Lower bounds for norms on certain algebras (Q1913493)

Sufficient conditions are given, which imply that all algebra norms \(|\;|\) on a Banach algebra \(A\) dominate the complete norm \(|\;|_A\) with which \(A\) is equipped (\(|x|_A\leq C|x|\), for all \(x\in A\)). These conditions apply whenever \(A\) is a \(C^*\)-algebra, \(A\) is the algebra of all bounded linear operators on a Banach space \(X\), \(A\) is the Calkin algebra on \(X= \ell_p\) or \(X= c_0\) or \(A\) is the weak Calkin algebra on \(X= L^1\). A somewhat weaker conclusion is reached for all algebra norms on the weak Calkin algebra on the Banach space \(X= C(\Omega)\), \(\Omega\) a compact metric space. When combined with a result of \textit{B. E. Johnson} [J. Lond. Math. Soc. 42, 537-541 (1967; Zbl 0152.32805)]\ this leads to strong uniqueness of norm theorems for some of these algebras.