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Let $A$ be a unital, injective $C^*$-algebra.

Recall that $A$ is called injective if whenever $S\subseteq T$ is an embedding of operator systems, and $\phi: S\to A$ is a unital completely positive map, then $\phi$ extends to a unital completely positive map $T\to A$.

In the paper "Topological boundaries of unitary representations" ,in the proof of Proposition 3.16, the authors use the fact that injective $C^*$-algebras are generated by their projections. I could not find a reference for this argument (injective $C^*$-algebras are monotone complete, and therefore also AW* algebras, but I don't know wether it helps).

Thank you for any help.

Shirly Geffen
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1 Answers1

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This holds for $AW^*$-algebras, which include injective $C^*$-algebras as you mentioned. One common way $AW^*$-algebras are defined is that they're the $C^*$-algebras $A$ such that (1) every masa in $A$ is generated by its projections, and (2) every set of orthogonal projections in $A$ has a least upper bound. (Where "is generated by" can be taken to mean "is the norm-closed span of".) (See this note by Saito and Wright for a discussion of the variants on this definition and why some of them are equivalent.)

Now take an element $x$ in an $AW^*$-algebra $A$. Since $x = \frac 12(x+x^*) + i ((\frac{1}{2i})(x-x^*))$, it's okay to assume $x$ is self-adjoint, which implies that the $C^*$-subalgebra of $A$ generated by $x$ is commutative. By Zorn's Lemma, every commutative $C^*$-subalgebra of $A$ is contained in a masa. Applying (1) gives an approximation of $x$ by a linear combination of projections in $A$, which finishes the proof.

Alex Bearden
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