Abstract
Let K be a compact Hausdorff space, <TEX>$\mathfrak{A}$</TEX> a commutative complex Banach algebra with identity and <TEX>$\mathfrak{C}(\mathfrak{A})$</TEX> the set of characters of <TEX>$\mathfrak{A}$</TEX>. In this note, we study the class of functions <TEX>$f:K{\rightarrow}\mathfrak{A}$</TEX> such that <TEX>${\Omega}_{\mathfrak{A}}{\circ}f$</TEX> is continuous, where <TEX>${\Omega}_{\mathfrak{A}}$</TEX> denotes the Gelfand representation of <TEX>$\mathfrak{A}$</TEX>. 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 <TEX>${\sigma}(\mathfrak{A},\mathfrak{C}(\mathfrak{A}))$</TEX>, are discussed. We also provide some results on its completeness under the norm defined by <TEX>${\mid}{\parallel}f{\parallel}{\mid}={\parallel}{\Omega}_{\mathfrak{A}}{\circ}f{\parallel}_{\infty}$</TEX>.
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