Certain classes of <Emphasis Type="Italic">C</Emphasis>*-algebras preserved by tracial approximation

We show that the following classes of C*-algebras in the classes Ω are inherited by simple unital C*-algebras in the classes TAΩ: (1) simple unital purely infinite C*-algebras, (2) unital isometrically rich C*-algebras, (3) unital Riesz interpolation C*-algebras.     

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.     

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

In this paper we study C*-algebras generated by a commuting family of isometric operators. Such algebras naturally generalize the Toeplitz algebra. We investigate *-automorphisms and ideals of C*-algebras generated by semigroups.     

The Proof of Inseparable Prime AW^＊—Algebras Being Primitive C^＊—Algebras

The relation between the inseparable prime C^*-algebras and primitive C^*-algebras is studied,and we prove that prime AW^*-algebras are all primitive C^*-algebras.     

<Emphasis Type="Italic">C</Emphasis>*-Non-Linear Second Quantization

We construct an inductive system of C*-algebras each of which is isomorphic to a finite tensor product of copies of the one-mode n-th degree polynomial extension of the usual Weyl algebra constructed in our previous paper (Accardi and Dhahri in Open Syst Inf Dyn 22(3):1550001, 2015). We prove that the inductive limit C*-algebra is factorizable and has a natural localization given by a family of C*-sub-algebras each of which is localized on a bounded Borel subset of \({\mathbb{R}}\). Finally, we prove that the corresponding family of Fock states, defined on the inductive family of C*-algebras, is projective if and only if n = 1. This is a weak form of the no-go theorems which emerge in the study of representations of current algebras over Lie algebras.     

<Emphasis Type="Italic">K</Emphasis>-homology of <Emphasis Type="Italic">C</Emphasis><Superscript><Emphasis Type="Italic">*</Emphasis></Superscript>-categories and symmetric spectra representing <Emphasis Type="Italic">K</Emphasis>-homology

We define K-homology groups K _* () for small C ^* -categories in terms of Hilbert modules over the C ^* -category . We also define a functor A ^f from the category of small C ^* -categories into the category of C ^* -algebras and show that there is a natural isomorphism . In addition, we give an easy construction of a functor from the category of C ^* -algebras into the category of symmetric spectra which represents K-homology, i.e. we show that the functor comes with a natural isomorphism for C ^* -algebras A. It then follows that the composition A ^f provides a functor that can be used in the Davis-Lück approach for constructing the Baum-Connes map.     

Partial isometries and an invariant of <Emphasis Type="Italic">C</Emphasis>*-algebras

In this paper, we will discuss some properties of biprojection-commutative elements which are relevant to the classification of certain infinite C*-algebras, and define an important invariant s(A) of C*-algebra A as well as give some basic properties with regard to s(A). Moreover we prove that the invariant s(A) has continuity.     

A Class of <Emphasis Type="Italic">C</Emphasis>*-algebras with non-stable <Emphasis Type="Italic">K</Emphasis><Subscript>1</Subscript>-group property

In this paper, we give a class of C*-algebras with non-stable K ₁-group property which include the example non-simple tracial topological rank zero and stable rank two C*-algebra given by Lin and Osaka.     

Commutative <Emphasis Type="Italic">C</Emphasis>*-Algebras of Toeplitz Operators on Complex Projective Spaces

We prove the existence of commutative C*-algebras of Toeplitz operators on every weighted Bergman space over the complex projective space \mathbbPⁿ\mathbb(C){{\mathbb{P}^n}\mathbb{(C)}}. The symbols that define our algebras are those that depend only on the radial part of the homogeneous coordinates. The algebras presented have an associated pair of Lagrangian foliations with distinguished geometric properties and are closely related to the geometry of \mathbbPⁿ\mathbb(C){{\mathbb{P}^n}\mathbb{(C)}}.     

State spaces of <Emphasis Type="Italic">JB</Emphasis><Superscript><Emphasis Type="Italic">*</Emphasis></Superscript>-triples<Superscript>*</Superscript>

An atomic decomposition is proved for Banach spaces which satisfy some affine geometric axioms compatible with notions from the quantum mechanical measuring process. This is then applied to yield, under appropriate assumptions, geometric characterizations, up to isometry, of the unit ball of the dual space of a JB^*-triple, and up to complete isometry, of one-sided ideals in C^*-algebras.Mathematics Subject Classification (2000):17C65, 46L07Both authors are supported by NSF grant DMS-0101153     

<Emphasis Type="Italic">C</Emphasis>*-Independence,Product States and Commutation

Let D be a unital C*-algebra generated by C*-subalgebras A and B possessing the unit of D. Motivated by the commutation problem of C*-independent algebras arising in quantum field theory, the interplay between commutation phenomena, product type extensions of pairs of states and tensor product structure is studied. Rooss theorem [11] is generalized in showing that the following conditions are equivalent: (i) every pair of states on A and B extends to an uncoupled product state on D; (ii) there is a representation of D such that (A) and (B) commute and is faithful on both A and B; (iii) is canonically isomorphic to a quotient of D.The main results involve unique common extensions of pairs of states. One consequence of a general theorem proved is that, in conjunction with the unique product state extension property, the existence of a faithful family of product states forces commutation. Another is that if D is simple and has the unique product extension property across A and B then the latter C*-algebras must commute and D be their minimal tensor product.Communicated by Klaus Fredenhagensubmitted 03/12/03, accepted 26/04/04     

Limit Automorphisms of the <Emphasis Type="Italic">C</Emphasis>*-Algebras Generated by Isometric Representations for Semigroups of Rationals

We consider inductive sequences of Toeplitz algebras whose connecting homomorphisms are defined by collections of primes. The inductive limits of these sequences are C*-algebras generated by representations for semigroups of rationals. We study the limit endomorphisms of these C*-algebras induced by morphisms between copies of the same inductive sequences of Toeplitz algebras. We establish necessary and sufficient conditions for these endomorphisms to be automorphisms of the algebras.     

Property (T) for topological groups and <Emphasis Type="Italic">C</Emphasis>*-algebras

The aim of the present (mostly expository) paper is to show the relationship of a generalization of Kazhdan’s property (T) for C*-algebras introduced in our recent paper to that of B. Bekka. It is shown that our definition coincides with Bekka’s definition for group C*-algebras of locally compact groups, whereas, in general, these definitions are distinct. Criteria for a C*-algebra to possess our property (T) are given. A number of examples of C*-algebras with and without property (T) are considered. Relations to K-theory are studied. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 8, pp. 171–192, 2007.     

On Inductive Limits for Systems of <Emphasis Type="Italic">C</Emphasis>*-Algebras

We consider a covariant functor from the category of an arbitrary partially ordered set into the category of C*-algebras and their *-homomorphisms. In this case one has inductive systems of algebras over maximal directed subsets. The article deals with properties of inductive limits for those systems. In particular, for a functor whose values are Toeplitz algebras, we show that each such an inductive limit is isomorphic to a reduced semigroup C*-algebra defined by a semigroup of rationals. We endow an index set for a family of maximal directed subsets with a topology and study its properties. We establish a connection between this topology and properties of inductive limits.     

On topologically finite-dimensional simple <Emphasis Type="Italic">C</Emphasis>*-algebras

We show that, if a simple C*-algebra A is topologically finite-dimensional in a suitable sense, then not only K₀(A) has certain good properties, but A is even accessible to Elliott’s classification program. More precisely, we prove the following results:If A is simple, separable and unital with finite decomposition rank and real rank zero, then K₀(A) is weakly unperforated.If A has finite decomposition rank, real rank zero and the space of extremal tracial states is compact and zero-dimensional, then A has stable rank one and tracial rank zero. As a consequence, if B is another such algebra, and if A and B have isomorphic Elliott invariants and satisfy the Universal coefficients theorem, then they are isomorphic.In the case where A has finite decomposition rank and the space of extremal tracial states is compact and zero-dimensional, we also give a criterion (in terms of the ordered K₀-group) for A to have real rank zero. As a byproduct, we show that there are examples of simple, stably finite and quasidiagonal C*-algebras with infinite decomposition rank.Supported by: EU-Network Quantum Spaces - Noncommutative Geometry (Contract No. HPRN-CT-2002-00280) and Deutsche Forschungsgemeinschaft (SFB 478).     

Generalized inverses in<Emphasis Type="Italic">C</Emphasis><Superscript>*</Superscript>-algebras, 2

Sea é un elemento di unaC ^*-algebra tale chepa=a per qualche proiezionep, alloraa é una unitá inversa generalizzata, chiamata inversa di Moore-Penrose, e si denotata cona ⁺. Per leC ^*-algebras, si ottiene altrest un risultato sulla minimizzazione di ∥axb−c∥.      

<Emphasis Type="Italic">C</Emphasis><Superscript>*</Superscript>-Subalgebras Generated by a Single Operator in <Emphasis Type="Italic">B</Emphasis>(<Emphasis Type="Italic">H</Emphasis>)

In this paper, we characterize a C ^*-subalgebra C ^*(x) of B(H), generated by a single operator x. We show that if x is polar-decomposed by aq, where a is the partial isometry part and q is the positive operator part of x, then C ^*(x) is *-isomorphic to the groupoid crossed product algebra A_q×_a\mathbbG_a\mathcal{A}_{q}\times_{\alpha }\mathbb{G}_{a} , where A_q=C^*(q)\mathcal{A}_{q}=C^{*}(q) and \mathbbG_a\mathbb{G}_{a} is the graph groupoid induced by a partial isometry part a of x.     

Simplicity of <Emphasis Type="Italic">C</Emphasis>*-algebras associated to row-finite locally convex higher-rank graphs

In a previous work, the authors showed that the C*-algebra C*(Λ) of a row-finite higher-rank graph Λ with no sources is simple if and only if Λ is both cofinal and aperiodic. In this paper, we generalise this result to row-finite higher-rank graphs which are locally convex (but may contain sources). Our main tool is Farthing’s “removing sources” construction which embeds a row-finite locally convex higher-rank graph in a row-finite higher-rank graph with no sources in such a way that the associated C*-algebras are Morita equivalent.     

<Emphasis Type="Italic">C</Emphasis><Superscript>*</Superscript>-Algebras of Bergman Type Operators with Piecewise Continuous Coefficients

The C ^*-algebra generated by the n poly-Bergman and m antipoly-Bergman projections and by the operators of multiplication by piecewise continuous functions on the Lebesgue space L ²(Π) over the upper half-plane is studied. Making use of a local principle, limit operators techniques, and the Plamenevsky results on two-dimensional convolution operators with symbols admitting homogeneous discontinuities we reduce the study to simpler C ^*-algebras associated with points and pairs . Applying a symbol calculus for the abstract unital C ^*-algebras generated by N orthogonal projections sum of which equals the unit and by M = n + m one-dimensional orthogonal projections and using relations for the Gauss hypergeometric function, we study local algebras at points being the discontinuity points of coefficients. A symbol calculus for the C ^*-algebra is constructed and a Fredholm criterion for the operators is obtained.     

Inequality for a trace on a unital <Emphasis Type="Italic">C</Emphasis>*-algebra

A new inequality for a trace on a unital C*-algebra is established. It is shown that the inequality obtained characterizes the traces in the class of all positive functionals on a unital C*-algebra. A new criterion for the commutativity of unital C*-algebras is proved.