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 Banach Function Algebras of n-Times Continuously Differentiable Functions on R^d Vanishing at Infinity and Their BSE-Extensions J. Korean Math. Soc.Published online April 10, 2019 Jyunji Inoue and Sin_Ei Takahasi Hokkaido University, Yamagata University Abstract : In authors' paper in 2007, it was shown that the BSE-extension 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 $k=0,...,n$, 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 $k=0,...,n-1$, 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 Banach function algebra, Lipschitz algebra, BSE-extension, BSE-algebra, BED-algebra, bounded approximate identity, multiplier algebra MSC numbers : Primary 46J15; Secondary 46J40, 46J20 Full-Text :