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For a compact Hausdorff space X and Banach dual E*, denote by C(X, (E*, a*)) the Banach space of all continuous functions on X to E* when the latter space is provided with its weak* topology. We show that if El., i = 1, 2, belong to a class of Banach duals satisfying a condition involving the space of multipliers on E, , then the existence of an isomorphism T mapping C(Xi, (El, a*)) onto C(X2, (E2, a*)) with 11 TIh 11 T-I small implies that XI and X2 are homeomorphic. Ultraproducts of Banach spaces and the notion of e-multipliers play key roles in obtaining this result.

Ultraproducts, $\epsilon$-multipliers, and isomorphisms