Bulletin of the
Korean Mathematical Society BKMS

Let $X$ be a hemicompact space with $(K_n)$ as an admissible exhaustion, and for each $n \in \Bbb N$, $A_n$ a Banach function algebra on $K_n$ with respect to $\| \cdot \|_n$ such that $A_{n+1}|_{K_n} \subset A_n$ and $\|f|_{K_n}\|_n\le \|f\|_{n+1}$ for all $f \in A_{n+1}$. We consider the subalgebra $A = \{ f\in C(X) : f|_{K_n}\in A_n, \forall n \in \Bbb N\}$ of $C(X)$ as a Fr\'echet function algebra and give a result related to its spectrum when each $A_n$ is natural. We also show that if $X$ is moreover noncompact, then any closed subalgebra of $A$ cannot be topologized as a regular Fr\'echet $Q$-algebra. As an application, the Lipschitz algebra of infinitely differentiable functions is considered.

Keywords: Frechet Lipschitz algebra, admissible exhaustion, Lipschitz algebra, Fr\'echet algebra

MSC numbers: Primary 46J10; Secondary 46M40