neumann algebra.
Let $M$ be the von

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