The authors establish various criteria for the equality of two elements in some unital Banach algebras. The first part of the paper is devoted to uniform algebras. Let be a uniform algebra on a compact Hausdorff space , and let be two elements of .
Among other results, the authors show that, if there exist and such that for all -peaking functions , then . Recall that a nonzero element is called a -peaking function if its peripheral spectrum is a singleton.
In the second part, the authors consider standard operator algebras. Let be a unital standard operator algebra on a Banach space and let be in . The authors show that, if there exist and such that for all , where denotes the spectral radius, then . Other identification criteria are given.