可能行无限图C~*-代数

本文研究可能行无限有向图的C~*-代数。对于一个可能行无限的有向图E,通过引进集合S(μ,v),将行无限点上的算子拓扑强收敛关系代数化表示出来,并由此构造了一个结构丰富的非零*-代数H_E;进而利用H_E证明了一个由Cuntz-Krieger E-族{s_e,p_v}生成的泛C~*-代数 C~*(E)的存在性,并且证明了H_E和 C~*(E)在图同构意义下不依赖于E的选择,从而是可能行无限有向图的同构不变量。     

<Emphasis Type="Italic">C</Emphasis>*-Algebras Generated by Mappings. Classification of Invariant Subspaces

We continue the study of an operator algebra associated with a self-mapping ? on a countable setX which can be represented as a directed graph. This C*-algebra belongs to a class of operator algebras, generated by a family of partial isometries satisfying some relations on their source and range projections. Earlier we have formulated the irreducibility criterion of such algebras, which give us a possibility to examine the structure of the corresponding Hilbert space. We will show that for reducible algebras the underlying Hilbert space can be represented either as an infinite sum of invariant subspaces or as a tensor product of a finite-dimensional Hilbert space with l²(Z). In the first case we present a conditions under which the studied algebra has an irreducible representation into a C*-algebra generated by a weighted shift operator. In the second case, the algebra has the irreducible finite-dimensional representations indexed by the unit circle.     

Stacey Crossed Products Associated to Exel Systems

There are many different crossed products by an endomorphism of a C*-algebra, and constructions by Exel and Stacey have proved particularly useful. Here we consider Exel crossed products associated to transfer operators which extend to be unital on the multiplier algebra. We show that every Exel crossed product is isomorphic to a Stacey crossed product, though by a different endomorphism of a different C*-algebra. We apply this result to a variety of Exel systems, including those associated to shifts on the path spaces of directed graphs.     

<Emphasis Type="Italic">C</Emphasis>*-algebras generated by cancellative semigroups

A C*-algebra generated by a commuting family of isometries is a natural generalization of the Toeplitz algebra. We study the *-automorphisms and invariant ideals of the C*-algebra geerated by a semigroup.     

Universal Operator Algebras Associated to Contractive Sequences of Non-Commuting Operators

We characterize the minimal isometric dilation of a non-commutativecontractive sequence of operators as a universal object forcertain diagrams of completely positive maps. A non-spatialconstruction of the minimal isometric dilation is also given,using Hilbert modules over C*-algebras. It is shown that the non-commutative disc algebras A_n (n2) arethe universal algebras generated by contractive sequences ofoperators and the identity, and C*(S₁, ..., S_n) (n2), the extensionthrough compact operators of the Cuntz algebra O_n, is the universalC*-algebra generated by a contractive sequence of isometries.It is also shown that the algebras A_n and C*(S₁, ..., S_n) arecompletely isometrically isomorphic to some free operator algebrasconsidered by D. Blecher. In particular, the universal operatoralgebra of a row (respectively column) contraction is identifiedwith a subalgebra of C*(S₁, ..., S_n). The internal characterizationof the matrix norm on a universal algebra leads to some factorizationtheorems.     

On a Class of Graded Ideals of Semigroup <Emphasis Type="Italic">C</Emphasis>*-Algebras

We present general results about graded C*-algebras and continue the previously initiated research of the C*-algebra generated by the left regular representation of an abelian semigroup. We study the invariant ideals of this C*-algebra invariant with respect to the representation of a compact group G in the automorphism group of this algebra. We prove that the invariance of the ideal is equivalent to the fact that this ideal is graded C*-algebra, that there is a maximum of all invariant ideals, and it is the commutator ideal. Separately we study a class of graded primitive ideals generated by a single projector.     

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.     

Chern Character for the Schwartz Algebra of p-Adic GL(n)

Let G denote the p-adic group GL(n), and let S(G) denote theSchwartz algebra of G. We construct a Chern character from theK-theory of the reduced C*-algebra of G to the periodic cyclichomology of S(G) which becomes an isomorphism after tensoringover Z with C.     

Ideals in Graph Algebras

We show that the graph construction used to prove that a gauge-invariant ideal of a graph C ^???-algebra is isomorphic to a graph C ^???-algebra, and also used to prove that a graded ideal of a Leavitt path algebra is isomorphic to a Leavitt path algebra, is incorrect as stated in the literature. We give a new graph construction to remedy this problem, and prove that it can be used to realize a gauge-invariant ideal (respectively, a graded ideal) as a graph C ^???-algebra (respectively, a Leavitt path algebra).     

BANACH ALGEBRAS WITH LARGE GROUPS OF UNITARY ELEMENTS

We study unitary Banach algebras, as defined by M. L. Hansenand R. V. Kadison in 1996, as well as some related conceptslike maximal or uniquely maximal Banach algebras. We show thata norm-unital Banach algebra is uniquely maximal if and onlyif it is unitary and has minimality of the equivalent norm.We prove that every unitary semisimple commutative complex Banachalgebra has a conjugate-linear involution mapping each unitaryelement to its inverse, and that, endowed with such an involution,becomes a hermitian *-algebra. The possibility of removing therequirement of commutativity in the above statement is alsoconsidered. The paper concludes by translating to real algebrassome results previously known in the complex case. In particular,we show that every maximal semisimple finite-dimensional realBanach algebra is isometrically isomorphic to a real C*-algebra.     

Property A, partial translation structures, and uniform embeddings in groups

We define the concept of a partial translation structure ona metric space X and show that there is a natural C*-algebraC*() associated with it, which is a subalgebra of the uniformRoe algebra C*_u(X). We introduce a coarse invariant of the metricwhich provides an obstruction to embedding the space in a group.When the space is sufficiently group-like, as determined byour invariant, properties of the Roe algebra can be deducedfrom those of C*(). We also give a proof of the fact that theuniform Roe algebra of a metric space is a coarse invariantup to Morita equivalence.     

The graded structure of Leavitt path algebras

A Leavitt path algebra associates to a directed graph a ?-graded algebra and in its simplest form it recovers the Leavitt algebra L(1, k). In this note, we first study this ?-grading and characterize the (?-graded) structure of Leavitt path algebras, associated to finite acyclic graphs, C _n -comet, multi-headed graphs and a mixture of these graphs (i.e., polycephaly graphs). The last two types are examples of graphs whose Leavitt path algebras are strongly graded. We give a criterion when a Leavitt path algebra is strongly graded and in particular characterize unital Leavitt path algebras which are strongly graded completely, along the way obtaining classes of algebras which are group rings or crossed-products. In an attempt to generalize the grading, we introduce weighted Leavitt path algebras associated to directed weighted graphs which have natural ⊕?-grading and in their simplest form recover the Leavitt algebras L(n, k). We then show that the basic properties of Leavitt path algebras can be naturally carried over to weighted Leavitt path algebras.     

On the classification of nonsimple graph C *-algebras

We prove that a graph C ^*-algebra with exactly one proper nontrivial ideal is classified up to stable isomorphism by its associated six-term exact sequence in K-theory. We prove that a similar classification also holds for a graph C ^*-algebra with a largest proper ideal that is an AF-algebra. Our results are based on a general method developed by the first named author with Restorff and Ruiz. As a key step in the argument, we show how to produce stability for certain full hereditary subalgebras associated to such graph C ^*-algebras. We further prove that, except under trivial circumstances, a unique proper nontrivial ideal in a graph C ^*-algebra is stable.     

The C*-envelope of the Tensor Algebra of a Directed Graph

Given an arbitrary countable directed graph G we prove the C*- envelope of the tensor algebra coincides with the universal Cuntz- Krieger algebra associated with G. Our approach is concrete in nature and does not rely on Hilbert module machinery. We show how our results extend to the case of higher rank graphs, where an analogous result is obtained for the tensor algebra of a row-finite k-graph with no sources.     

On the <Emphasis Type="Italic">K</Emphasis>-Theory of Graph <Emphasis Type="Italic">C</Emphasis><Superscript>*</Superscript>-Algebras

We classify graph C ^*-algebras, namely, Cuntz-Krieger algebras associated to the Bass-Hashimoto edge incidence operator of a finite graph, up to strict isomorphism. This is done by a purely graph theoretical calculation of the K-theory of the C ^*-algebras and the method also provides an independent proof of the classification up to Morita equivalence and stable equivalence of such algebras, without using the boundary operator algebra. A direct relation is given between the K ₁-group of the algebra and the cycle space of the graph. We thank Jakub Byszewski for his input in Sect. 2.8. The position of the unit in K ₀( _Ч) was guessed based on some example calculations by Jannis Visser in his SCI 291 Science Laboratory at Utrecht University College.     

Operator space projective tensor product of $C^*$-algebras

For a finite dimensional -algebra A and any -algebra B, we determine a constant of equivalence of operator space projective norm and the Banach space projective norm on . We also discuss the *-Banach algebra . Received May 12, 1999; in final form September 8, 1999 / Published online April 12, 2001     

Representations of C*-algebras in spaces with indefinite metric

The problem is considered of orthogonalization of J-symmetric representations of C^*-algebras in the Pontryagin spaces π_x. It is proved that in spaces with finite rank of indefiniteness, every such representation is similar to a *-representation in a Hilbert space. Necessary and sufficient conditions are established for the existence of an invariant dual pair of subspaces for a J-symmetric operator algebra.     

Automorphism groups of graphs and edge-contraction

If a class C of finite graphs is closed under contraction and forming subgraphs, and if every finite abstract group occurs as the automorphism group of some graph in C, then C contains all finite graphs (up to isomorphism). Also related results concerning automorphism groups of graphs on given surfaces are mentioned.