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

#### Abstract

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

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 -algebra, are allequivalent. Then it is shown that if is a topological-algebra and is an lmc semisimple algebra which isadvertibly complete, then every surjective homomorphism has a closed graph. In particular, if is a Q-algebra with acomplete metrizable topology, and is a semisimple Fréchet algebra, then every surjective homomorphism isautomatically continuous.

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