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


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 topologicalQ-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.

DOI Code: 10.1285/i15900932v28n2p57

Keywords: automatic continuity; topological algebra; Fr´echet algebra; Q-algebra; spectrum function; spectral radius; upper semicontinuity; advertibly complete

Classification: 46H40; 46H05

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