Let be an increasing sequence of non-negative real numbers with and if . The Marcinkiewicz sequence space is the space of all bounded sequences such that , and its subspace contains all elements such that . The space equipped with the norm is rearrangement invariant and is a predual of a Lorentz space. Given a complex Banach space , denote by the Banach algebra of all functions which are continuous and bounded on , the closed unit ball of , and holomorphic on the interior of . By denote the Banach algebra of functions in which are uniformly continuous on . A subset of is said to be a boundary for if for all . The Šilov boundary of is the minimal closed boundary. A point is a peak point of if there is such that for all .
Characterizations of extreme, complex extreme and exposed points of are given. For instance, it is proved that is complex extreme if and only if is a peak point of . Applying those characterizations, a condition is found which is necessary and sufficient for a subset of to be a boundary for . It is also shown that it is possible that a set of peak points of is a boundary for , yet it is not the Šilov boundary for .