Let ▫$A$▫ and ▫$B$▫ be unital semisimple Banach algebras. If ▫$\phi \colon M_2(A)\to B$▫ is a bijective spectrum-preserving linear map, then ▫$\phi$▫ is a Jordan homomorphism.
COBISS.SI-ID: 16067673
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
Given a sequence of bounded operators ▫$a_j$▫ on a Hilbert space ▫$\mathcal{H}$▫ with ▫$\sum_{j=1}^\infty a_j^\ast a_j = 1 = \sum_{j=1}^\infty a_ja_j^\ast$▫, we study the map ▫$\Psi$▫ defined on ▫$B(\mathcal{H})$▫ by ▫$\Psi(x) = \sum_{j=1}^\inftya_j^\ast xa_j$▫ and its restriction ▫$\Phi$▫ to the Hilbert-Schmidt class ▫$C^2(\mathcal{H})$▫. In the case when the sum ▫$\sum_{j=1}^\infty a_j^\ast a_j$▫ is norm-convergent we show in particular that the operator ▫$\Phi-1$▫ is not invertible if and only if the C▫$^\ast$▫-algebra ▫$A$▫ generated by ▫$\{a_j\}_{j=1}^\infty $▫ has an amenable trace. This is used to show that ▫$\Psi$▫ may have fixed points in ▫$B(\mathcal{H})$▫ which are not in the commutant ▫$A'$▫ of ▫$A$▫ even in the case when the weak▫$^\ast$▫ closure of ▫$A$▫ is injective. However, if ▫$A$▫ is abelian, then all fixed points of ▫$\Psi$▫ are in ▫$A'$▫ even if the operators ▫$a_j$▫ are not positive.
COBISS.SI-ID: 16227673
We describe the general form of bijective comparability preserving transformations of the Hilbert space effect algebra, thus improving several known characterizations of ortho-order automorphisms.
COBISS.SI-ID: 16568409
Let ▫$H$▫ be an infinite dimensional separable complex Hilbert space and ▫${\cal U}$▫ be the group of all unitary operators on $H$. Motivated by the algebraic properties of surjective isometries of ▫$\cal U$▫ that have recently been revealed, and also by some classical results related to automorphisms of the unitary groups of operator algebras, we determine the structures of bijective transformations of ▫$\cal U$▫ that respect certain algebraic operations. These are, among others, the usual product of operators, the Jordan triple product, the inverted Jordan triple product, and the multiplicative commutator. Our basic approach to obtain these results is the use of commutativity preserving transformations on the unitary group.
COBISS.SI-ID: 16568153