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.     

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.     

Bivariant K-Theories for C*-Algebras

A new framework for bivariant K-theory is developed. Various types of homology-cohomology theories are discussed. Our techniques can be used for producing natural elements in E-theory out of continuous fields with non-isomorphic fibers. An alternative definition for the Kasparov product in E-theory is proposed.     

C*-Algebras and Controlled Topology

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

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.     

Hilbert C*-Modules over Monotone Complete C*-Algebras

The aim of the present paper is to describe self-duality and C*-reflexivity of Hilbert A-modules ?? over monotone complete C*-algebras A by the completeness of the unit ball of ?? with respect to two types of convergence being defined, and by a structural criterion. The derived results generalize earlier results ofH. Widom [Duke Math. J. 23, 309-324, MR 17 # 1228] and W. L. Paschke [Trans. Amer. Mat. Soc. 182 , 443-468, MR 50 # 8087, Canadian J. Math. 26, 1272-1280, MR 57 # 10433]. For Hilbert C*-modules over commutative AW*-algebras the equivalence of the self-duality property and of the Kaplansky-Hilbert property is reproved, (cf. M. Ozawa [J. Math. Soc. Japan 36, 589-609, MR 85 # 46068]). Especially, one derives that for a C*-algebra A the A-valued inner product of every Hilbert A-module ?? can be continued to an A-valued inner product on it's A-dual Banach A-module ??' turning ??' to a self-dual Hilbert A-module if and only if A is monotone complete (or, equivalently, additively complete) generalizing a result of M. Hamana [Internat. J. Math. 3 (1992), 185 - 204]. A classification of countably generated self-dual Hilbert A-modules over monotone complete C*-algebras A is established. The set of all bounded module operators End ′(??) on self-dual Hilbert A-modules ?? over monotone complete C*-algebras A is proved again to be a monotone complete C*-algebra. Applying these results a Weyl-Berg type theorem is proved.     

Decomposition Rank of Subhomogeneous C*-Algebras

We analyze the decomposition rank (a notion of covering dimensionfor nuclear C*-algebras introduced by E. Kirchberg and the author)of subhomogeneous C*-algebras. In particular, we show that asubhomogeneous C*-algebra has decomposition rank n if and onlyif it is recursive subhomogeneous of topological dimension n,and that n is determined by the primitive ideal space. As an application, we use recent results of Q. Lin and N. C.Phillips to show the following. Let A be the crossed productC*-algebra coming from a compact smooth manifold and a minimaldiffeomorphism. Then the decomposition rank of A is dominatedby the covering dimension of the underlying manifold. 2000 MathematicsSubject Classification 46L85, 46L35.     

Spectral Properties of Abelian C*-Algebras

Let be an Abelian unital C ^*-algebra and let denote its Gelfand spectrum. We give some necessary and sufficient conditions for a nondegenerate representation of to be unitarily equivalent to a representation in which the elements of act multiplicatively, by their Gelfand transforms, on a space L ²( ,), where is a positive measure on the Baire sets of . We also compare these conditions with the multiplicity-free property of a representation.     

Morita Equivalence for Locally C*-Algebras

The concept of Morita equivalence is generalized to the contextof locally C^*-algebras. This generalizes a well-known theoremof Brown, Green and Rieffel, Pacific J. Math. 71 (1977) 349–363.2000 Mathematics Subject Classification 46L08, 46L05.     

On C*-Algebras from Interval Maps

Given a unimodal interval map f, we construct partial isometries acting on Hilbert spaces associated to the orbit of each point. Then we prove that such partial isometries give rise to representations of a C*-algebra associated to the subshift encoding the kneading sequence of the critical point. This construction has the advantage of incorporating maps with a non necessarily Markov partition (e.g. Fibonacci unimodal map). If we are indeed in the presence of a finite Markov partition, then we prove that these new representations coincide with the (previously considered by the authors) representations arising from the Cuntz–Krieger algebra of the underlying (finite) transition matrix.     

C*-Algebras Characterized by Ergodic Actions of the n-Torus

Using basic harmonic analysis on the n-torus, it is shown thatthe non-commutative n-tori are the only primitive C*-algebras(up to isomorphism) on which the n-torus Tⁿ acts freely andergodically. An explicit construction recovers the generatingunitaries in a natural way from the action. 2000 MathematicsSubject Classification 46L55.     

C*-Algebras with Dense Nilpotent Subalgebras

In a beautiful result, Herrero (D. A. Herrero, ‘Normallimits of nilpotent operators’, Indiana Univ. Math. J.23 (1973/74) 1097–1108) showed that a normal operatoron l² lies in the closure of the set of nilpotent operatorsif and only if its spectrum is connected and contains zero.In the quest for an automatic continuity result for algebrahomomorphisms between C* -algebras, Dales showed that, if adiscontinuous algebra homomorphism : A u exists between C*-algebrasA and u, and if (A) is dense in u, then there is a C*-algebrau₂ with a dense subalgebra N u² such that every x N is quasinilpotent(see p. 685 of H. G. Dales, Banach algebras and automatic continuity,London Mathematical Society Monographs 24, Oxford UniversityPress, 2001). (A discontinuous homomorphism ₂: A₂ u₂ can bedefined with the same basic properties as , but the revisedtarget space u₂ has a dense subalgebra consisting of quasinilpotentelements.) As remarked by Dales, no such C*-algebra was thenknown; but here we present one. Indeed, using the full powerof Herrero's result, one may arrange that every x N is nilpotent.The C*-algebra is constructed in a ‘neat’ way; itis most naturally constructed as a non-separable, concrete C*-algebraof operators on a separable Hilbert space K but one can arrangethat the algebra u itself be separable if desired. 2000 MathematicsSubject Classification 47C15, 46H40 (primary), 47A10, 46L06,46L05, 46H35 (secondary).     

C*-代数的迹迹秩

引入C*-代数迹迹秩的概念,讨论它的基本性质.另外,迹迹秩为零和迹拓扑秩为零的C*-代数等价,同时讨论这类代数的拟对角扩张性质.设0→I→ A→A/I→0是拟对角扩张的短正合列,证明如果TTR(I)≤k且TTR(A/I)=0,则TTR(A)≤k.     

Unbounded C*-Seminorms and Biweights on Partial *-Algebras

Unbounded C*-seminorms generated by families of biweights on a partial *-algebra are considered and the admissibility of biweights is characterized in terms of unbounded C*-seminorms they generate. Furthermore, it is shown that, under suitable assumptions, when the family of biweights consists of all those ones which are relatively bounded with respect to a given C*-seminorm q, it can be obtained an expression for q analogous to that one which holds true for the norm of a C*-algebra.     

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.     

Elementary C*-Algebras and Haagerup Tensor Products

In this note,we give a new characterization of elementary C*-algebras in terms of completely compact maps and Haagerup tensor products.     

Conorm and Essential Conorm in C*-Algebras

Let A be a unital C^*-algebra with non-zero socle (soc(A)). We introduce the essential conorm of an element a in A (denoted by γ _e (a)), as the conorm of the element π(a), where π denotes the canonical projection of A onto . It is established that for every von Neumann regular element , γ _e (a) = max . We characterize the continuity points of the conorm and essential conorm for extremally rich C^*-algebras. Some formulae for the distance from zero to the generalized spectrum and Atkinson spectrum are also obtained. Authors partially supported by I+D MEC projects no. MTM2005-02541 and MTM2007-65959, and Junta de Andalucía grants FQM0199 and FQM1215.     

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.     

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.