The spine of a Fourier-Stieltjes algebra

We define the spine A *(G) of the Fourier–Stieltjes algebraB (G) of a locally compact group G. This algebra encodes informationabout much of the fine structure of B (G), particularly informationabout certain homomorphisms and idempotents. We show that A *(G) is graded over a certain semi-lattice, thatof non-quotient locally precompact topologies on G. We computethe spine's spectrum G*, which admits a semi-group structure.We discuss homomorphisms from A *(G) to B (H) where H is anotherlocally compact group; and we show that A *(H) contains theimage of every completely bounded homomorphism from the Fourieralgebra A (H) of any amenable group G. We also show that A *(G)contains all of the idempotents in B (G). Finally, we computeexamples for vector groups, abelian lattices, minimally almostperiodic groups and the (ax + b)-group; and we explore the complexityof A *(G) for the discrete rational numbers and free groups.     

Spectra of Analytic Toeplitz Tuples on Hardy Spaces

Let A be a complex algebra (with unit e), X a left A moduleand a = (a₁, ..., a_m) a commuting tuple of elements in A. FollowingTaylor [8], we have to consider the Koszul complex K_•(a,X) in order to define the joint spectrum of the tuple a. 1991Mathematics Subject Classification 47A13.     

Polynomial and Polygonal Connections between Idempotents in Finite-Dimensional Real Algebras

Let P (A) be the set of idempotents in a finite-dimensionalreal algebra A. Let p and q be idempotents that lie in the samecomponent of P (A). Then, among the continuous paths connectingp and q in P (A), there exist a polynomial path of degree atmost 3 and a polygonal path consisting of at most three segments.2000 Mathematics Subject Classification 46H10, 16P10.     

Tiling Infinite-Dimensional Normed Spaces

The present paper deals with real infinite-dimensional normedspaces; some of the main concepts here make sense, and havebeen treated in the literature, in the general context of topologicalHausdorff linear spaces over reals. A subset of a normed space X is a body if it is different fromX itself and is the closure of its non-empty interior. A coveringof X by bodies is called a tiling of X whenever any two differentmembers of it have disjoint interiors. The elements of sucha covering are called tiles. A tiling is bounded (respectivelyconvex) whenever each tile is bounded (respectively convex).1991 Mathematics Subject Classification 46B20.     

Estimates for the Number of Sums and Products and for Exponential Sums in Fields of Prime Order

Our first result is a ‘sum-product’ theorem forsubsets A of the finite field F_p, p prime, providing a lowerbound on max (|A + A|, |A · A|). The second and mainresult provides new bounds on exponential sums     

Structural Projections on JBW*-Triples

A linear projection R on a Jordan*-triple A is said to be structuralprovided that, for all elements a, b and c in A, the equality{Rab Rc} = R{a Rbc} holds. A subtriple B of A is said to becomplemented if A = B + Ker(B), where Ker(B) = {aA: {B a B}= 0}. It is shown that a subtriple of a JBW*-triple is complementedif and only if it is the range of a structural projection. A weak* closed subspace B of the dual E* of a Banach space Eis said to be an N*-ideal if every weak* continuous linear functionalon B has a norm preserving extension to a weak* continuous linearfunctional on E* and the set of elements in E which attain theirnorm on the unit ball in B is a subspace of E. It is shown thata subtriple of a JBW*-triple A is complemented if and only ifit is an N*-ideal, from which it follows that complemented subtriplesof A are weak* closed, and structural projections on A are weak*continuous and norm non-increasing. It is also shown that everyN*-ideal in A possesses a triple product with respect to whichit is a JBW*-triple which is isomorphic to a complemented subtripleof A.     

Operator amenability of Fourier-Stieltjes algebras, II

We give an example of a non-compact, locally compact group Gsuch that its Fourier–Stieltjes algebra B (G) is operatoramenable. Furthermore, we characterize those G for which A *(G),the spine of B (G) as introduced by M. Ilie and N. Spronk, isoperator amenable and show that A *(G) is operator weakly amenablefor each G.     

A Sequential Algorithm for an Inventory Selection Problem

Assume that we have m finished products in an inventory. Eachfinished product is characterized by two measurements P andQ. A customer specifies a purchase order by the requirementsof characteristics P and Q. A product is qualified to satisfya purchase order if and only if it possesses better measurementsof both P and Q than the customer requires. For a given batchof n purchase orders, the inventory selection problem is tochoose n finished products from the inventory to satisfy allpurchase orders with a minimum cost. This problem can be formulatedas a large-scale transportation problem. When the cost functionof selecting a product to satisfy an order exhibits certainstructure, we develop a fast sequential algorithm to solve thisproblem. Possible extensions and related problems are also discussedin this paper.     

Best Approximation in an Asymmetrically Weighted L1 Measure

Let f(x) be a given, real-valued, continuous function definedon an interval [a,b]of the real line. Given a set of m real-valued,continuous functions _j(x) defined on [a,b], a linear approximatingfunction can be formed with any real setA = {a₁, a₂,..., a_m}. We present results for determining A sothat F(A, x) is a best approximation to(x) when the measureof goodness of approximation is a weighted sum of |F(A, x)–f(x)|,the weights being positive constants, w, when F(A, x) f(x)and w2 otherwise (when w, = w2 = 1, the measure is the L₁, norm).The results are derived from a linear programming formulationof the problem. In particular, we give a theorem which shows when such bestapproximations interpolate the function at fixed ordinates whichare independent of f(x). We show how the fixed points can becalculated and we present numerical results to indicate thatthe theorem is quite robust.     

Conical Uniqueness Sets for the Spherical Radon Transform

Let K be a cone in Rⁿ. Then K is a uniqueness set for the sphericalRadon transform if and only if it is not contained in the zeroset of any (nontrivial) homogeneous harmonic polynomial. A localversion of this result is also proved. 1991 Mathematics SubjectClassification 44A12.     

Boundedness and Existence of Periodic Solutions of Quasi-linear Systems

Systems of the form x' = A (t, x)x+B(t, x) are considered, andresults are obtained concerning boundedness and periodicityby using corresponding information from linear systems of theform x' = A (t, f(t))x+B(t, f(t)).]Here f(t) belongs to theclosed ball of a suitable Banach space.     

Two New Constructions of the O'Nan Group

A construction is described of a 153-dimensional representationof the triple cover of O'Nan's sporadic simple group, over thefield of order 4. This is then extended to a 306-dimensionalrepresentation of 3·O'N:2 over GF(2). A similar constructionin characteristic 3 gives an unrelated representation of theO'Nan group in 154 dimensions over GF(3), which extends to O'N:2provided that the field is extended to GF(9).     

When is a Coxeter System Determined by its Coxeter Group?

A Coxeter system is a pair (W, S) where W is a group and whereS is a set of involutions in W such that W has a presentationof the form W=     

The Minimal Base Size of Primitive Solvable Permutation Groups

A base of a permutation group G is a sequence B of points fromthe permutation domain such that only the identity of G fixesB pointwise. Answering a question of Pyber, we prove that allprimitive solvable permutation groups have a base of size atmost four.     

Bounded-Norm Matrix-Inverse Mappings

This paper introduces the concept of bounded-norm matrix-inversemappings, i.e. mappings µ : R^mxnR^nxm such that, for allnonzero mxn matrices A, the matrix µ(A) is a generalizedinverse of A and ||µ(A)||> k/s(A), where K < 0 isa constant and s(A) is the nonzero singular value of A havingsmallest absolute value. It is shown how the definition of suchmappings is motivated by the need to ensure finite terminationof the inner-iterations of generalized elimination methods forthe solution of nonlinearly constrained optimization problems.The main result of the paper is that the mapping defined byµ(A) = A^b is a bounded-norm matrix-inverse mapping, providedthat the basic inverse A^b is calculated using Gaussian eliminationwith complete pivoting. The concept of bounded-norm matrix-inversemappings is then extended to that of boundednorm least-squaresmatrix-inverse mappings. It is proved that the mapping definedby µ(A) = A^ß is a bounded-norm least-squaresmatrix-inverse mapping, provided that the basic least-squaresinverse A^ß is calculated using the QR decompositionwith column pivoting.     

Formulas for Liapunov Functions for Systems of Linear Difference Equations

Explicit quadratic Liapunov functions that provide necessaryand sufficient conditions for the asymptotic stability of thesystem of linear difference equations x (t + 1) = Ax(t) areconstructed by transforming the original systems to y (t + 1)= Gy(t), where G is a companion matrix associated with the characteristicpolynomial of A. A necessary and sufficient condition for allroots of the characteristic polynomial to lie in the unit circle|z| < 1 on the complex plane is also derived. 2000 MathematicalsSubject Classification 39A11, 93D05.     

Zel'Manov's Theorem for Primitive Jordan-Banach Algebras

The following result is well known and easy to prove (see [14,Theorem 2.2.6]). Theorem 0. If A is a primitive associative Banach algebra, thenthere exists a Banach space X such that A can be seen as a subalgebraof the Banach algebra BL(X) of all bounded linear operatorson X in such a way that A acts irreducibly on X and the inclusionABL(X) is continuous. In fact, if X is any vector space on which the primitive Banachalgebra A acts faithfully and irreducibly, then X can be convertedin a Banach space in such a way that the requirements in Theorem0 are satisfied and even the inclusion ABL(X) is contractive. Roughly speaking, the aim of this paper is to prove the appropriateJordan variant of Theorem 0.     

Arens Regularity for Quotients Ap(E) of the Herz Algebra

If the subset T supports a synthesizable p-pseudofunction T,then the restriction A_p(E) to E of the Herz algebra on T isnot Arens regular. The proof is direct, by brute force, anddoes not show the existence of Day points for A_p(E). 2000 MathematicsSubject Classification 43A15, 43A10 (primary), 46L10 (secondary).     

Multiplicative Error Analysis of Matrix Transformation Algorithms

The author's recently introduced relative error measure forvectors is applied to the error analysis of algorithms whichproceed by successive transformation of a matrix. Instead ofmodelling the roundoff errors at each stage by A: = T(A)+E onemodels them by A: =e^E T(A) where E is a small linear transformation.This can simplify analyses considerably. Applications to theparallel Jacobi method for eigenvalues, and to Gaussian elimination,are given.     

Lower Bounds for Morse Index of Constant Mean Curvature Tori

In the paper, three lower bounds are given for the Morse indexof a constant mean curvature torus in Euclidean3-space, in termsof its spectral genus g. The first two lower bounds grow linearlyin g and are stronger for smaller values of g, while the thirdgrows quadratically in g but is weaker for smaller values ofg. 2000 Mathematics Subject Classification 53A10, 53A35.