Article
Kyungpook Mathematical Journal 2020; 60(1): 117-125
Published online March 31, 2020
Copyright © Kyungpook Mathematical Journal.
The Maximal Ideal Space of Extended Differentiable Lipschitz Algebras
Mohammad Ali Abolfathi∗ and Ali Ebadian
Department of Mathematics, Urmia University, P. O. Box 165, Urmia, Iran
e-mail : m.abolfathi@urmia.ac.ir and a.ebadian@urmia.ac.ir
Received: May 14, 2018; Revised: December 25, 2018; Accepted: December 26, 2018
Abstract
In this paper, we first introduce new classes of Lipschitz algebras of infinitely differentiable functions which are extensions of the standard Lipschitz algebras of infinitely differentiable functions. Then we determine the maximal ideal space of these extended algebras. Finally, we show that if
Keywords: Banach function algebra, diff,erentiable Lipschitz algebras, extended Lipschitz algebra, maximal ideal space, rational functions. This work was supported by Urmia University.
1. Introduction
Let
Let (
The subalgebra of those functions
is denoted by
It is interesting to note that
Let
exists. We call
Definition 1.1
Let
K is calledregular if for eachz 0 ∈K there exists a constantC such that for allz ∈K ,δ (z ,z 0) ≤C |z −z 0|.K is calleduniformly regular if there exists a constantC such that for allz ,w ∈K ,δ (z ,w ) ≤C |z −w |.
Definition 1.2
Let
Let
For each
This inequality implies that
Definition 1.3
The algebra of complex-valued functions
The algebra of functions
Let
Whenever we refer to
Definition 1.4
Let
For
For convenience, we regard
2. The Maximal Ideal Space of Extended Differentiable Lipschitz Algebras
We show that, the completeness of
Theorem 2.1
Obviously
By letting
Therefore for every
which implies (
In the following let
It is easy to see that
Theorem 2.2
By the above, the inclusion
Obviously
So for every
Hence
Corollary 2.3
By the naturality of
Hence
By similarly way the algebra
Corollary 2.4
Lemma 2.5
Dn +1(X,K )⊆ Lipn (X,K , 1)⊆ ℓipn (X,K, α ),The standard norms of Dn +1(X,K )and Lipn (X,K , 1)are equivalent on Dn +1(X,K ),Dn +1(X,K )is close subalgebra of Lipn (X,K , 1).
(i) Obviously
consequently,
Hence
(ii) Let
Now for
and hence, |
Hence for all
Therefore standard norms of
(iii) It follows immediate from (ii).
Definition 2.6
Let
where
It is easy to see that
Theorem 2.7
See [5].
Lemma 2.8
For
and it is also known that
therefore
and the proof is complete.
Corollary 2.9
Straightforward calculations show that
for all
hence
Therefore
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