Pták type theorem for \(C^ *\)-algebras (Q1095382)

A famous theorem due to \textit{V. Pták} [Czech. Math. J. 12(87), 555-557 (1962; Zbl 0116.253)] states that (the spectral radius) \(\| T\| =1\) iff \(\| T^ n\| =1\) for an \(n\times n\) matrix T with (the spectral norm) \(\| T\| \leq 1\). The present note offers how to settle it in a dimension-free context. The idea is to replace the size number n of T in the statement by the rank \(\rho\) (T) due to \textit{H. Kraljević} [Lect. Notes Math. 948, 98-117 (1982; Zbl 0497.46030)], and thus we have Theorem. \(r(T)=1\) iff \(\| T^{\rho (T)+1}\| =1\) for an element T of a \(C^ *\)-algebra with \(\| T\| \leq 1\). In general the above \(\rho (T)+1\) cannot be reduced.