Order structure, multipliers, and Gelfand representation of vector-valued function algebras (Q728021)

Let \(A\) be a Banach algebra and \(X\) a locally compact Hausdorff space. As usual, \(C_0(X, A)\) denotes the algebra of all continuous functions \(f:X \to A\) vanishing at infinity. The paper studies the properties of both multipliers of \(C^*\)-Segal algebras and the Gelfand representation of the algebra \(C_0(X, A)\). It is shown that a \(C^*\)-Segal algebra \(A\) has an order unitization if and only if \(C_0(X,A)\) has an order unitization. Also, a description of a multiplier module of \(C_0(X, A)\) is given through some algebra of continuous functions with values in the multiplier module of \(A\).