CONTINUITY OF BANACH ALGEBRA VALUED FUNCTIONS

Title & Authors
CONTINUITY OF BANACH ALGEBRA VALUED FUNCTIONS
Rakbud, Jittisak;

Abstract
Let K be a compact Hausdorff space, $\small{\mathfrak{A}}$ a commutative complex Banach algebra with identity and $\small{\mathfrak{C}(\mathfrak{A})}$ the set of characters of $\small{\mathfrak{A}}$. In this note, we study the class of functions $\small{f:K{\rightarrow}\mathfrak{A}}$ such that $\small{{\Omega}_{\mathfrak{A}}{\circ}f}$ is continuous, where $\small{{\Omega}_{\mathfrak{A}}}$ denotes the Gelfand representation of $\small{\mathfrak{A}}$. The inclusion relations between this class, the class of continuous functions, the class of bounded functions and the class of weakly continuous functions relative to the weak topology $\small{{\sigma}(\mathfrak{A},\mathfrak{C}(\mathfrak{A}))}$, are discussed. We also provide some results on its completeness under the norm defined by \${\mid}{\parallel}f{\parallel}{\mid}
Keywords
Banach algebra;Gelfand representation;character;
Language
English
Cited by
References
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