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.     

The Daugavet Property of C*-Algebras and Non-commutative Lp-Spaces

We prove that a C ^*-algebra A or a predual N _* of a von Neumann algebra N has the Daugavet property if and only if A (or N) is non-atomic. We also prove a similar (although somewhat weaker) result for non-commutative L _p-spaces corresponding to non-atomic von Neumann algebras.     

Property (T) for C*-Algebras

A notion of Property (T) is defined for an arbitrary unitalC*-algebra A admitting a tracial state. This is extended toa notion of Property (T) for a pair (A, B) where B is a C*-subalgebraof A. Let be a discrete group and its reduced algebra. We show that has Property (T) if and only if the group has Property (T).More generally, given a subgroup of , the pair has Property (T) if and only if the pair of groups(, ) has Property (T). 2000 Mathematics Subject Classification46L05, 22D25.     

The Tracial Rokhlin Property for Automorphisms on Non-simple C^＊-Algebras

Let A be a unital AF-algebra （simple or non-simple） and let α be an automorphism of A. Suppose that α has certain Rokhlin property and A is α-simple. Suppose also that there is an integer J ≥ 1 such that J α＊^J0 =idKo（A）. The author proves that A α Z has tracial rank zero.     

C*-Algebras with the Approximate Positive Factorization Property

We say that a unital -algebra has the approximate positive factorization property (APFP) if every element of is a norm limit of products of positive elements of . (There is also a definition for the nonunital case.) T. Quinn has recently shown that a unital AF algebra has the APFP if and only if it has no finite dimensional quotients. This paper is a more systematic investigation of -algebras with the APFP. We prove various properties of such algebras. For example: They have connected invertible group, trivial , and stable rank 1. In the unital case, the group separates the tracial states. The APFP passes to matrix algebras, and if is an ideal in such that and have the APFP, then so does . We also give some new examples of -algebras with the APFP, including type factors and infinite-dimensional simple unital direct limits of homogeneous -algebras with slow dimension growth, real rank zero, and trivial group. Simple direct limits of homogeneous -algebras with slow dimension growth which have the APFP must have real rank zero, but we also give examples of (nonsimple) unital algebras with the APFP which do not have real rank zero. Our analysis leads to the introduction of a new concept of rank for a -algebra that may be of interest in the future.

     

实C~*-代数中*运算的唯一性

本文指出：如果有两个*运算使得同一个实Banach代数均成为实C*-代数,则这两个*运算必然是相同的,即实C*-代数中*运算是唯一的．     

Hopf C*-Algebras

In this paper we define and study Hopf C^*-algebras. Roughlyspeaking, a Hopf C^*-algebra is a C^*-algebra A with a comultiplication: A M(A A) such that the maps a b (a)(1 b) and a (a 1)(b)have their range in A A and are injective after being extendedto a larger natural domain, the Haagerup tensor product A _hA. In a purely algebraic setting, these conditions on are closelyrelated to the existence of a counit and antipode. In this topologicalcontext, things turn out to be much more subtle, but neverthelessone can show the existence of a suitable counit and antipodeunder these conditions. The basic example is the C^*-algebra C₀(G) of continuous complexfunctions tending to zero at infinity on a locally compact groupwhere the comultiplication is obtained by dualizing the groupmultiplication. But also the reduced group C^*-algebra of a locally compact group with thewell-known comultiplication falls in this category. In factall locally compact quantum groups in the sense of Kustermansand the first author (such as the compact and discrete ones)as well as most of the known examples are included. This theory differs from other similar approaches in that thereis no Haar measure assumed. 2000 Mathematics Subject Classification: 46L65, 46L07, 46L89.     

Real C*-Algebras, United KK-Theory, and the Universal Coefficient Theorem

We define united KK-theory for real C*-algebras A and B such that A is separable and B is -unital, extending united K-theory in the sense that KK^CRT( , B) = K^CRT(B). United KK-theory combines real, complex, and self-conjugate KK-theory; but unlike unaugmented KK-theory for real C*-algebras, it admits a Universal Coefficient Theorem. For all separable A and B in which the complexification of A is in the bootstrap category, KK^CRT(A,B) appears as the middle term of a short exact sequence whose outer terms involve the united K-theory of A and B. As a corollary, we prove that united K-theory classifies KK-equivalence for real C*-algebras whose complexification is in the bootstrap category.Mathematics Subject Classification (2000): 19K35, 46L80.     

C~*-代数的实完全等距映射

C*-代数的*-同构一定是(完全)等距映射,反之不然.本文证明了C*-代数的实完全等距映射能够完全决定C*-代数*-同构的结论.     

Asymptotic Homomorphisms of C *-Algebras and C *-Extensions

We present a brief introduction to two theories in the category of C ^*-algebras—theory of asymptotic homomorphisms and theory of extensions—and explain how these theories are related to each other.     

Intrinsic Covering Dimension for Nuclear C*-Algebras with Real Rank Zero

For operators belonging either to a class of global bisingular pseudodifferential operators on \({{\mathbb{R}^{m}} \times {\mathbb{R}^{n}}}\) or to a class of bisingular pseudodifferential operators on a product \({M \times N}\) of two closed smooth manifolds, we show the equivalence of their ellipticity (defined by the invertibility of certain operator-valued, homogeneous principal symbols) and their Fredholm mapping property in associated scales of Sobolev spaces. We also prove the spectral invariance of these operator classes and then extend these results to larger classes of Toeplitz type operators.     

C*-Algebras and Controlled Topology

We describe some aspects of the relationship between the controlled topology and C*-algebra approaches to the Novikov conjecture.     

The Tracial Topological Rank of C*-Algebras

We introduce the notion of tracial topological rank for C*-algebras.In the commutative case, this notion coincides with the coveringdimension. Inductive limits of C*-algebrasof the form PM_n(C(X))P,where X is a compact metric space with dim X k, and P is aprojection in M_n(C(X)), have tracial topological rank no morethan k. Non-nuclear C*-algebras can have small tracial topologicalrank. It is shown that if A is a simple unital C*-algebra withtracial topological rank k (< ), then

(i) A is quasidiagonal,

(ii) A has stable rank 1,

(iii) A has weakly unperforatedK₀(A),

(iv) A has the following Fundamental Comparabilityof Blackadar:if p, q A are two projections with (p) < (q)for all tracialstates on A, then p q

. 2000 MathematicsSubject Classification: 46L05, 46L35.     

Normal Elements of C*-Algebras of Real Rank Zero without Finite-Spectrum Approximants

We investigate inductive limits of Toeplitz-type C*-algebras.One example, which has real-rank zero, is the middle term ofan exact sequence where is a Bunce-Deddens algebra and I is AF. Using Berg's technique,we produce a normal element N that is not the limit of finite-spectrumnormals. Moreover, this is an example of a normal element inan inductive limit that is not the limit of normal elementsof the approximating subalgebras. A second example is an embedding of C() ( the closed disk) into , where is a simple AF algebra and is the Toeplitz algebra.Let _n, for n 2, be the CW complex obtained as the quotientof by an n-fold identification of the boundary. (So ₂ = RP².)Regarding C(_n) as a subalgebra of C(), we find nontrivial embeddingsof C(_n) into type I inductive limits. From this, we producea *-homomorphism, for n odd, C₀(_n\{pt}) _{n + 1}, that inducesan isomorphism on K-theory. More generally, for X a connectedCW complex minus a point, and for n odd, we show that the map is a split surjection.     

C*-Bundles for Certain Liminal C*-Algebras

It is shown that certain liminal C*-algebras whose limit sets in their primitive ideal space are discrete can be described as algebras of continuous sections of a C*-bundle associated with them. Their multiplier algebras are also described in a similar manner. The class of C*-algebras under discussion includes all the liminal C*-algebras with Hausdorff primitive ideal spaces but also many other liminal algebras. A large sub-class of examples is examined in detail.      

Infinitesimal Generators of C *-Algebras

We develop the method introduced previously, to construct infinitesimal generators on locally compact group C ^*-algebras and on tensor product of C ^*-algebras. It is shown in particular that there is a C ^* -algebra A such that the C ^*-tensor product of A and an arbitrary C ^*-algebra B can have a non-approximately inner strongly one parameter group of ^*-automorphisms.     

Projective Convex Combinations in C *-Algebras with the Unitary Factorization Property

Mathematical Notes -