Upper semicontinuity of the spectrum function and automatic continuity in topological $Q$-algebras

Abstract

In 1993, M. Fragoulopoulou applied thetechnique of Ransford and proved that if $E$ and $F$ are lmcalgebras such that $E$ is a Q-algebra, $F$ is semisimple andadvertibly complete, and $(E,F)$ is a closed graph pair, then eachsurjective homomorphism $\varphi:E\longrightarrow F$ is continuous. Later onin 1996, it was shown by Akkar and Nacir that if $E$ and $F$ areboth LFQ-algebras and $F$ is semisimple then evey surjectivehomomorphism $\varphi:E\longrightarrow F$ is continuous. In this work weextend the above results by removing the lmc property from $E$.

We first show that in a topological algebra, the uppersemicontinuity of the spectrum function, the upper semicontinuityof the spectral radius function, the continuity of the spectralradius function at zero, and being a $Q$-algebra, are allequivalent. Then it is shown that if $A$ is a topological$Q$-algebra and $B$ is an lmc semisimple algebra which isadvertibly complete, then every surjective homomorphism $T:A\longrightarrow B$ has a closed graph. In particular, if $A$ is a Q-algebra with acomplete metrizable topology, and $B$ is a semisimple Fréchet algebra, then every surjective homomorphism $T:A\longrightarrow B$ isautomatically continuous.