# PROJECTIVE LIMIT OF A SEQUENCE OF BANACH FUNCTION ALGEBRAS AS A FRECHET FUNCTION ALGEBRA

• Published : 2002.05.01
• 53 15

#### Abstract

Let X be a hemicompact space with ($K_{n}$) as an admissible exhaustion, and for each n $\in$ N, $A_{n}$ a Banach function algebra on $K_{n}$ with respect to $\parallel.\parallel_n$ such that $A_{n+1}\midK_{n}$$\subsetA_n and{\parallel}f{\mid}K_n{\parallel}_n{\leq}{\parallel}f{\parallel}_{n+1} for all f\in$$A_{n+1}$, We consider the subalgebra A = { f $\in$ C(X) : $\forall_n\;{\epsilon}\;\mathbb{N}$ of C(X) as a frechet 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 Frechet Q-algebra. As an application, the Lipschitzalgebra of infinitely differentiable functions is considered.d.

#### Keywords

Frechet Lipschitz algegra;admissible exhaustion;Lipschitz algebra;Frechet algebra

#### References

1. E. Beckenstein, L. Narici, and C. Suffel, Topological Algebras, Notas Mat., vol. 60, North-Holland, Amsterdam, 1977.
2. T. G. Honary and H. Mahyar, Approximation in Lipschitz algebras of infinitely differentiable functions, Bull. Korean Math. Soc. 36 (1999), 629-636.
3. D. R. Sherbert, The structure of ideals and point derivations in Banach algebras of Lipschitz functions, Trans. Amer. Math. Soc. 111 (1964), 240-272. https://doi.org/10.1090/S0002-9947-1964-0161177-1
4. H. Goldmann, Uniform Fréchet Algebras, North-Holland, Amsterdam, 1990.
5. H. G. Dales, Banach Algebras and Automatic Continuity, Oxford University Press, Oxford, 2000.
6. H. G. Dales and A. M. Davie, Quasianalytic Banach function algebras, J. Functional. Analysis 13 (1973), 28-50. https://doi.org/10.1016/0022-1236(73)90065-7
7. F. Sady, Relations between Banach function algebras and Fréchet function algebras, Honam Math. J. 20 (1998), 79-88.

#### Cited by

1. Separating maps on Fréchet algebras vol.37, pp.1, 2014, https://doi.org/10.2989/16073606.2013.779603