Full spectrum positive elements of a $C^*$-algebra

Question

I would like to know criteria for a C*-algebra $A$ to have a positive contraction $a$ with full spectrum, ie $\sigma(a) = [0,1]$. I am particularly interested in the simple case. I believe that if a C*-algebra is simple, unital, and nonelementary (ie not the compacts) then this should be true. Is there a good reference or easy proof of this?

Answer 1

I claim that a C*-algebra $A$ lacks such an element if and only if every self-adjoint element of $A$ has countable spectrum. Such C*-algebras are called "scattered"; a good reference is Ghasemi and Koszmider, Noncommutative Cantor-Bendixson derivatives and scattered C*-algebras. Scattered implies that minimal projections exist, so the algebras $K(H)$ for arbitrary Hilbert spaces $H$ are the only simple scattered C*-algebras.

My claim is an easy consequence of functional calculus. Suppose $A$ contains a self-adjoint element $x$ whose spectrum is uncountable. Since the spectrum is closed, this means that it contains a copy of the Cantor set. So there is a continuous function $f$ from $\sigma(X)$ onto $[0,1]$, and $f(x)$ is then the positive element with spectrum exactly $[0,1]$.