what's spectral axiom

Question

I encounter a proposition in an article:

For any $0\leq x\leq 1$ in a C*-algebra enjoying the spectral axiom, there are projections $(e_n)$ such that $$x=\sum_{n=1}^{\infty}\frac{1}{2^n}e_n.$$

The author says this can be found in p.367, Operator algebras by Stratila and Szido, but I can't find this reference. Can anyone provide some information about it? Or, give the definition of spectral axiom?

Answer 1

The only book by Stratila and Zsido, as far as I know, is "Lectures on von Neumann algebras"; page 367 of that book pertains to the bibliography list.

The statement, with the exact same notation, though, does appear in the aforementioned book in Corollary 2.23. But only for $B(H)$.

Now, this is my guess. The key property to prove what you want is the following:

If $x\geq0$ and $\alpha>0$, there exists a projection $e$ such that $$xe\geq\alpha e, \ \ \ x(1-e)\leq\alpha (1-e). $$

So I would venture that this property is the "spectral axiom" (never seen those words, cannot find them in any C$^*$-algebra text). It's probably a bit weaker than having Borel functional calculus, but strong enough to do many things with projections.