The article as is doesn't shed much light on much. We need expansions and examples here. Some questions to answer when someone knowledgeable gets to edits this:
Is closedness not required in the definition?
What is the relationship between characters on A, the space X, and the "support" of A?
Give some examples. Take C(X), X compact Hausdorff. Take a closed *-ideal corresponding to some closed set of X. Forget the C*-structure, so we have a Banach function algebra. What is the character space? Is it the obvious guess, the quotient space?
How about examples different from above? Take, for instance, analytic polynomials on the circle, what is the character space? Presumably this is bigger than the circle and related to the Shilov boundary somehow? Mct mht (talk) 00:16, 1 June 2012 (UTC)