Determining elements in -algebras through spectral properties

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