Continuity of Banach algebra valued functions
Commun. Korean Math. Soc. 2014 Vol. 29, No. 4, 527-538
https://doi.org/10.4134/CKMS.2014.29.4.527
Printed October 1, 2014
Jittisak Rakbud
Silpakorn University
Abstract : Let $K$ be a compact Hausdorff space, $\mathscr{A}$ a commutative complex Banach algebra with identity and $\mathscr{C}(\mathscr{A})$ the set of characters of $\mathscr{A}$. In this note, we study the class of functions $f:K\rightarrow\mathscr{A}$ such that $\Omega_{\mathscr{A}}\circ f$ is continuous, where $\Omega_{\mathscr{A}}$ denotes the Gelfand representation of $\mathscr{A}$. 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 $\sigma(\mathscr{A},\mathscr{C}(\mathscr{A}))$, are discussed. We also provide some results on its completeness under the norm defined by $\n{f}=\N{\Omega_{\mathscr{A}}\circ f}_{\infty}$.
Keywords : Banach algebra, Gelfand representation, character
MSC numbers : Primary 46E40; Secondary 46E15
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