Table of Contents
ISRN Mathematical Analysis
Volume 2012 (2012), Article ID 187952, 10 pages
http://dx.doi.org/10.5402/2012/187952
Research Article

Some Dense Linear Subspaces of Extended Little Lipschitz Algebras

Department of Mathematics, Faculty of Science, Arak University, Arak 38156-8-8349, Iran

Received 18 November 2011; Accepted 3 January 2012

Academic Editors: S. Anita and H. Hedenmalm

Copyright © 2012 Davood Alimohammadi and Sirous Moradi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let ( 𝑋 , 𝑑 ) be a compact metric space. In 1987, Bade, Curtis, and Dales obtained a sufficient condition for density of a subspace 𝑃 of little Lipschitz algebra l i p ( 𝑋 , 𝛼 ) in this algebra and in particular showed that L i p ( 𝑋 , 1 ) is dense in l i p ( 𝑋 , 𝛼 ) , whenever 0 < 𝛼 < 1 . Let 𝐾 be a compact subset of 𝑋 . We define new classes of Lipchitz algebras L i p ( 𝑋 , 𝐾 , 𝛼 ) for 𝛼 ∈ ( 0 , 1 ] and l i p ( 𝑋 , 𝐾 , 𝛼 ) for 𝛼 ∈ ( 0 , 1 ) , consisting of those continuous complex-valued functions 𝑓 on 𝑋 such that 𝑓 | 𝐾 ∈ L i p ( 𝐾 , 𝛼 ) and 𝑓 | 𝐾 ∈ l i p ( 𝐾 , 𝛼 ) , respectively. In this paper we obtain a sufficient condition for density of a linear subspace 𝑃 of extended little Lipschitz algebra l i p ( 𝑋 , 𝐾 , 𝛼 ) in this algebra and in particular show that L i p ( 𝑋 , 𝐾 , 1 ) is dense in l i p ( 𝑋 , 𝐾 , 𝛼 ) , whenever 0 < 𝛼 < 1 .