Rings of quotients of finite $AW^*$-algebras. Representation and algebraic approximation

We show that Berberian's $*$-regular extension of a finite $AW^*$-algebra admits a faithful representation, matching the involution with adjunction, in the $\mathbb C$-algebra of endomorphisms of a closed subspace of some ultrapower of a Hilbert space. It also turns out that this extension is a homomorphic image of a regular subalgebra of an ultraproduct of matrix $*$-algebras $\mathbb C^{n\times n}$.