I’m in no mood to do something challenging after this last ditch effort to learn analysis before my prelim, so I’ll do something nice (functional analytic like I promised) that never ceases to amaze me.
Theorem: If is a complex homomorphism on a Banach algebra A, then the norm of , as a linear functional, is at most 1.
Recall that a Banach algebra is just a Banach space (complete normed linear space) with a multiplication that satisfies , associativity, distributivity, and for any scalar .
Complex homomorphisms are just linear functionals that preserve multiplication and .
Assume not, i.e. there exists such that . To simplify notation, let and let . Then and .
Now so form a Cauchy sequence. Now A is a Banach space and hence complete, so there exists such that . But now factor to see that and take the limit to get . Now take the homomorphism of both sides, and we have a contradiction (in particular ).
So some reasons why this may not be all that shocking: we require these to be complex, and complex things tend to work out nicer than real. Also, these are pretty stringent conditions on what constitutes a Banach algebra and what constitutes a homomorphism. We should be able to get some nice structure with all the tools available. It isn’t like we got a lot. Really we’re just saying that these things are bounded and hence continuous, which isn’t all that surprising.
OK. I’ll stop down playing it. It does surprise me.