The positivity of linear functionals on Cuntz algebras associated to unit vectors

We study the linear functional on the Cuntz algebra associated to a sequence of unit vectors in that is a generalization of the Cuntz state. We prove that is positive if and only if is a constant sequence.

     

AF-embeddability of crossed products of Cuntz algebras

We investigate crossed products of Cuntz algebras by quasi-free actions of abelian groups. We prove that our algebras are AF-embeddable when actions satisfy a certain condition. We also give a necessary and sufficient condition that our algebras become simple and purely infinite, and consequently our algebras are either purely infinite or AF-embeddable when they are simple.     

Certain Cuntz Semigroup Properties of Certain Crossed Product C*-algebras

We show that the following properties of the C^*-algebras in a class Ω are inherited by simple unital C-algebras in the class TAΩ:(1)(m,n)-decomposable,(2) weakly(m,n)-divisible,(3) weak Riesz interpolation.As an application,let A be an infinite dimensional simple unital C-algebra such that A has one of the above-listed properties.Suppose that α:G→Aut(A) is an action of a finite group G on A which has the tracial Rokhlin property.Then the crossed product C^*-algebra C^*(G,A,α) also has the property under consideration.     

单C~*-代数α-比较性的一种等价刻画

给出C~*-代数α-比较性的等价刻画:对于单的含单位元的稳定有限的C~*-代数A而言,A具有α-比较性,当且仅当对于任意的a,b∈W(A),若α·d_r(a)d_τ(b)(_τ∈QT(A)),则a≤b在Cuntz半群W(A)中成立.利用此刻画,证明了具有α-比较性的C~*-代数一定具有弱比较性;若A具有α-比较性,其中α=m+1,则A具有正元的强迹m-比较性;对于满足Kirchberg-R?rdam条件的C~*-代数,E-稳定、严格比较、α-比较性(α=m+1)、强迹m-比较性、弱比较性以及局部弱比较性彼此等价;若α:=inf{α′∈(1,∞)|A具有α′-比较}∞,则A具有α-比较性.     

Mean dimension and AH-algebras with diagonal maps

Mean dimension for AH-algebras with diagonal maps is introduced. It is shown that if a simple unital AH-algebra with diagonal maps has mean dimension zero, then it has strict comparison on positive elements. In particular, the strict order on projections is determined by traces. Moreover, a lower bound of the mean dimension is given in term of Toms' comparison radius. Using classification results, if a simple unital AH-algebra with diagonal maps has mean dimension zero, it must be an AH-algebra without dimension growth. Two classes of AH-algebras with diagonal maps are shown to have mean dimension zero: the class of AH-algebras with at most countably many extremal traces, and the class of AH-algebras with numbers of extreme traces which induce same state on the _K0${\mathrm{K}}_0$-group being uniformly bounded (in particular, this class includes AH-algebras with real rank zero).     

Standard noncommuting and commuting dilations of commuting tuples

We introduce a notion called `maximal commuting piece' for tuples of Hilbert space operators. Given a commuting tuple of operators forming a row contraction, there are two commonly used dilations in multivariable operator theory. First there is the minimal isometric dilation consisting of isometries with orthogonal ranges, and hence it is a noncommuting tuple. There is also a commuting dilation related with a standard commuting tuple on boson Fock space. We show that this commuting dilation is the maximal commuting piece of the minimal isometric dilation. We use this result to classify all representations of the Cuntz algebra coming from dilations of commuting tuples.

     

Minimal Representations of Cuntz Algebras

Suppose that O_n(2 ≤ n ≤∞) is a Cuntz algebra. There exists a number 1 ≤ k(π, Ω) ≤∞such that if the cyclic representations(H, π, Ω) and(H, π, Ω) of the Cuntz algebra O_n are unitary equivalent, k(π, Ω) = k(π, Ω). Applying the number, one can define a minimal representation of O_n and give a sufficient condition of the minimality for a representation of O_n. Moreover, the relations between minimal states and minimal representations of Cuntz algebras and the relations between the minimality and the irreducibility of the representation of O_n are investigated, respectively.     

Lifting fixed points of completely positive mappings

Let 1?n?∞, and let be a row contraction on some Hilbert space H. Let F(T) be the space of all X∈B(H) such that . We show that, if non-zero, this space is completely isometric to the commutant of the Cuntz part of the minimal isometric dilation of .     

A relation between certain interpolated Cuntz algebras and interpolated free group factors

We investigate von Neumann algebras generated by the real parts of generators of Toeplitz extensions of interpolated Cuntz algebras on sub-Fock spaces. We show that some of them are isomorphic to interpolated free group factors . For example, in case of the golden number the corresponding number is .

     

拟对角扩张Cuntz半群的某些性质

设O→J→A→B→O是一个拟对角扩张.作者证明如果J和B具有Cuntz半群的某些性质,则A也具有相同的半群性质.     

On the nuclearity of certain Cuntz‐Pimsner algebras

In the present paper, we give a short proof of the nuclearity property of a class of Cuntz‐Pimsner algebras associated with a Hilbert ??‐bimodule ??, where ?? is a separable and nuclear C *‐algebra. We assume that the left ??‐action on the bimodule ?? is given in terms of compact module operators and that ?? is direct summand of the standard Hilbert module over ?? (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

非单C~*-代数α-比较性的等价刻画

给出非单C~*-代数α-比较性的等价刻画:当每个τ∈QT(■H)均为忠实时,■具有α-比较性,当且仅当对于任意的〈a〉,〈b〉∈Cu(■)且∝,若α·d_τ(a)≤在Cu(■)中成立;一般地,当QT(■H)≠Φ时,■具有α-比较性,当且仅当对于任意的,∈Cu(■),若存在η>0,使得d_τ(α)≤(α~(-1)-η)d_τ(b)(_τ∈QT(■H)),则≤在Cu(■)中成立.     

Minimal Cuntz-Krieger dilations and representations of Cuntz-Krieger algebras

Given a contractive tuple of Hilbert space operators satisfying certainA-relations we show that there exists a unique minimal dilation to generators of Cuntz-Krieger algebras or its extension by compact operators. This Cuntz-Krieger dilation can be obtained from the classical minimal isometric dilation as a certain maximalA-relation piece. We define a maximal piece more generally for a finite set of polynomials inn noncommuting variables. We classify all representations of Cuntz-Krieger algebrasO _A obtained from dilations of commuting tuples satisfyingA-relations. The universal properties of the minimal Cuntz-Krieger dilation and the WOT-closed algebra generated by it is studied in terms of invariant subspaces.     

Dilating covariant representations of the non-commutative disc algebras

Let φ be an isometric automorphism of the non-commutative disc algebra An for n?2. We show that every contractive covariant representation of (An,φ) dilates to a unitary covariant representation of (On,φ). Hence the C^∗-envelope of the semicrossed product Anφ_×Z⁺ is Onφ_×Z.     

Simplicial normalization in the entire cyclic cohomology of Banach algebras

We show that the entire cyclic cohomology of Banach algebras defined by Connes has the simplicial normalization property. A key tool in the proof is the notion and properties of supertraces on the Cuntz algebraQ A. As an example of further applications of this technique, we give a proof of the homotopy invariance of entire cyclic cohomology.Address after 1st October 1995: Department of Mathematics, University of Exeter, North Park Road, Exeter EX4 4QE, U.K.     

GENERALIZED SIMPLE NONCOMMUTATIVE TORI

The generalized noncommutative torus Tkp of rank n was defined in [4] by the crossed product Am/k ×a3 Z ×a4 … ×an Z, where the actions ai of Z on the fibre Mk(C) of a rational rotation algebra Am/k are trivial, and C*(kZ × kZ) ×a3 Z ×a4 ... ×an Z is a completely irrational noncommutative torus Ap of rank n. It is shown in this paper that Tkp is strongly Morita equivalent to Ap, and that Tkp (?) Mp∞ is isomorphic to Ap (?) Mk(C) (?) Mp∞ if and only if the set of prime factors of k is a subset of the set of prime factors of p.     

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

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