Let ▫$A$▫ be a Banach algebra. By ▫$\sigma(x)$▫ and ▫$r(x)$▫ we denote the spectrum and the spectral radius of ▫$x \in A$▫, respectively. We consider the relationship between elements ▫$a,b \in A$▫ that satisfy one of the following two conditions: (1) ▫$\sigma(ax) = \sigma(bx)$▫ for all ▫$x \in A$▫, (2) ▫$r(ax) \le r(bx)$▫ for all ▫$x \in A$▫. In particular we show that (1) implies ▫$a=b$▫ if ▫$A$▫ is a ▫$C^\ast$▫-algebra, and (2) implies ▫$a \in \mathbb{C}b$▫ if ▫$A$▫ is a prime ▫$C^\ast$▫-algebra. As an application of the results concerning the conditions (1)and (2) we obtain some spectral characterizations of multiplicative maps.
COBISS.SI-ID: 16287833
We classify, up to isomorphism, all gradings by an arbitrary abelian group on simple finitary Lie algebras of linear transformations (special linear, orthogonal and symplectic) on infinite-dimensional vector spaces over an algebraically closed field of characteristic different from 2.
COBISS.SI-ID: 16339545
Lie superautomorphisms of prime associative superalgebras are considered. A definitive result is obtained for central simple superalgebras: their Lie superautomorphisms are of standard forms, except when the dimension of the superalgebra in question is 2 or 4.
COBISS.SI-ID: 16299353