The(**)-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.     

关于核型$C^{*}$代数单调积的注记

Given two nuclear C^＊-algebras A1 and A2 with states φ1 and φ2, we show that the monotone product C^＊-algebra A1 △→ A2 is still nuclear. Furthermore, if both the states φ1 and φ2 are faithful, then the monotone product ,A1 △→ A2 is nuclear if and only if the C^＊-algebras ,A1 and A2 both are nuclear.     

纯无限单的C*-代数通过某些C*-代数扩张的非稳定K-理论

设0→B■E■A→0是有单位元C~*-代数E的一个扩张,其中A是有单位元纯无限单的C~*-代数,B是E的闭理想.当B是E的本性理想并且同时是单的、可分的而且具有实秩零及性质(PC)时,证明了K_0(E)={[p]| p是E\B中的投影};当B是稳定C~*-代数时,证明了对任意紧的Hausdorff空间X,有■(C(X,E))/■_0(C(X,E))≌K_1(C(X,E)).     

On Markushevich bases in preduals of von Neumann algebras

We prove that the predual of any von Neumann algebra is 1-Plichko, i.e., it has a countably 1-norming Markushevich basis. This answers a question of the third author who proved the same for preduals of semifinite von Neumann algebras. As a corollary we obtain an easier proof of a result of U. Haagerup that the predual of any von Neumann algebra enjoys the separable complementation property. We further prove that the selfadjoint part of the predual is 1-Plichko as well.     

$C^*$-动力系统的Haagerup性质

令$A$是一个单位$C^*$-代数, $\tau$是它的一个态, $\alpha$是一个离散群$G$在$A$上保持$\tau$的作用. 首先, 我们通过考虑 $C^*$-代数的态, 推广了动力系统的Haagerup性质, 并且证明了动力系统有 Haagerup性质当且仅当它的约化交叉积有Haagerup性质. 然后, 我们引入了$G$在$A$上关于$\tau$的拟顺从作用. 最后, 利用上面的结果, 我们证明了如果$\alpha$是$G$在$A$上关于$\tau$的拟顺从作用, 那么$(A,\tau)$有Haagerup性质当且仅当$(A\rtimes_{\alpha,r}G,\tau'')$有Haagerup性质, 其中$\tau''$是由$\tau$诱导的$A\rtimes_{\alpha,r}G$上的态. 本文的主要结论推广了一些经典情况下的已知结果.     

Persistence Approximation Property for Maximal Roe Algebras

Persistence approximation property was introduced by Hervé Oyono-Oyono and Guoliang Yu. This property provides a geometric obstruction to Baum-Connes conjecture. In this paper, the authors mainly discuss the persistence approximation property for maximal Roe algebras. They show that persistence approximation property of maximal Roe algebras follows from maximal coarse Baum-Connes conjecture. In particular, let X be a discrete metric space with bounded geometry, assume that X admits a fibred coarse embedding into Hilbert space and X is coarsely uniformly contractible, then C_(max)~*(X) has persistence approximation property. The authors also give an application of the quantitative K-theory to the maximal coarse Baum-Connes conjecture.     

极大谱距离的注记

Two equivalent conditions are given for the non-commutative unital C^＊-subalgebras of B（H） algebras A and B to be quantum C^＊ independent.     

Generalisations of the Haagerup approximation property to arbitrary von Neumann algebras

The notion of the Haagerup approximation property, originally introduced for von Neumann algebras equipped with a faithful normal tracial state, is generalised to arbitrary von Neumann algebras. We discuss two equivalent characterisations, one in term of the standard form and the other in term of the approximating maps with respect to a fixed faithful normal semifinite weight. Several stability properties, in particular regarding the crossed product construction are established and certain examples are introduced.     

Compressions of free products of von Neumann algebras

A reduction formula for compressions of von Neumann algebra II–factors arising as free products is proved. This shows that the fundamental group is for some such algebras. Additionally, by taking a sort of free product with an unbounded semicircular element, continuous one parameter groups of trace scaling automorphisms on II–factors are constructed; this produces type III factors with core , where can be a full II–factor without the Haagerup approximation property. Received: 26 October 1998 / in final form 18 March 1999     

Hilbert空间中的Bessel列构成的 $C^*$-代数 $B_H(I)$

Let H be a separable Hilbert space, B H(I), B(H) and K(H) the sets of all Bessel sequences {f i}i∈I in H, bounded linear operators on H and compact operators on H, respectively. Two kinds of multiplications and involutions are introduced in light of two isometric linear isomorphisms αH : B H(I) → B(?2), β : B H(I) → B(H), respectively, so that B H(I) becomes a unital C*-algebra under each kind of multiplication and involution. It is proved that the two C*-algebras(B H(I), ?, ?) and(B H(I), ·, *) are *-isomorphic. It is also proved that the set F H(I) of all frames for H is a unital multiplicative semi-group and the set R H(I) of all Riesz bases for H is a self-adjoint multiplicative group, as well as the set K H(I) := β-1(K(H)) is the unique proper closed self-adjoint ideal of the C*-algebra B H(I).     

Joint Reducing Subspaces of Multiplication Operators and Weight of Multi-variable Bergman Spaces

This paper mainly concerns a tuple of multiplication operators defined on the weighted and unweighted multi-variable Bergman spaces, their joint reducing subspaces and the von Neumann algebra generated by the orthogonal projections onto these subspaces. It is found that the weights play an important role in the structures of lattices of joint reducing subspaces and of associated von Neumann algebras. Also, a class of special weights is taken into account. Under a mild condition it is proved that if those multiplication operators are defined by the same symbols, then the corresponding von Neumann algebras are $*$-isomorphic to the one defined on the unweighted Bergman space.     

TOEPLITZ C^≠-ALGEBRA OVER THE L-SHAPED DOMAIN IN C^n

The authors investigate the Toeplitz C^≠-algebra over the L-shaped domain in C^n by the groupoid approach, and successfully parameterize the unit space of the groupoid assocated to the Toeplitz C^≠-algebra.     

WHEN CAN THE STABLE ALGEBRA DETERMINE THE STRUCTURE OF A C^＊-ALGEBRA ？

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von Neumann代数中套子代数的摄动

本文主要讨论ｖｏｎ　Ｎｅｕｍａｎｎ代数中套子代数的摄动．给出了因子ｖｏｎ　Ｎｅｕｍａｎｎ代数中套相似的一个充分条件．证明了任何因子ｖｏｎ　Ｎｅｕｍａｎｎ代数中相邻的套子代数经由一个邻近于单位元的可逆算子是相似的．     

von Neumann 代数上保持混合Jordan三重积的非线性映射

本文证明了在没有中心交换投影的von Neumann代数上的一个双映射$\Phi$如果保持混合Jordan三重积, 则$\Phi(I)\Phi$是一个线性*-同构和一个共轭线性*-同构的和, 其中$\Phi(I)$是中心自伴元素且$\Phi(I)^{2}=I$. 同时给出了因子von Neumann 代数上保持混合Jordan三重积映射的结构.     

eO-代数簇公理

扩张Ockham代数簇$e{\bf O}$是由所有$(L;\wedge,\vee, f, k,0,1)$所组成的代数类,其中$(L;\wedge,\vee,0,1)$是有界分配格, $f$是$L$上的偶同态, $k$是$L$ 是$L$上的同态且满足条件: $fk=kf$. 在本文中,我们把Urquhart定理推广到$e{\bf O}$-代数类,并特别考虑$e{\bf O}$-代数的子代数类 $e_2{\bf M}$.在子代数类$e_2{\bf M}$中, $f$和$k$满足条件: $f^{2}=id_L$及$k^{2}=id_L$. 我们证明: 在子代数类$e_2{\bf M}$中,有19个非等价公理.同时我们给出其蕴含关系的表达图式.     

Refinement monoids with weak comparability and applications to regular rings and -algebras

We prove a cancellation theorem for simple refinement monoids satisfying the weak comparability condition, first introduced by K.C. O'Meara in the context of von Neumann regular rings. This result is then applied to von Neumann regular rings and -algebras of real rank zero via the monoid of isomorphism classes of finitely generated projective modules.

     

Co-amenability and Connes’s embedding problem

A longstanding open question of Connes asks whether any finite von Neumann algebra embeds into an ultraproduct of finite-dimensional matrix algebras.As of yet,algebras verified to satisfy the Connes’s embedding property belong to just a few special classes (e.g.,amenable algebras and free group factors).In this article,we prove that von Neumann algebras satisfying Popa’s co-amenability have Connes’s embedding property.     

The Monge-Ampère Equation for Strictly $(n−1)$-Convex Functions with Neumann Condition

A $C^2$ function on $\mathbb{R}^n$ is called strictly $(n-1)$-convex if the sum of any $n-1$ eigenvalues of its Hessian is positive. In this paper, we establish a global $C^2$ estimates to the Monge-Ampère equation for strictly $(n-1)$-convex functions with Neumann condition. By the method of continuity, we prove an existence theorem for strictly $(n-1)$-convex solutions of the Neumann problems.     

Normal state spaces of Jordan and von Neumann algebras

Normal state spaces of Jordan and von Neumann algebras are characterized among convex sets. The normal state spaces of JBW-algebras are precisely those which are spectral and elliptic; along these the normal state spaces of von Neumann algebras are distinguished by the global 3-ball property.