Full Crossed Products by Coactions, C0(X)-Algebras and C*-Bundles

Let be a coaction of a locally compact group G on a C*-algebraA. We show that if I is a -invariant ideal in A, then for full crossed products, as Landstadet al. have done for spatial crossed products by coactions.We prove that for suitable coactions, the crossed products ofC₀(X)-algebras are again C₀(X)-algebras, and the crossed productsof continuous C*-bundles by a locally compact group are againcontinuous C*-bundles. 1991 Mathematics Subject Classification46L55.     

Commutative Subalgebras and Sphericity in Z2-Graded Lie Algebras

Let g = g₀g₁ be a Z₂-graded simple Lie algebra. Fix a Borelsubalgebra b₀g₀. Let ag₁ be a b₀-stable subalgebra. Then a isautomatically commutative. It is known that if is a spherical G-variety, then G₀.a is a sphericalG₀-variety. We describe all Z₂-gradings having the propertythat is a spherical G-varietyfor any a. 2000 Mathematics Subject Classification 14L30, 17B20,20E46.     

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).     

Some New and Old Asymptotic Representations of the Jost Solution and the Weyl m-Function For Schrodinger Operators on the Line

For the general one-dimensional Schrödinger operator –d²/dx²+q(x) with real q L₁(R), this paper presents a new series representationof the Jost solution which, in turn, implies a new asymptoticrepresentation of the Weyl m-function for locally summable q.This representation is then applied to smooth potentials q toobtain Weyl m-function power asymptotics. The condition q^(N) L₁(x₀, x₀ + ), for N N₀, allows one to derive the (N + 1)term for almost all x [x₀, x₀ + ), thereby refining a relevantresult by Danielyan, Levitan and Simon. 2000 Mathematics SubjectClassification 34E05, 34L40 (primary), 34B20, 34L25 (secondary).     

A Class of Infinite Dimensional Simple Lie Algebras

Let A be an abelian group, F be a field of characteristic 0,and , ß be linearly independent additive maps fromA to F, and let ker()\{0}. Then there is a Lie algebra L = L(A,, ß, ) = _xA Fe_x under the product [e_x, e_y]]=(x–y)e_x+y+(ß) (x, y) e_x+y–. If, further, ß() = 1, and ß(A) = Z, thereis a subalgebra L⁺:=L(A⁺, , ß, ) = _xA⁺ Fe_x, whereA⁺ = {xA|ß(x)0}. The necessary and sufficient conditionsare given for L' = [L, L] and L⁺ to be simple, and all semi-simpleelements in L' and L⁺ are determined. It is shown that L' andL⁺ cannot be isomorphic to any other known Lie algebras andL' is not isomorphic to any L⁺, and all isomorphisms betweentwo L' and all isomorphisms between two L⁺ are explicitly described.     

Generators and defining relations for ring of invariants of commuting locally nilpotent derivations or automorphisms

Let A be an algebra over a field K of characteristic zero andlet ₁, ..., _sDer _K(A) be commuting locally nilpotent K-derivationssuch that _i(x_j) equals _ij, the Kronecker delta, for some elementsx₁, ..., x_sA. A set of generators for the algebra is found explicitly and a set of defining relationsfor the algebra A is described. Similarly, let ₁, ..., _s Aut_K(A)be commuting K-automorphisms of the algebra A is given suchthat the maps _i – id_A are locally nilpotent and _i (x_j)= x_j + _ij, for some elements x₁, ..., x_s A. A set of generatorsfor the algebra A: = {a A | ₁(a) = ... = _s(a) = a} is foundexplicitly and a set of defining relations for the algebra Ais described. In general, even for a finitely generated non-commutativealgebra A the algebras of invariants A and A are not finitelygenerated, not (left or right) Noetherian and a minimal numberof defining relations is infinite. However, for a finitely generatedcommutative algebra A the opposite is always true. The derivations(or automorphisms) just described appear often in many differentsituations (possibly) after localization of the algebra A.     

Crystal Bases for Quantum Generalized Kac-Moody Algebras

In this paper, we develop the crystal basis theory for quantumgeneralized Kac–Moody algebras. For a quantum generalizedKac–Moody algebra U_q(g), we first introduce the categoryO_int of U_q(g)-modules and prove its semisimplicity. Next, wedefine the notion of crystal bases for U_q(g)-modules in thecategory O_int and for the subalgebra . We then prove the tensor product rule and the existence theoremfor crystal bases. Finally, we construct the global bases forU_q(g)-modules in the category O_int and for the subalgebra . 2000 Mathematics Subject Classification17B37, 17B67.     

Exponential stability and maximal attractors for a one-dimensional nonlinear thermoviscoelasticity

This paper is concerned with the global existence, exponentialstability of solutions and associated nonlinear C₀-semigroupas well as the existence of maximal attractors in Hⁱ (i = 1,2, 4) for a nonlinear one-dimensional thermoviscoelasticitydescribing a kind of solid-like material. Some new ideas andmore delicated estimates are employed to prove the global existenceand exponential stability of solutions. The important featurefor the existence of maximal attractors in Hⁱ₊ (i = 1, 2, 4)is that the metric spaces H¹₊, H²₊ and H⁴₊ we work with arethree incomplete metric spaces, as can be seen from the physicalconstraints, i.e. > 0 and u > 0, with and u being absolutetemperature and deformation gradient (strain). For any positiveparameters ₁, ₂, ..., ₅ verifying some conditions, a sequenceof closed subspaces Hⁱ Hⁱ+ (i = 1, 2, 4) is found, and theexistence of maximal attractors in Hⁱ (i = 1, 2, 4) is established.     

Isotropy Representations, Eigenvalues of a Casimir Element, and Commutative Lie Subalgebras

Let h be a reductive subalgebra of a semisimple Lie algebrag and C_h U(h) be the Casimir element determined by the restrictionof the Killing form on g to h. The paper studies eigenvaluesof C_h on the isotropy representation mg/h. Some general estimatesconnecting the eigenvalues and the Dynkin indices of m are given.If h is a symmetric subalgebra, it is shown that describingthe maximal eigenvalue of C_h on exterior powers of m is connectedwith possible dimensions of commutative Lie subalgebras in m,thereby extending a result of Kostant. In this situation, aformula is also given for the maximal eigenvalue of C_h on m.More generally, a similar picture arises if h = g, where isan automorphism of finite order m and m is replaced by the eigenspaceof corresponding to a primitive mth root of unity.     

The Minimum Index of a Non-Congruence Subgroup of SL2 Over an Arithmetic Domain. II: The Rank Zero Cases

Let K be a function field of genus g with a finite constantfield F_q. Choose a place of K of degree and let C be the arithmeticDedekind domain consisting of all elements of K that are integraloutside . An explicit formula is given (in terms of q, g and) for the minimum index of a non-congruence subgroup in SL₂(C).It turns out that this index is always equal to the minimumindex of an arbitrary proper subgroup in SL₂(C). The minimumindex of a normal non-congruence subgroup is also determined.     

Great time delay in a system with material cycling and delayed biomass growth

In this paper, we consider the effects of time delay on thelinear stability of the positive equilibrium of a reaction–diffusionmodel with time delayed nutrient recycling. We take the twodelay kernels as f(s) = (s), g(s) = p(0) + (1 – p)(r),s (0, +), r [0, +), 0 < p < 1, where is the Dirac deltafunction. When both N₁ and N₂ are space-independent, we canobserve that the increasing time delay destabilizes the positiveequilibrium, and stability-switches from stability to instabilityto stability occur. That is, for some special time delay kernels,the equilibrium is stable for small time delay, bifurcates towardsstable oscillations when the time delay is increased, then regainsstability for larger values. When both N₁ and N₂ are space-dependent,we are concerned with the effects of the delay kernels as wellas diffusion on the dynamics of the system.     

An error bound for the modified successive overrelaxation method

In this paper we consider the modified successive overrelaxation(MSOR)methodto appropriate the solution of the linear system D^-1/2 Ax =D^-1/2b, where A is a symmetric, positive definite and consistentlyordered matrix and D is a diagonal matrix with the diagonalidentical to that of A. The main purpose of this paper is to obtain some theoreticalresults, namely a bound for the norm of _n = v –v_n in termsof the norms _n –v_n-1, _n+1 –v_n and their inner product,where v =D^-1/2 x and v_n is the nth iteration vector, obtainedusing the (MSOR)method.     

Embedding l{infty} into the Space of Bounded Operators on Certain Banach Spaces

Sufficient conditions are given on a Banach space X which ensurethat embeds in L(X), the space of all bounded linear operatorson X. A basic sequence e_n is said to be quasisubsymmetric iffor any two increasing sequences (k_n) and (l_n) of positive integerswith (k_n) (l_n) for all n, () dominates (). If a Banach space X has a seminormalized quasisubsymmetric basis then l embedsin l(X). 2000 Mathematics Subject Classification 46B28 (primary),46B03 (secondary).     

On Positive Multipeak Solutions of a Nonlinear Elliptic Problem

In this paper we continue our investigation in [5, 7, 8] onmultipeak solutions to the problem –²u+u=Q(x)|u|^q–2u, xR^N, uH¹(R^N) (1.1) where = ^N_i=1²/x²_i is the Laplace operator in R^N, 2 < q < for N = 1, 2, 2 < q < 2N/(N–2) for N3, and Q(x)is a bounded positive continuous function on R^N satisfying thefollowing conditions. (Q₁) Q has a strict local minimum at some point x₀R^N, that is,for some > 0 Q(x)>Q(x₀) for all 0 < |x–x₀| < . (Q₂) There are constants C, > 0 such that |Q(x)–Q(y)|C|x–y| for all |x–x₀| , |y–y₀| . Our aim here is to show that corresponding to each strict localminimum point x₀ of Q(x) in R^N, and for each positive integerk, (1.1) has a positive solution with k-peaks concentratingnear x₀, provided is sufficiently small, that is, a solutionwith k-maximum points converging to x₀, while vanishing as 0 everywhere else in R^N.     

A Large Deviation Principle for Polynomial Hypergroups

Let S_n, n 1, be a random walk on a polynomial hypergroup (N₀,*), that is, a Markov chain on the nonnegative integers withstationary transition probabilities P_ij = _i* µ({j}), whereµ is a fixed probability measureon N₀. Under certain conditionson this measure, the principle of large deviations is shownfor the distributions of S_n/n. This result comprises the largedeviation principle for birth and death random walks associatedwith the polynomials generating the polynomial hypergroup.     

The Heat Flow of the CCR Algebra

Let Pf(x) = –if'(x) and Qf(x) = xf(x) be the canonicaloperators acting on an appropriate common dense domain in L²(R).The derivations D_P(A) = i(PA–AP) and D_Q(A) = i(QA–AQ)act on the *-algebra A of all integral operators having smoothkernels of compact support, for example, and one may considerthe noncommutative ‘Laplacian’, L = + , as a linear mapping of A into itself. L generates a semigroup of normal completely positive linearmaps on B(L₂(R)), and this paper establishes some basic propertiesof this semigroup and its minimal dilation to an E₀-semigroup.In particular, the author shows that its minimal dilation ispure and has no normal invariant states, and he discusses thesignificance of those facts for the interaction theory introducedin a previous paper. There are similar results for the canonical commutation relationswith n degrees of freedom, where 1 n < . 2000 MathematicsSubject Classification 46L57 (primary), 46L53, 46L65 (secondary).     

On {sigma}-Normal C*-Algebras

A C*-algebra A is said to be monotone (respectively monotone-) complete if every increasing net (respectively increasingsequence) of elements in the ordered space A_h of all hermitianelements of A has a supremum in A_h. It is straightforward toverify that every monotone complete C*-algebra is an AW*-algebra.For type I AW*-algebras, the converse is known to be true. However,for general AW*-algebras, this question is still open, althoughan impressive attack on the problem was made by E. Christensenand G. K. Pedersen, who showed that properly infinite AW*-algebrasare monotone -complete [4]. 1991 Mathematics Subject Classification46L05, 46L06.     

Grothendieck's Inequalities for Real and Complex JBW*-Triples

We prove that, if and >0, if V and W are complex JBW*-triples (with preduals V_* andW_*, respectively), and if U is a separately weak*-continuousbilinear form on V x W, then there exist norm-one functionals₁, ₂ V_* and ₁, ₂ W_* satisfying for all (x, y) V x W. Here, for a norm-one functional on acomplex JB*-triple V, |·| stands for the prehilbertianseminorm on V associated to given by for all x W, where z V^** satisfies z = |z| =1. We arrive at this form of ‘Grothendieck's inequality’through results of C.-H. Chu, B. Iochum, and G. Loupias, andan amended version of the ‘little Grothendieck's inequality’for complex JB*-triples due to T. Barton and Y. Friedman. Wealso obtain extensions of these results to the setting of realJB*-triples. 2000 Mathematical Subject Classification: 17C65,46K70, 46L05, 46L10, 46L70.     

Finite-dimensional H{infty} control of flexible beam equation systems

In this paper, we consider H control problem with measurementfeedback for flexible beam equation systems. The aim is to constructa finite-dimensional H controller with a given level for theflexible beam equation system. For that purpose, we first formulatethe system as an infinite-dimensional system in l² and derivea finite-dimensional reduced-order system for the infinite-dimensionalsystem. Then, an H controller with level d less than is constructedfor the reduced-order model. The finite-dimensional controllertogether with a residual mode filter plays a role of a finite-dimensionalH controller with level for the original flexible beam equationsystem, if the order of the residual mode filter is chosen sufficientlylarge.     

The Free-boundary Problem for Gravity-driven Unidirectional Viscous Flows

If a layer of viscous fluid drains steadily under gravity, whilein contact with an upward-moving vertical cylinder of a generalcross section with boundary C_B its velocity w(x, y) satisfiesa Poisson equation w_xx+w_yy = g/v = constant. There are two boundaries,the given curve C_B on which w takes a prescribed constant value,and a free boundary C_p marking the edge of the layer, whoseshape we wish to determine. In order to do this, we need twoboundary conditions on C_F; one is the zero-stress conditionw/n = 0, and we assume that the other is that w is constantaround C_F. The latter condition is shown to be equivalent toa requirement that the volume flux in the layer is maximal.The properties of the resulting free-boundary problem are discussed,and numerical solutions obtained for some special cases. Forexample, a solution for the case when C_B is a 270? corner showsthat the layer thickness at the corner is reduced to 76% ofthe thickness on the flat part of the boundary. The presentanalysis is relevant to industrial coating problems, such asedge effects in continuous galvanizing of sheet steel.