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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.
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참고문헌 (9)
참고문헌 신청
F. T. Birtel / 1962 / On a Commutative Extension of a Banach Algebra / Proceedings of the American Mathematical Society 13(5):815~822
H. G. Dales / 2000 / Banach Algebras and Automatic Continuity / The Clarendon Press, Oxford University Press
John T. Daly / 1976 / A Banach Algebra of Functions With Bounded nth Differences / Transactions of the American Mathematical Society 223:279~294
Jyunji Inoue / 2007 / On characterizations of the image of the Gelfand transform of commutative Banach algebras / Mathematische Nachrichten 280(1-2):105~126
Jyunji Inoue / 2014 / Segal algebras in commutative Banach algebras / Rocky Mountain Journal of Mathematics 44(2):539~589
H. G. Dales / 2000 / Banach Algebras and Automatic Continuity / The Clarendon Press, Oxford University Press
John T. Daly / 1976 / A Banach Algebra of Functions With Bounded nth Differences / Transactions of the American Mathematical Society 223:279~294
Jyunji Inoue / 2007 / On characterizations of the image of the Gelfand transform of commutative Banach algebras / Mathematische Nachrichten 280(1-2):105~126
Jyunji Inoue / 2014 / Segal algebras in commutative Banach algebras / Rocky Mountain Journal of Mathematics 44(2):539~589
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