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.     

Purely infinite C*-Algebras: Ideal-preserving zero homotopies

We show that if A is a separable, nuclear, -absorbing (or strongly purely infinite) C*-algebra which is homotopic to zero in an ideal-system preserving way, then A is the inductive limit of C*-algebras of the form where Γ is a finite connected graph (and is the algebra of continuous functions on Γ that vanish at a distinguished point ).We show further that if B is any separable, nuclear C*-algebra, then is isomorphic to a crossed product where D is an inductive limit of C*-algebras of the form (and D is -absorbing and homotopic to zero in an ideal-system preserving way).Received: December 2003 Revision: July 2004 Accepted: July 2004     

Complete Positivity of Elementary Operators on C*-Algebras

Let be a C*-algebra. We obtain some conditions that are equivalent to the statement that every n-positive elementary operator on is completely positive.     

Approximately Linear Mappings in Banach Modules over a C^＊-algebra

Let X and Y be vector spaces. The authors show that a mapping f ： X →Y satisfies the functional equation 2d f（∑^2d j=1（-1）^j＋1xj/2d）=∑^2dj=1（-1）^j＋1f（xj） with f（0） = 0 if and only if the mapping f ： X→ Y is Cauchy additive, and prove the stability of the functional equation （≠） in Banach modules over a unital C^＊-algebra, and in Poisson Banach modules over a unital Poisson C＊-algebra. Let A and B be unital C^＊-algebras, Poisson C^＊-algebras or Poisson JC^＊- algebras. As an application, the authors show that every almost homomorphism h ： A →B of A into is a homomorphism when h（（2d-1）^nuy） =- h（（2d-1）^nu）h（y） or h（（2d-1）^nuoy） = h（（2d-1）^nu）oh（y） for all unitaries u ∈A, all y ∈ A, n = 0, 1, 2,.... Moreover, the authors prove the stability of homomorphisms in C^＊-algebras, Poisson C^＊-algebras or Poisson JC^＊-algebras.     

C*-Algebras of Integral Operators with Piecewise Slowly Oscillating Coefficients and Shifts Acting Freely

We establish a symbol calculus for the C*-subalgebra of generated by the operators of multiplication by slowly oscillating and piecewise continuous functions and the operators where is the Cauchy singular integral operator and The C*-algebra is invariant under the transformations

Metric Geometry in Homogeneous Spaces of the Unitary Group of a C*-Algebra. Part II. Geodesics Joining Fixed Endpoints

This article focuses on the study of the metric geometry of homogeneous spaces (the unitary group of a C*-algebra modulo the unitary group of a C*-subalgebra ) where the invariant Finsler metric in is induced by the quotient norm of Under the assumption that is of compact type, i.e. when the unitary group is relatively compact in the strong operator topology, this work presents local and global versions of Hopf-Rinow-like theorems: given points there exists a minimal uniparametric group curve joining ρ₀ and ρ₁.     

C*-Modular Vector States

Let be a C*-algebra and X a Hilbert C* -module. If is a projection, let be the p-sphere of X. For φ a state of with support p in and consider the modular vector state φ_x of given by The spheres provide fibrations

Multipliers of Fractional Cauchy Transforms

Let B_n denote the unit ball of , n ≥ 2. Given an α > 0, let denote the class of functions defined for by integrating the kernel against a complex-valued measure on the sphere . Let denote the space of holomorphic functions in the ball. A function is called a multiplier of provided that for every . In the present paper, we obtain explicit analytic conditions on which imply that g is a multiplier of . Also, we discuss the sharpness of the results obtained. This research was supported by RFBR (grant no. 08-01-00358-a), by the Russian Science Support Foundation and by the programme “Key scientific schools NS 2409.2008.1”.     

Graph C *-Algebras and Their Ideals Defined by Cuntz-Krieger Family of Possibly Row-Infinite Directed Graphs

Let E be a possibly row-infinite directed graph. In this paper, first we prove the existence of the universal C^*-algebra C^*(E) of E which is generated by a Cuntz-Krieger E-family {s_e, p_v}, and the gauge-invariant uniqueness theorem and the Cuntz-Krieger uniqueness theorem for the ideal of C^*(E). Then we get our main results about the ideal structure of Finally the simplicity and the pure infiniteness of is discussed.     

Ratios of Norms for Polynomials and Connected n-width Problems

Let be a bounded simply connected domain with boundary Γ and let be a regular compact set with connected complement. In this paper we investigate asymptotics of the extremal constants:

Hyperbolic Function Theory in the Clifford Algebra {\mathcal {C}}\ell_{n+1, 0}

The aim of this paper is to give the basic principles of hyperbolic function theory on the Clifford algebra . The structure of the theory is quite similar to the case of Clifford algebras with negative generators, but the proofs are not obvious. The (real) Clifford algebra is generated by unit vectors with positive squares e²_i = + 1. The hyperbolic Dirac operator is of the form where Q₀f is represented by the composition . If is a solution of H_kf = 0, then f is called k-hypergenic in Ω, where is an open set. We introduce some basic results of hyperbolic function theory and give some representation theorems on . Received: October, 2007. Accepted: February, 2008.     

A Locally Trivial Quantum Hopf Fibration

The irreducible *-representations of the polynomial algebra of the quantum3-sphere introduced by Calow and Matthes are classified. The K-groups of its universal C^*-algebra are shown to coincide with their classical counterparts. The U(1)-action on corresponding for p=1=q to the classical Hopf fibration is proven to be Galois (free). The thus obtained locally trivial Hopf–Galois extension is shown to be equivariantly projective (admitting a strong connection) and non-cleft. The latter is proven by determining an appropriate pairing of cyclic cohomology and K-theory. Presented by S. L. Woronowicz Mathematics Subject Classifications (2000) 16W30, 46L87.     

Spherical Monogenics: An Algebraic Approach

The space of spherical monogenics in can be regarded as a model for the irreducible representation of Spin(m) with weight . In this paper we construct an orthonormal basis for . To describe the symmetry behind this procedure, we define certain Spin(m − 2)-invariant representations of the Lie algebra (2) on . Received: October, 2007. Accepted: February, 2008.     

On the enumeration of positive cells in generalized cluster complexes and Catalan hyperplane arrangements

Let Φ be an irreducible crystallographic root system with Weyl group W and coroot lattice , spanning a Euclidean space V. Let m be a positive integer and be the arrangement of hyperplanes in V of the form for and . It is known that the number of bounded dominant regions of is equal to the number of facets of the positive part of the generalized cluster complex associated to the pair by S. Fomin and N. Reading. We define a statistic on the set of bounded dominant regions of and conjecture that the corresponding refinement of coincides with the $h$-vector of . We compute these refined numbers for the classical root systems as well as for all root systems when m = 1 and verify the conjecture when Φ has type A, B or C and when m = 1. We give several combinatorial interpretations to these numbers in terms of chains of order ideals in the root poset of Φ, orbits of the action of W on the quotient and coroot lattice points inside a certain simplex, analogous to the ones given by the first author in the case of the set of all dominant regions of . We also provide a dual interpretation in terms of order filters in the root poset of Φ in the special case m = 1. 2000 Mathematics Subject Classification Primary—20F55; Secondary—05E99, 20H15     

Peripherally Monomial-Preserving Maps between Uniform Algebras

Let and be uniform algebras and p(z,w) = z^mwⁿ a twovariable monomial. We characterize maps T from certain subsets of into such that holds for all f and g in the domain of T; peripherally monomial-preserving maps. Furthermore and are proved to be isometrical isomorphic as Banach algebras. If the greatest common divisor of m and n is 1, then T is extended to an isometrical linear isomorphism; a weighted composition operator. An example of peripherally monomial-preserving surjections between uniform algebras which is not linear, nor multiplicative, nor injective is given when the greatest common divisor is strictly greater than 1. The first, third and fourth authors were partly supported by the Grantsin-Aid for Scientific Research, The Ministry of Education, Science, Sports and Culture, Japan.     

Convexities of partial monounary algebras

In the present paper we prove that the collection of all convexities of partial monounary algebras is finite; namely, it has exactly 23 elements. Further, we show that for each element there exists a subset of such that is generated by and card . This work was supported by the Science and Technology Assistance Agency under the contract No. APVT-20-004104. Supported by Grant VEGA 1/3003/06.     

Algebraic cycles on the relative symmetric powers and on the relative Jacobian of a family of curves. I

In this paper we construct and study the actions of certain deformations of the Lie algebra of Hamiltonians on the plane on the Chow groups (resp., cohomology) of the relative symmetric powers and the relative Jacobian of a family of curves . As one of the applications, we show that in the case of a single curve C this action induces a -form of a Lefschetz sl₂- action on the Chow groups of C ^[N]. Another application gives a new grading on the ring CH ₀(J) of 0-cycles on the Jacobian J of C (with respect to the Pontryagin product) and equips it with an action of the Lie algebra of vector fields on the line. We also define the groups of tautological classes in CH and in CH and prove for them analogs of the properties established in the case of the Jacobian of a single curve by Beauville in [5]. We show that our algebras of operators preserve the subrings of tautological cycles and act on them as some explicit differential operators. This work was partially supported by the NSF grant DMS-0601034.     

On the branch point index of minimal surfaces

Let Γ be a closed, sufficiently smooth Jordan curve in and denote by the class of disk-type surfaces which map ∂B continuously and monotonically onto Γ. Then any minimal surface possesses only finitely many branch points in , and the order of any such point is well-defined, and also the index of an interior branch point is defined in a natural way if X is nonplanar. We show that also the index of boundary branch points can be defined if the curvature κ and the torsion τ of Γ are strictly nonzero. Secondly we derive upper bounds for the index of any branch point in terms of the total curvature of Γ or of its cut number. Dedicated to Professor Heinz K?nig on the occasion of his 80^th birthday     

On range tripotents in JBW*-triples

Let B be a JBW*-triple, let A be a JB*-subtriple of B and let be the set of range tripotents relative to A. It is shown that, under certain conditions, the supremum of a family of range tripotents in coincides with that in the complete lattice of all tripotents in B. As a consequence, a sufficient condition for a tripotent to be a range tripotent relative to A is obtained. The action of isomorphisms on range tripotents is investigated, and an analysis of the suprema of families of spectral range tripotents leads to a generalization of a result known for open projections in W*-algebras. Received: 8 July 2008     

The Spectrum of the C*-Algebra of Pseudodifferential Boundary Value Problems on a Manifold with Smooth Edges

The C ^*-algebra generated by the operators of pseudodifferential boundary value problems on a manifold with smooth closed disjoint edges and boundary is studied. The operators act in the space L ₂( ) L ₂( ). The goal of this paper is to describe all (up to an equivalence) irreducible representations of the algebra Bibliography: 12 titles.