On spectral and central states of Banach algebras
The article deals with trace like functionals on general Banach algebras. Central and spectral states were considered in the lecture notes \textit{F. F. Bonsall} and \textit{J. Duncan,} Numerical ranges of operators on normed spaces and of elements of normed algebras (1971; Zbl 0207.448). On \(C^*\)-algebras the two types of states coincide. \textit{H. K. Wimmer} (Amer. Math. Monthly 81, 625-627 (1974; Zbl 0291.15021)] proved that \[ (*)\quad \lim_{n\to \infty}| \tau(a^ n)|^{1/n}=\rho(a) \] for every matrix a where \(\tau\) denotes the trace and \(\rho\) stands for the spectral radius. This formula does not hold in infinite dimensional finite factors while any commutative separable Banach algebra has a state \(\tau\) satisfying (*). A result is related to the paper of S. \textit{Kowalski} and \textit{Z. Słodkowski} [Stud. Math. 67, 215-223 (1980; Zbl 0456.46041)]. Assume that the Banach algebra A has a unique spectral state and let \(\tau\) be a complex function on A such that \(\tau(0)=0\) and \(\tau(a)-\tau(b)\in conv Sp(a-b)\). Then \(\tau\) is the spectral state of A.
trace like functionals on general Banach algebras
scientific article; zbMATH DE number 3842510
scientific article