### Pushing down Loeb measures.

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The aim of this paper is an investigation of topological algebras with an orthogonal sequence which is total. Closed prime ideals or closed maximal ideals are kernels of multiplicative functionals and the continuous multiplicative functionals are given by the “coefficient functionals”. Our main result states that an orthogonal total sequence in a unital Fréchet algebra is already a Schauder basis. Further we consider algebras with a total sequence ${\left({x}_{n}\right)}_{n\in \mathbb{N}}$ satisfying ${x}_{n}^{2}={x}_{n}$ and ${x}_{n}{x}_{n+1}={x}_{n+1}$ for all n ∈ ℕ.

A systematic investigation of algebras of holomorphic functions endowed with the Hadamard product is given. For example we show that the set of all non-invertible elements is dense and that each multiplicative functional is continuous, answering some questions in the literature.

Let $H\left({G}_{1}\right)$ be the set of all holomorphic functions on the domain ${G}_{1}.$ Two domains ${G}_{1}$ and ${G}_{2}$ are called Hadamard-isomorphic if $H\left({G}_{1}\right)$ and $H\left({G}_{2}\right)$ are isomorphic algebras with respect to the Hadamard product. Our main result states that two admissible domains are Hadamard-isomorphic if and only if they are equal.

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