### A characteristic property of the algebra $C{\left(\text{\Omega}\right)}_{\beta}$.

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In this paper we extend the characterization of characters given in [1], [2] and [8] onto m-pseudoconvex algebras. As a consequence (and a generalization) we give a characterization of continuous homomorphisms from m-pseudoconvex algebras into commutative semisimple m-pseudoconvex algebras.

We prove that a real or complex F-algebra has all left and right ideals closed if and only if it is noetherian.

A question of Warner and Whitley concerning a nonunital version of the Gleason-Kahane-Żelazko theorem is considered in the context of nonnormed topological algebras. Among other things it is shown that a closed hyperplane M of a commutative symmetric F*-algebra E with Lindelöf Gel'fand space is a maximal regular ideal iff each element of M belongs to some closed maximal regular ideal of E.

We prove that a real or complex unital F-algebra is a Q-algebra if and only if all its maximal one-sided ideals are closed.

We construct two examples of complete multiplicatively convex algebras with the property that all their maximal commutative subalgebras and consequently all commutative closed subalgebras are Banach algebras. One of them is non-metrizable and the other is metrizable and non-Banach. This solves Problems 12-16 and 22-24 of [7].

We construct a complete multiplicatively pseudoconvex algebra with the property announced in the title. This solves Problem 25 of [6].

We compare the singular spectrum of a Banach algebra element with the usual spectrum and exponential spectrum.

Denote by any set of cardinality continuum. It is proved that a Banach algebra A with the property that for every collection ${a}_{\alpha}:\alpha \in \subset A$ there exist α ≠ β ∈ such that ${a}_{\alpha}\in {a}_{\beta}{A}^{}$ is isomorphic to ${\u2a01}_{i=1}^{r}(\u2102\left[X\right]/{X}^{{d}_{i}}\u2102\left[X\right])\oplus E$, where $d\u2081,...,{d}_{r}\in \mathbb{N}$, and E is either $X\u2102\left[X\right]/{X}^{d\u2080}\u2102\left[X\right]$ for some d₀ ∈ ℕ or a 1-dimensional ${\u2a01}_{i=1}^{r}\u2102\left[X\right]/{X}^{{d}_{i}}\u2102\left[X\right]$-bimodule with trivial right module action. In particular, ℂ is the unique non-zero prime Banach algebra satisfying the above condition.

We deal with the algebras consisting of the quotients that produce bounded evaluation on suitable ideals of the multiplication algebra of a normed semiprime algebra A. These algebras of quotients, which contain A, are subalgebras of the bounded algebras of quotients of A, and they have an algebra seminorm for which the relevant inclusions are continuous. We compute these algebras of quotients for some norm ideals on a Hilbert space H: 1) the algebras of quotients with bounded evaluation of the ideal...

We give a simple complex-variable proof of an old result of Zemánek and Le Page on the radical of a Banach algebra. Incidentally, the argument also proves a recent result of Harris and Kadison.

For a given unital Banach algebra A we describe joint spectra which satisfy the one-way spectral mapping property. Each spectrum of this class is uniquely determined by a family of linear subspaces of A called spectral subspaces. We introduce a topology in the space of all spectral subspaces of A and utilize it to the study of the properties of the spectra.