### Topological Algebras with Finitely-Generated Bases.

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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).

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.

Let $X$ be a completely regular topological space. Let $A\left(X\right)$ be a ring of continuous functions between ${C}^{\ast}\left(X\right)$ and $C\left(X\right)$, that is, ${C}^{\ast}\left(X\right)\subset A\left(X\right)\subset C\left(X\right)$. In [9], a correspondence ${\mathcal{Z}}_{A}$ between ideals of $A\left(X\right)$ and $z$-filters on $X$ is defined. Here we show that ${\mathcal{Z}}_{A}$ extends the well-known correspondence for ${C}^{\ast}\left(X\right)$ to all rings $A\left(X\right)$. We define a new correspondence ${\mathcal{Z}}_{A}$ and show that it extends the well-known correspondence for $C\left(X\right)$ to all rings $A\left(X\right)$. We give a formula that relates the two correspondences. We use properties of ${\mathcal{Z}}_{A}$ and ${\mathcal{Z}}_{A}$ to characterize ${C}^{\ast}\left(X\right)$ and $C\left(X\right)$ among...

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