Von Neumann Regular Rings Satisfying Weak Comparability

The notion of weak comparability was first introduced by K.C. O’Meara, to prove that directly finite simple regular rings satisfying weak comparability must be unit-regular. In this paper, we shall treat (non-necessarily simple) regular rings satisfying weak comparability and give some interesting results. We first show that directly finite regular rings satisfying weak comparability are stably finite. Using the result above, we investigate the strict cancellation property and the strict unperforation property for regular rings satisfying weak comparability, and we show that these rings have the strict unperforation property, which means that nA ≺ nB implies A ≺ B for any finitely generated projective modules A, B and any positive integer n.      

On Regular Rings Satisfying Almost Comparability

In this article, we study regular rings satisfying almost comparability. We first show that, for regular rings, almost comparability is inherited by finitely generated projective modules and finite matrix rings, and, as a main result, we prove that the strict cancellation property holds for the family of all finitely generated projective modules over directly finite regular rings satisfying almost comparability.     

Regular Rings Satisfying Generalized Almost Comparability

We study regular rings satisfying generalized almost comparability. First, we determine the forms of regular rings satisfying generalized almost comparability. Next, using the above result, we treat the strict cancellation property and the strict unperforation property for regular rings satisfying generalized almost comparability.     

Left SF-Rings and Regular Rings

A ring R is called a left (right) SF-ring if all simple left (right) R-modules are flat. It is known that von Neumann regular rings are left and right SF-rings. In this article, we study the regularity of left SF-rings and we prove the following: 1) if R is a left SF-ring whose all complement left (right) ideals are W-ideals, then R is strongly regular; 2) if R is a left SF-ring whose all maximal essential right ideals are GW-ideals, then R is regular.     

Annihilator Ideal-Based Zero-Divisor Graphs Over Multiplication Modules

For a commutative ring R with identity, an ideal-based zero-divisor graph, denoted by Γ _I (R), is the graph whose vertices are {x ∈ R?I | xy ∈ I for some y ∈ R?I}, and two distinct vertices x and y are adjacent if and only if xy ∈ I. In this article, we investigate an annihilator ideal-based zero-divisor graph by replacing the ideal I with the annihilator ideal Ann(M) for a multiplication R-module M. Based on the above-mentioned definition, we examine some properties of an R-module over a von Neumann regular ring, and the cardinality of an R-module associated with Γ _Ann(M)(R).     

On Quotient Rings in Alternative Rings

We introduce a notion of left nonsingularity for alternative rings and prove that an alternative ring is left nonsingular if and only if every essential left ideal is dense, if and only if its maximal left quotient ring is von Neumann regular (a Johnson-like Theorem). Finally, we obtain a Gabriel-like Theorem for alternative rings.     

A von Neumann type inequality for certain domains in

Strict contractions on a Hilbert space have a functional calculus with functions that are analytic in the unit disc of the complex plane; an estimate of the norm is then provided by von Neumann's inequality. We consider functions that satisfy related inequalities with respect to multioperators connected to certain domains in ; a representation formula and a Nevanlinna-Pick type theorem are obtained.

     

正则环与YJ-内射性

文献 [1]中 ,Ming.R.Y.C引进了 YJ 内射模的概念 ,且指出正则环上的每个模均是 YJ 内射模 ,那么反之如何呢 ?文 [1]中做了一些结果 ,本文拟就这个问题作进一步讨论 .     

Rings of quotients of C(X) induced by points

We study two rings of quotients of C(X), the ring of continuous functions on the Tychonoff space X. The first, F(X), is the ring of quotients induced by the filter of ideals consisting of dense finite intersections of fixed maximal ideals. The second, C[F], is the ring of quotients induced by the filter of dense cofinite subspaces of X. After some preliminary information we explicitly describe in §2 and §3 the constructions of these rings of quotients. In the third section, we use F(X)and C[F] to define and study the class of h-points and h-spaces. In particular, we show that C-spaces and P-spaces are h-spaces. In the last section we construct an ideal of C(X)which will be used to give an ideal theoretic characterization of h-points. This revised version was published online in June 2006 with corrections to the Cover Date.     

**-Haagerup Property for C*-Algebras

Extending the notion of Haagerup property for finite von Neumann algebras to the general von Neumann algebras, the authors define and study the $(**)$-Haagerup property for $C^{*}$-algebras in this paper. They first give an answer to Suzuki''s question (2013), and then obtain several results of $(**)$-Haagerup property parallel to those of Haagerup property for $C^{*}$-algebras. It is proved that a nuclear unital $C^{*}$-algebra with a faithful tracial state always has the $(**)$-Haagerup property. Some heredity results concerning the $(**)$-Haagerup property are also proved.     

L 2-Cohomology for Von Neumann Algebras

We study L ²-Betti numbers for von Neumann algebras, as defined by D. Shlyakhtenko and A. Connes in [CoS]. We give a definition of L ²-cohomology and show how the study of the first L ²-Betti number can be related to the study of derivations with values in a bi-module of affiliated operators. We show several results about the possibility of extending derivations from sub-algebras and about uniqueness of such extensions. In particular, we show that the first L ²-Betti number of a tracial von Neumann algebra coincides with the corresponding number for an arbitrary weakly dense sub-C*-algebra. Along the way, we prove some results about the dimension function of modules over rings of affiliated operators which are of independent interest. Received: March 2006 Revision: October 2006 Accepted: October 2006     

因子von Neumann代数上导子的等价刻画

设R是含非平凡幂等元P的素环,C∈R,C=PC.本文证明可加映射△:R→R在C可导,即△(AB)=△(A)B+A△(B),A,B∈R,AB=C当且仅当存在导子δ:R→R,使得△(A)=δ(A)+△(I)A,A∈R.没有I_1型中心直和项的von Neumann代数上的可导映射也有类似结论.利用该结论证明了,若非零算子C∈B(X),使得ran(C)或ker(C)在X中可补,则可加映射△:B(X)→B(X)在C可导当且仅当它是导子.特别地,证明了因子von Neumann代数上的可加映射在任意但固定的非零算子可导当且仅当它是导子.     

A semi-finite algebra associated to a subfactor planar algebra

We canonically associate to any planar algebra two type II_∞ factors M_±. The subfactors constructed previously by the authors in Guionnet et al. (2010) [6] are isomorphic to compressions of M_± to finite projections. We show that each M_± is isomorphic to an amalgamated free product of type I von Neumann algebras with amalgamation over a fixed discrete type I von Neumann subalgebra. In the finite-depth case, existing results in the literature imply that M₊≅M₋ is the amplification a free group factor on a finite number of generators. As an application, we show that the factors Mj constructed in Guionnet et al. (in press) [6] are isomorphic to interpolated free group factors L(F(rj)), rj=1+2δ−2j(δ−1)I, where δ² is the index of the planar algebra and I is its global index. Other applications include computations of laws of Jones-Wenzl projections.     

素环上强保持2-Jordan乘积的映射

对于给定的正整数k≥1,环R上的元x,y的k-Jordan乘积定义为{x,y}_k={{x,y}_(k-1),y}_1,其中{x,y}_0=x,{x,y}_1=xy+yx.假设R是包含有单位元与一非平凡幂等元的素环.本文证明了R上的满射f满足{f(x),f(y)}2={x,y}_2对所有x,y∈R成立当且仅当存在λ∈l(R的可扩展中心)且λ~3=1,使得下列之一成立:(1)若R的特征不为2,则f(x)=λx对所有x∈R成立;(2)若R的特征为2,则f(x)=λx+μ(x)对所有x∈R成立,其中μ:R→l是一个映射.作为应用,得到了因子von Neumann代数上保持上述性质映射的结构.     

BOUNDED SOLUTIONS AND PERIODIC SOLUTIONS OF VISCOUS POLYTROPIC GAS EQUATIONS

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Von Neumann equivalence and group exactness

We will show that group exactness is a von Neumann equivalence invariant. This result generalizes the previously known fact stating that group exactness is stable under measure equivalence and W*-equivalence.     

Perturbations of certain crossed product algebras by free groups

Given two von Neumann algebras M and N   acting on the same Hilbert space, d(M,N)$d(M,N)$ is defined to be the Hausdorff distance between their unit balls. The Kadison–Kastler problem asks whether two sufficiently close von Neumann algebras are spatially isomorphic. In this article, we show that if _P0$P_0$ is an injective von Neumann algebra with a cyclic tracial vector, G is a free group, α is a free action of G   on _P0$P_0$ and N   is a von Neumann algebra such that d(N,_P0?αG)<1/7×10⁻⁷$d(N,P_0{?}_{\alpha }G)<1/7\times {10}^{-7}$, then N   and _P0?αG$P_0{?}_{\alpha }G$ are spatially isomorphic. Suitable choices of the actions give the first examples of infinite noninjective factors for which this problem has a positive solution.     

Hyperfinite Von Neumann games

Let ${}^1\text{[V(}\text{R}\text{)]}$ be an ω₁-saturated enlargement in the sense of Keisler (1977) and let $\text{F}$ be a hyperfinite finite set in ¹$\text{N}$. Following the suggestion of Wesley (1971) we define a class of hyperfinite games of the form: $\text{Γ}{}_{\mathrm{<mtext>F</mtext>}}\text{(}{}^1\text{υ)=〈Φ,}\text{A}\text{(}\text{F}\text{),}{}^1\text{υ〉}$, and show that measure-theoretic analogues of the kernel and bargaining set exist in this nonstandard setting such that their standard parts Loeb-measurable measurable on the Loeb space generated by the internal ¹finitely additive measure $\text{u}{}_{\mathrm{<mtext>F</mtext>}}\text{:}\text{A}\text{(}\text{F}\text{)→}{}^1\text{R}{}_{\mathrm{+}}$.