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Structure spaces for rings of continuous functions with applications to realcompactifications

Lothar RedlinSaleem Watson — 1997

Fundamenta Mathematicae

Let X be a completely regular space and let A(X) be a ring of continuous real-valued functions on X which is closed under local bounded inversion. We show that the structure space of A(X) is homeomorphic to a quotient of the Stone-Čech compactification of X. We use this result to show that any realcompactification of X is homeomorphic to a subspace of the structure space of some ring of continuous functions A(X).

A multiplier theorem for Fourier series in several variables

Nakhle AsmarFlorence NewbergerSaleem Watson — 2006

Colloquium Mathematicae

We define a new type of multiplier operators on L p ( N ) , where N is the N-dimensional torus, and use tangent sequences from probability theory to prove that the operator norms of these multipliers are independent of the dimension N. Our construction is motivated by the conjugate function operator on L p ( N ) , to which the theorem applies as a particular example.

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