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THE RADICAL SEMISIMPLICITY
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2-sided ideal applies arbitrary associated assume Banach algebra Banach space belongs bounded called carrier space choose closed compact completely regular completes the proof complex condition consider consists contains continuous functions COROLLARY defined definition denote dense discussion divisor of zero dual equal equivalent example exists extended fact field finite fixed follows functions Furthermore given hand Hausdorff hence hermitian Hilbert space homomorphism identity element immediate implies involution irreducible isomorphism left ideal Lemma linear mapping Math maximal modular minimal modulo multiplication neighborhood non-zero normed algebra Note Observe obtain obviously operators particular positive preceding primitive PROOF proper properties proved quasi-regular radical regular relation relative representation respect result rings scalar semi-simple separates sequence shows spectrum strongly structure subalgebra subset sufficient theorem theory topological vector