On a problem of Fell and Doran
Various proofs of the Factorization Theorem for representations of Banach algebras are compared with its original proof due to P. Cohen.
Let s be the space of rapidly decreasing sequences. We give the spectral representation of normal elements in the Fréchet algebra L(s',s) of so-called smooth operators. We also characterize closed commutative *-subalgebras of L(s',s) and establish a Hölder continuous functional calculus in this algebra. The key tool is the property (DN) of s.