Multiplicative maps into the spectrum
Tom 239 / 2017
Studia Mathematica 239 (2017), 55-66
MSC: Primary 46H05, 46L05; Secondary 46L10.
DOI: 10.4064/sm8705-1-2017
Opublikowany online: 17 May 2017
Streszczenie
We consider the converse of a famous result of W. Żelazko et al. which characterizes multiplicative functionals amongst the dual space members of a complex unital Banach algebra . Specifically, we investigate when a continuous multiplicative map \phi :A\rightarrow \mathbb C, with values \phi (x) belonging to the spectrum of x, is automatically linear. Our main result states that if A is a C^\star -algebra, then \phi always generates a corresponding character \psi _\phi of A. It is then shown that \phi shares many linear properties with its induced character. Moreover, if A is a von Neumann algebra, then it turns out that \phi itself is linear, and that it corresponds to its induced character.
