### A characterization of ${B}^{*}$-algebras.

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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.

A properly infinite C*-algebra has no non-zero traces. We construct properly infinite Banach *-algebras with non-zero, bounded traces, and show that there are even such algebras which are fairly "close" to the class of C*-algebras, in the sense that they can be hermitian or *-semisimple.

For a locally convex *-algebra A equipped with a fixed continuous *-character ε (which is roughly speaking a generalized F*-algebra), we define a cohomological property, called property (FH), which is similar to character amenability. Let ${C}_{c}\left(G\right)$ be the space of continuous functions with compact support on a second countable locally compact group G equipped with the convolution *-algebra structure and a certain inductive topology. We show that $({C}_{c}\left(G\right),{\epsilon}_{G})$ has property (FH) if and only if G has property (T). On...