Banach function algebras of $n$-times continuously differentiable functions on $\mathbf{R}^d$ vanishing at infinity and their BSE-extensions
J. Korean Math. Soc. 2019 Vol. 56, No. 5, 1333-1354
https://doi.org/10.4134/JKMS.j180665
Published online September 1, 2019
Jyunji Inoue, Sin-Ei Takahasi
Hokkaido University; Yamagata University
Abstract : In authors' paper in 2007, it was shown that the BSE-exten\-sion of $C^1_0(\mathbf{R})$, the algebra of continuously differentiable functions $f$ on the real number space $\mathbf{R}$ such that $f$ and $df/dx$ vanish at infinity, is the Lipschitz algebra $Lip_1(\mathbf{R})$. This paper extends this result to the case of $C^n_0(\mathbf{R}^d)$ and $C^{n-1,1}_b(\mathbf{R}^d)$, where $n$ and $d$ represent arbitrary natural numbers. Here $C^n_0(\mathbf{R}^d)$ is the space of all $n$-times continuously differentiable functions $f$ on $\mathbf{R}^d$ whose $k$-times derivatives are vanishing at infinity for $k=0,\ldots,n$, and $C^{n-1,1}_b(\mathbf{R}^d)$ is the space of all $(n-1)$-times continuously differentiable functions on $\mathbf{R}^d$ whose $k$-times derivatives are bounded for $k=0, \ldots ,n-1$, and $(n-1)$-times derivatives are Lipschitz. As a byproduct of our investigation we obtain an important result that $C^{n-1,1}_b(\mathbf{R}^d)$ has a predual.
Keywords : natural commutative Banach function algebra, Lipschitz algebra, BSE-extension, BSE-algebra, BED-algebra, bounded approximate identity, multiplier algebra
MSC numbers : Primary 46J15; Secondary 46J40, 46J20
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