Title: PROJECTIVE LIMIT OF A SEQUENCE OF BANACH FUNCTION ALGEBRAS AS A FRECHET FUNCTION ALGEBRA
Abstract:Let X be a hemicompact space with (<TEX>$K_{n}$</TEX>) as an admissible exhaustion, and for each n <TEX>$\in$</TEX> N, <TEX>$A_{n}$</TEX> a Banach function algebra on <TEX>$K_{n}$</TEX> with respect t...Let X be a hemicompact space with (<TEX>$K_{n}$</TEX>) as an admissible exhaustion, and for each n <TEX>$\in$</TEX> N, <TEX>$A_{n}$</TEX> a Banach function algebra on <TEX>$K_{n}$</TEX> with respect to <TEX>$\parallel.\parallel_n$</TEX> such that <TEX>$A_{n+1}\midK_{n}$</TEX><TEX>$\subsetA_n$</TEX> and<TEX>${\parallel}f{\mid}K_n{\parallel}_n{\leq}{\parallel}f{\parallel}_{n+1}$</TEX> for all f<TEX>$\in$</TEX><TEX>$A_{n+1}$</TEX>, We consider the subalgebra A = { f <TEX>$\in$</TEX> C(X) : <TEX>$\forall_n\;{\epsilon}\;\mathbb{N}$</TEX> of C(X) as a frechet function algebra and give a result related to its spectrum when each <TEX>$A_{n}$</TEX> is natural. We also show that if X is moreover noncompact, then any closed subalgebra of A cannot be topologized as a regular Frechet Q-algebra. As an application, the Lipschitzalgebra of infinitely differentiable functions is considered.d.Read More