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Let $A$ and $B$ be two finite-dimensional C*-algebras. Let $\gamma$ denote the projective Banach space tensor product norm on the algebraic tensor product $A\odot B$, so $\gamma(t)=\inf\{\sum_{i}\|a_ …

A useful reference is Berberian's book "Baer *-rings". 1 - If $x^*x=1$, then $x$ is certainly a partial isometry, since $xx^*x=x$. By taking $e=1$ and $f=xx^*$ in Proposition 1.9 of Berberian, we obt …

The following might be useful. For a C*-algebra $A$, the following statements are equivalent: 1) $A$ is scattered; 2) the spectrum of any self-adjoint element in $A$ is countable 3) all maximal com …

I cannot answer your question, but maybe the following remarks are useful. Firstly, it is important to specify what is meant with "up to isomorphism." I assume that you mean *-isomorphisms if you ment …

I am not sure whether your definition of a Baer *-ring is correct. In the book of Berberian on Baer *-rings a ring $A$ is called a Baer *-ring if 1) $A$ is a *-ring (not necessarily an algebra), i.e. …

Yes, this follows from Theorem 4.2 in Dye's Theorem and Gleason's Theorem for AW*-algebras by Jan Hamhalter. Link: https://arxiv.org/abs/1408.4597 Here the statement is: given an AW*-algebra $A$ with …