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.     

A boolean transfer principle from L*-Algebras to AL*-Algebras

Just as Kaplansky [4] has introduced the notion of an AW*-module as a generalization of a complex Hilbert space, we introduce the notion of an AL*-algebra, which is a generalization of that of an L*-algebra invented by Schue [9, 10]. By using Boolean valued methods developed by Ozawa [6–8], Takeuti [11–13] and others, we establish its basic properties including a fundamental structure theorem. This paper should be regarded as a continuation or our previous paper [5], the familiarity with which is presupposed. MSC: 03C90, 03E40, 17B65, 46L10.     

C*-Algebras and Controlled Topology

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

Generalized inverses inC*-Algebras

Viene estesa al contesto delle algebreC* la teoria di base degli inversi generalizzati. In particolare il risultato di Penrose in relazione alla riduzione di ‖AX - C‖2 alla normaC*; e vengono date condizioni modificate in modo tale che risulti almeno un inverso Moore-Penrose di un dato elemento.     

Derivations on LMC*-Algebras

It is shown that derivations on LMC*-algebras are always continuous and generate a continuous one-parameter group of automorphisms. The structure of the derivation and the automorphism group on LMC*-algebras is investigated.     

Prime Non-Commutative JB*-Algebras

We prove that if A is a prime non-commutative JB*-algebra whichis neither quadratic nor commutative, then there exist a primeC*-algebra B and a real number with < 1 such that A =B as involutive Banach spaces, and the product of A is relatedto that of B (denoted by , say) by means of the equality xy= xy+(1 – )yx. 2000 Mathematics Subject Classification46K70, 46L70.     

Dual Pairs of Hopf *-Algebras

If A is a Hopf *-algebra, the dual space A' is again a *-algebra.There is a natural subalgebra A° of A' that is again a Hopf*-algebra. In many interesting examples, A° will be largeenough (to separate points of A). More generally, one can considera pair (A, B) of Hopf *-algebras and a bilinear form on A xB with conditions such that, if the pairing is non-degenerate,one algebra can be considered as a subalgebra of the dual ofthe other. In these notes, we study such pairs of Hopf *-algebras. We startfrom the notion of a Hopf *-algebra A and its reduced dual A°.We give examples of pairs of Hopf *-algebras, and discuss theproblem of non-degeneracy. The first example is an algebra pairedwith itself. The second example is the pairing of a Hopf *-algebra(due to Jimbo) and the twisted SU(n) of Woronowicz. We alsodiscuss the notion of the quantum double of Drinfeld in thisframework of dual pairs.     

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.     

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.     

Spectral Properties of Partial *-Algebras

We continue our study of topological partial *algebras focusing our attention to some basic spectral properties. The special case of partial *-algebras of operators is examined first, in order to find sufficient hints for the study of the abstract case. The outcome consists in the selection of a class of topological partial *-algebras (partial GC*-algebras) that behave well from the spectral point of view and that allow, under certain conditions, a faithful realization as a partial O*-algebra.     

W~*-代数中的C~*-端点

本文讨论Ｗ＊－代数Ａ中闭单位球Ａ１的Ｃ＊－端点问题．主要结果是：是一个投影｝；一个对称元｝；其中，Ａ～Ｓ１和Ａ～Ｐ１分别表Ａ１中自伴元全体所构成的集合和Ａ１中正元全体所构成的集合．     

PrO-C*-代数的顺从性和核性

研究了Pro—C^*-代数的顺从性和核性．主要证明了(1)顺从Pro—C^*代数的闭理想是顺从的；(2)核Pro—C^*代数类对归纳极限封闭；(3)交换σ-C^*-代数和核C^*-代数都是核，σ-C^*-代数并且核σ-C^*-代数类对于商运算、张量积运算和可数逆向极限封闭．进一步得到核，σ-C^*-代数的扩张保持核性的条件。     

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.     

Homomorphisms between Poisson JC*-Algebras

It is shown that every almost linear mapping of a unital Poisson JC*-algebra to a unital Poisson JC*-algebra is a Poisson JC*-algebra homomorphism when h(2 ⁿ uy) = h(2 ⁿ u) h(y), h(3 ⁿ u y) = h(3 ⁿ u) h(y) or h(q ⁿ u y) = h(q ⁿ u) h(y) for all , all unitary elements and n = 0, 1, 2, · · · , and that every almost linear almost multiplicative mapping is a Poisson JC*-algebra homomorphism when h(2x) = 2h(x), h(3x) = 3h(x) or h(qx) = qh(x) for all . Here the numbers 2, 3, q depend on the functional equations given in the almost linear mappings or in the almost linear almost multiplicative mappings.Moreover, we prove the Cauchy–Rassias stability of Poisson JC*-algebra homomorphisms in Poisson JC*-algebras.*This work was supported by grant No. R05-2003-000-10006-0 from the Basic Research Program of the Korea Science & Engineering Foundation.     

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.     

Active lattices determine AW*-algebras

We prove that operator algebras that have enough projections are completely determined by those projections, their symmetries, and the action of the latter on the former. This includes all von Neumann algebras and all AW*-algebras. We introduce active lattices, which are formed from these three ingredients. More generally, we prove that the category of AW*-algebras is equivalent to a full subcategory of active lattices. Crucial ingredients are an equivalence between the category of piecewise AW*-algebras and that of piecewise complete Boolean algebras, and a refinement of the piecewise algebra structure of an AW*-algebra that enables recovering its total structure.     

Pro—C^＊—代数中的次正规元

探讨了Pro-C^*-代数中的次正规元,给出了具有余等距对Pro-C^*-代数中次正规元的一个代数特征。     

Domination theorems in non-commutativeC *-Algebras

LetA be a Banach algebra with unite. Ifu, v ₁, …,v _n are elements ofA thenu is said to be dominated byv ₁, …,v _n if there is γ>O such that for everyx∈A we have ‖ux‖≤γ(‖v ₁ x‖+…+‖v _n x‖). It is shown that for everyx∈A we have ‖ux‖≤γ(‖v ₁ x‖+…+‖v _n x‖). it is shown for finite-dimensional algebras that the above condition is sufficient foru to belong to the left ideal generated byv ₁, …,v _n in some superalgebra ofA. A similar result is proved for subalgebras of the algebra of all bounded operators on a Hilbert space.     

Locally Convex Quasi *-Algebras of Operators

This note is mainly concerned with locally convex quasi C*-normed *-algebras which arise as completions of C*-algebras of operators under certain topologies. Their importance is made clear by the representation theory of abstract locally convex quasi C*-normed *-algebras, investigated in previous papers and whose basic aspects are also overviewed here.