Spectral flow is the integral of one forms on the Banach manifold of self adjoint Fredholm operators

One may trace the idea that spectral flow should be given as the integral of a one form back to the 1974 Vancouver ICM address of I.M. Singer. Our main theorem gives analytic formulae for the spectral flow along a norm differentiable path of self adjoint bounded Breuer-Fredholm operators in a semifinite von Neumann algebra. These formulae have a geometric interpretation which derives from the proof. Namely we define a family of Banach submanifolds of all bounded self adjoint Breuer-Fredholm operators and on each submanifold define global one forms whose integral on a norm differentiable path contained in the submanifold calculates the spectral flow along this path. We emphasise that our methods do not give a single globally defined one form on the self adjoint Breuer-Fredholms whose integral along all paths is spectral flow rather, as the choice of the plural ‘forms’ in the title suggests, we need a family of such one forms in order to confirm Singer's idea. The original context for this result concerned paths of unbounded self adjoint Fredholm operators. We therefore prove analogous formulae for spectral flow in the unbounded case as well. The proof is a synthesis of key contributions by previous authors, whom we acknowledge in detail in the introduction, combined with an additional important recent advance in the differential calculus of functions of non-commuting operators.     

自同态与环的弱维数

利用模的自同态研究ｖｏｎ Ｎｅｕｍａｎｎ正则环与环的弱维数,给出了ｖｏｎ Ｎｅｕｍａｎｎ正则环的新的刻划,同时也刻划了弱维数≤ｎ（ｎ≥０）的环。     

A note on maximality of analytic crossed products

Let G be a compact abelian group with the totally ordered dual group which admits the positive semigroup . Let N be a von Neumann algebra and be an automorphism group of on N. We denote to the analytic crossed product determined by N and α. We show that if is a maximal σ-weakly closed subalgebra of , then induces an archimedean order in .     

Operator theory on noncommutative varieties, II

An -tuple of operators on a Hilbert space is called a -constrained row contraction if and

where is a WOT-closed two-sided ideal of the noncommutative analytic Toeplitz algebra and is defined using the -functional calculus for row contractions.

We show that the constrained characteristic function associated with and is a complete unitary invariant for -constrained completely non-coisometric (c.n.c.) row contractions. We also provide a model for this class of row contractions in terms of the constrained characteristic functions. In particular, we obtain a model theory for -commuting c.n.c. row contractions.

     

The Patch Topology and the Ultrafilter Topology on the Prime Spectrum of a Commutative Ring

Let R be a commutative ring and let Spec(R) denote the collection of prime ideals of R. We define a topology on Spec(R) by using ultrafilters and demonstrate that this topology is identical to the well-known patch or constructible topology. The proof is accomplished by use of a von Neumann regular ring canonically associated with R.     

Monotonicity properties of the Neumann heat kernel in the ball

A well-known conjecture of R. Laugesen and C. Morpurgo asserts that the diagonal of the Neumann heat kernel of the unit ball U⊂Rn is a strictly increasing radial function. In this paper we use probabilistic arguments to settle this conjecture and to prove some inequalities for the Neumann heat kernel in the ball.     

Idempotent and Nilpotent Submodules of Multiplication Modules

All rings are commutative with identity, and all modules are unital. The purpose of this article is to investigate multiplication von Neumann regular modules. For this reason we introduce the concept of nilpotent submodules generalizing nilpotent ideals and then prove that a faithful multiplication module is von Neumann regular if and only if it has no nonzero nilpotent elements and its Krull dimension is zero. We also give a new characterization for the radical of a submodule of a multiplication module and show in particular that the radical of any submodule of a Noetherian multiplication module is a finite intersection of prime submodules.     

New results on common properties of bounded linear operators RS and SR

Let X,Y be Banach spaces,R:X → Y and S:Y → X be bounded linear operators.When λ≠ 0,we investigate common properties of λⅠ SR and λⅠ RS.This work should be viewed as a continuation of researches of Barnes and Lin et al..We also apply these results obtained to B-Fredholm theory,extensions and Aluthge transforms.     

Malnormal subgroups of lattices and the Pukánszky invariant in group factors

Let G be a connected semisimple real algebraic group. Assume that G(R) has no compact factors and let Γ be a torsion-free uniform lattice subgroup of G(R). Then Γ contains a malnormal abelian subgroup A. This implies that the II₁ factor VN(Γ) contains a masa A with Pukánszky invariant {∞}.     

On finiteness of the set of intermediate subfactors

For type factors with , we show that the sets and are finite. Moreover, , the set of intermediate subfactors, is finite if and only if it is equal to . If is an irreducible subfactor, then we recover a result of Y. Watatani.

     

On the relative commutants of subfactors

Let be factors generated by a periodic tower of finite dimensional -algebras. We prove that for sufficiently large , is -isomorphic to a subalgebra of .

     

Common properties of bounded linear operators AC and BA: Spectral theory

Let X, Y be Banach spaces, and B, be bounded linear operators satisfying the operator equation . Recently, as extensions of Jacobson's lemma, Corach, Duggal and Harte studied common properties of and in algebraic viewpoint and also obtained some topological analogues. In this note, we continue to investigate common properties of AC and BA from the viewpoint of spectral theory. In particular, we give an affirmative answer to one question posed by Corach et al. by proving that has closed range if and only if has closed range.     

Common operator properties of the linear operators

Let and be bounded linear operators defined on Banach spaces, , . When , then the operators and have many basic operator properties in common. This situation is studied in this paper.

     

Estimates for the Approximation Numbers of One Class of Integral Operators. II

We obtain estimates for the Schatten–von Neumann norms of the approximation numbers of Hardy integral operator with variable limits of integration.     

A System of Four Matrix Equations over von Neumann Regular Rings and Its Applications

We consider the system of four linear matrix equations A1X = C1, XB2=C2, A3XB3=C3 and A4XB4 = C4 over h, an arbitrary von Neumann regular ring with identity. A necessary and sufficient condition for the existence and the expression of the general solution to the system are derived. As applications, necessary and sufficient conditions are given for the system of matrix equations A1X = C1 and A3X=C3 to have a bisymmetric solution, the system of matrix equations A1X = C1 and A3XB3 = C3 to have a perselfconjugate solution over h with an involution and char h≠2, respectively. The representations of such solutions are also presented. Moreover, some auxiliary resultson other systems over h are obtained. The previous known results on some systems of matrix equations are special cases of the new results.     

Links ore sets,and classical localizations

The notion of a simple ring D_Gderived from a group ring KG is introduced in case K is a field and G is an infinite residually finite group. The close link between D_Gand K_G is exploited in both directions: first, for a simple proof of the Kaplansky's conjecture concerning direct finiteness of KG. Second, to show that D_Gprovides counter-examples to some conjectures dealing with von Neumann regular rings and the rings all of whose one-sided ideals are generated by idempotents.     

算子代数上导子空间的拓扑自反性

It is proved that the spaces of derivations on some operator algebras are topologically reflexive in the weak operator topology.