### 2-normed Algebras-I

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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.

Let φ be a Jordan automorphism of an algebra . The situation when an element a ∈ satisfies $1/2(\phi \left(a\right)+{\phi}^{-1}\left(a\right))=a$ is considered. The result which we obtain implies the Kleinecke-Shirokov theorem and Jacobson’s lemma.

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].