Multiplicative maps into the spectrum

Cheick Touré¹ ; Francois Schulz² ; Rudi Brits¹

¹ Department of Mathematics University of Johannesburg Johannesburg, South Africa
² Department of Mathematics University of Johannesburg South Africa

Studia Mathematica, Tome 239 (2017) no. 1, pp. 55-66

Résumé

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 $A$. 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.

Détail

DOI : 10.4064/sm8705-1-2017

Keywords: consider converse famous result elazko which characterizes multiplicative functionals amongst dual space members complex unital banach algebra specifically investigate continuous multiplicative map phi rightarrow mathbb values phi belonging spectrum automatically linear main result states star algebra phi always generates corresponding character psi phi shown phi shares many linear properties its induced character moreover von neumann algebra turns out phi itself linear corresponds its induced character