= Johnson's continuity theorem for irreducible normed representations
{c}
{title2=$\|\pi(a)\|\le C\|a\|$}
Every <algebraically irreducible representation of a Banach algebra> that is normed is <continuous>. The <continuous> orbit <vectors> form an <invariant subspace>. If all nonzero orbit maps were discontinuous, apply the <countable annihilation sequence in an irreducible Banach module> and the <gliding-hump continuity principle> to right multiplication on the algebra and evaluation of the operator-valued representation. Off-diagonal maps vanish, whereas diagonal maps are discontinuous, a contradiction. Once every orbit map is <continuous>, the <Uniform boundedness principle> on the <Banach algebra> gives the common operator-norm bound. Completing the original normed representation space supplies the Banach operator space needed in the argument.
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