Determining elements in C∗C∗-algebras through spectral properties

Let AA be a unital C∗C∗-algebra and A″A″ its second dual. By σ(a)σ(a) and r(a)r(a) we denote the spectrum and the spectral radius of a∈Aa∈A, respectively. The following two statements hold for arbitrary a,b∈Aa,b∈A: (1) σ(ac)⊆σ(bc)∪{0}σ(ac)⊆σ(bc)∪{0} for every c∈Ac∈A if and only if there exists a central projection z∈A″z∈A″ such that a=zba=zb, (2) r(ac)≤r(bc)r(ac)≤r(bc) for every c∈Ac∈A if and only if there exists a central element zz in A″A″ such that a=zba=zb and ‖z‖≤1‖z‖≤1.