For an element A in a unital C^(*)-algebra B,the operator-valued 1-formωA(z)=(z-A)^(-1) dz is analytic on the resolvent setρ(A),which plays an important role in the functional calculus of A.This paper defines a clas...For an element A in a unital C^(*)-algebra B,the operator-valued 1-formωA(z)=(z-A)^(-1) dz is analytic on the resolvent setρ(A),which plays an important role in the functional calculus of A.This paper defines a class of Hermitian metrics onρ(A)through the coupling of the operator-valued(1,1)-formΩA=-ωA^(*)∧ωA with tracial and vector states.Its main goal is to study the connection between A and the properties of the metric concerning curvature,arc length,completeness and singularity.A particular example is when A is quasi-nilpotent,in which case the metric lives on the punctured complex plane C\{0}.The notion of the power set is defined to gauge the"blow-up"rate of the metric at 0,and examples are given to indicate a likely link with A’s hyper-invariant subspaces.展开更多
We prove that a surjective map(on the positive cones of unital C^(*)-algebras)preserves the minimum spectrum values of harmonic means if and only if it has a Jordan *-isomorphism extension to the whole algebra.We repr...We prove that a surjective map(on the positive cones of unital C^(*)-algebras)preserves the minimum spectrum values of harmonic means if and only if it has a Jordan *-isomorphism extension to the whole algebra.We represent weighted geometric mean preserving bijective maps on the positive cones of prime C^(*)-algebras in terms of Jordan *-isomorphisms of the algebras.We also characterize order isomorphisms and orthoisomorphisms of the projection lattice of the von Neumann algebra of all bounded linear operators on a Hilbert space,answering an open question arisen by Dye.Finally,we give a description for Fuglede-Kadison determinant preserving maps on the positive cone of a finite von Neumann algebra and improve Gaal and Nayak’s work on this topic.展开更多
基金The first author was partially supported by NSF grants DMS-0900938 and DMS-1162309The second author was partially supported by NSF grant DMS-1001625.
文摘We undertake a local analysis of combinatorial independence as it connects to topological entropy within the framework of actions of sofic groups.
文摘For an element A in a unital C^(*)-algebra B,the operator-valued 1-formωA(z)=(z-A)^(-1) dz is analytic on the resolvent setρ(A),which plays an important role in the functional calculus of A.This paper defines a class of Hermitian metrics onρ(A)through the coupling of the operator-valued(1,1)-formΩA=-ωA^(*)∧ωA with tracial and vector states.Its main goal is to study the connection between A and the properties of the metric concerning curvature,arc length,completeness and singularity.A particular example is when A is quasi-nilpotent,in which case the metric lives on the punctured complex plane C\{0}.The notion of the power set is defined to gauge the"blow-up"rate of the metric at 0,and examples are given to indicate a likely link with A’s hyper-invariant subspaces.
基金supported by Louisiana Christian University Carolyn and Adams Dawson Professorship Fund(2206251515302)the second named author was supported by the NSFC(Grant No.11101220)the Fundamental Research Funds for the Central Universities(Grant No.96172373)。
文摘We prove that a surjective map(on the positive cones of unital C^(*)-algebras)preserves the minimum spectrum values of harmonic means if and only if it has a Jordan *-isomorphism extension to the whole algebra.We represent weighted geometric mean preserving bijective maps on the positive cones of prime C^(*)-algebras in terms of Jordan *-isomorphisms of the algebras.We also characterize order isomorphisms and orthoisomorphisms of the projection lattice of the von Neumann algebra of all bounded linear operators on a Hilbert space,answering an open question arisen by Dye.Finally,we give a description for Fuglede-Kadison determinant preserving maps on the positive cone of a finite von Neumann algebra and improve Gaal and Nayak’s work on this topic.
