Morita Equivalence of Operator Algebras and Their C*-Envelopes

If two operator algebras A and B are strongly Morita equivalent(in the sense of [5]), then their C*-envelopes C*(A) and C*(B)are strongly Morita equivalent (in the usual C*-algebraic sensedue to Rieffel). Moreover, if Y is an equivalence bimodule fora (strong) Morita equivalence of A and B, then the operation,Y_hA–, of tensoring with Y, gives a bijection between theboundary representations of C*(A) for A and the boundary representationsof C*(B) for B. Thus the ‘noncommutative Choquet boundaries’of Morita equivalent A and B are the same. Other important objectsassociated with an operator algebra are also shown to be preservedby Morita equivalence, such as boundary ideals, the Shilov boundaryideal, Arveson's property of admissability, and the latticeof C*-algebras generated by an operator algebra. 1991 MathematicsSubject Classification 47D25, 46L05, 46M99, 16D90.     

Invariability of Trivial Extensions of Tilted Algebras Under Stable Equivalences

As a sequel to [8], we investigate here the behaviour of thetrivial extensions of tilted algebras under stable equivalence.It will be shown that if a finite-dimensional symmetric algebra is stably equivalent to the trivial extension of a tilted algebraB, then is the trivial extension of some tilted algebra A whichhas the same type as B.     

Irregularities of Point Distribution Relative to Convex Polygons III

Suppose that P is a distribution of N points in the unit squareU=[0, 1]². For every x=(x₁, x₂)U, let B(x)=[0, x₁]x[0, x₂] denotethe aligned rectangle containing all points y=(y₁, y₂)U satisfying0y₁x₁ and 0y₂x₂. Denote by Z[P; B(x)] the number of points ofP that lie in B(x), and consider the discrepancy function D[P; B(x)]=Z[P; B(x)]–Nµ(B(x)), where µ denotes the usual area measure.     

Rangs Stables De Certaines Extensions

M. A. Rieffel [24] a introduit le rang stable topologique (tsr),pour généraliser aux C*-algèbres, le conceptde dimension de recouvrement pour les espaces compacts, affirmantainsi le principe selon lequel une C*-algèbre est ‘unespace localement compact non commutatif’. II a montréque l'on a tsr ((A)) = [dim (Â)] + 1, pour toute C*-algèbre commutative A etque trs (B/J) tsr (B), pour toute C*-algèbre B et pourtout idéal bilatère fermé J dans B (généralisantle fait que, si X est un espace compact et F un sous-ensemblefermé dans X, alors on a dim (F) dim (X), oùdim(X) est la dimension de recouvrement de X [19]). D'autrepart, le rang stable topologique peut être utilisépour obtenir des théorèmes de ‘cancellation’pour les modules projectifs, comme ceci est fait dans [25, 2].Un peu plus tard, R. H. Herman et L. N. Vaserstein [14] ontmontré que pour toute C*-algèbre unitaire A, lerang stable topologique de A et le rang stable de Basse de Acoincident, done, pour toute C*-algèbre unitaire A, onnote sr(A) cette valeur commune appelée rang stable deA. Les C*-algèbres unitaires de rang stable 1 ont étéétudiée géométriquement par M. Rørdam[27], il a montré que l'on a sr(A) = 1 si et seulementsi l'enveloppe convexe des unitaires de A est égale àla boule unité fermé de A. D'autre part, Rieffel[24] avait introduit le rang stable connexe (csr) d'une C*-algèbre,sur lequel V. Nistor [18] a publié un article trésintéressant. Mon travail dans ce papier consiste àcompléter certains travaux déjà entreprisdans les articles qui sont cités ci-dessus.     

Multistep approximation of linear sectorial evolution equations

This paper concerns the linear multistep approximation of alinear sectorial evolution equation u_t = Au on a complex Banachspace X. Given a strictly A()-stable q-step method of orderp whose stability region includes a sectorial region containingthe spectrum of the operator A, the corresponding evolutionsemigroup for the method is Cⁿ(hA), n 0, defined on X^q, whereC(z) L (C^q) denotes the one-step map associated with the method.It is shown that for appropriately chosen V, Y: C C^q, basedon the principal right and left eigenvectors of C(z), Cⁿ(hA)approximates the semigroup V(hA)e^nhAY^H(hA) with optimal orderp.     

Multiple Positive Solutions of Semilinear Differential Equations with Singularities

The existence of positive solutions of a second order differentialequation of the form z'+g(t)f(z)=0 (1.1) with the separated boundary conditions: z(0) – ßz'(0)= 0 and z(1)+z'(1) = 0 has proved to be important in physicsand applied mathematics. For example, the Thomas–Fermiequation, where f = z^3/2 and g = t^–1/2 (see [12, 13, 24]),so g has a singularity at 0, was developed in studies of atomicstructures (see for example, [24]) and atomic calculations [6].The separated boundary conditions are obtained from the usualThomas–Fermi boundary conditions by a change of variableand a normalization (see [22, 24]). The generalized Emden–Fowlerequation, where f = z^p, p > 0 and g is continuous (see [24,28]) arises in the fields of gas dynamics, nuclear physics,chemically reacting systems [28] and in the study of multipoletoroidal plasmas [4]. In most of these applications, the physicalinterest lies in the existence and uniqueness of positive solutions.     

On the division of polynomial matrices

The main purpose of this paper is to determine two new algorithmsfor the division of the polynomial matrix B(s) R[s]^pxq by A(s) R[s]^pxp (a) based on the Laurent matrix expansion at s = =of the inverse of A(s), i.e. A(s)^–1, and (b) in a waysimilar to the one presented by Gantmacher (1959).     

Beardon's Diophantine Equations and Non-Free Mobius Groups

Let µ be a real number. The Möbius group G_µis the matrix group generated by It is known that G_µ is free if |µ| 2 (see [1])or if µ is transcendental (see [3, 8]). Moreover, thereis a set of irrational algebraic numbers µ which is densein (–2, 2) and for which G_µ is non-free [2, p. 528].We may assume that µ > 0, and in this paper we considerrational µ in (0, 2). The following problem is difficult. Let G^nf denote the set of all rational numbers µ in (0,2) for which G_µ is non-free. In 1969 Lyndon and Ullman[8] proved that G^nf contains the elements of the forms p/(p²+ 1) and 1/(p + 1), where p = 1, 2, ..., and that if µ₀ G^nf, then µ₀/p G^nf for p = 1, 2, .... In 1993 Beardon[2] studied problem (P) by means of the words of the form A^rB^s A^t and A^r B^s A^t B^u A^v, and he obtained a sufficient conditionfor solvability of (P), included implicitly in [2, pp. 530–531],by means of the following Diophantine equations: 1991 Mathematics SubjectClassification 20E05, 20H20, 11D09.     

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.     

Cluster Sets of Harmonic Functions at the Boundary of a Half-Space

The purpose of this paper is to answer some questions posedby Doob [2] in 1965 concerning the boundary cluster sets ofharmonic and superharmonic functions on the half-space D givenby D = R^n–1 x (0, + ), where n 2. Let f: D [–,+] and let Z D. Following Doob, we write B_Z (respectively C_Z)for the non-tangential (respectively minimal fine) cluster setof f at Z. Thus l B_Z if and only if there is a sequence (X_m)of points in D which approaches Z non-tangentially and satisfiesf(X_m) l. Also, l C_Z if and only if there is a subset E ofD which is not minimally thin at Z with respect to D, and whichsatisfies f(X) l as X Z along E. (We refer to the book byDoob [3, 1.XII] for an account of the minimal fine topology.In particular, the latter equivalence may be found in [3, 1.XII.16].)If f is superharmonic on D, then (see [2, 6]) both sets B_Z andC_Z are subintervals of [–, +]. Let denote (n –1)-dimensional measure on D. The following results are due toDoob [2, Theorem 6.1 and p. 123]. 1991 Mathematics Subject Classification31B25.     

Discontinuous Homomorphisms from Non-Commutative Banach Algebras

In the 1970s, a question of Kaplansky about discontinuous homomorphismsfrom certain commutative Banach algebras was resolved. Let Abe the commutative C*-algebra C(), where is an infinite compactspace. Then, if the continuum hypothesis (CH) be assumed, thereis a discontinuous homomorphism from C() into a Banach algebra[2, 7]. In fact, let A be a commutative Banach algebra. Then(with (CH)) there is a discontinuous homomorphism from A intoa Banach algebra whenever the character space _A of A is infinite[3, Theorem 3] and also whenever there is a non-maximal, primeideal P in A such that |A/P|=2^₀ [4, 8]. (It is an open questionwhether or not every infinite-dimensional, commutative Banachalgebra A satisfies this latter condition.) 1991 MathematicsSubject Classification 46H40.     

Bilinear isometries on subspaces of continuous functions

Let A and B be strongly separating linear subspaces of C₀(X) and C₀(Y), respectively, and assume that ?A ≠ ?? (?A stands for the set of generalized peak points for A) and ?B ≠ ??. Let T: A × B → C₀(Z) be a bilinear isometry. Then there exist a nonempty subset Z₀ of Z, a surjective continuous mapping h: Z₀ → ?A × ?B and a norm‐one continuous function a: Z₀ → K such that T (f, g)(z) = a (z)f (π_x (h (z))g (π_y (h (z)) for all z ∈ Z₀ and every pair (f, g) ∈ A × B. These results can be applied, for example, to non‐unital function algebras (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Weil Representations of Symplectic Groups Over Rings

We are interested in Weil representations of Sp(2n, R), whereR is the ring Z/p^lZ, p is an odd prime and l is a positive integer,or, more generally, R = O/p^l, where O is the ring of integersof a local field, p is the maximal ideal of O and O/p has oddcharacteristic. One reason for this interest is that a continuousfinite-dimensional complex representation of Sp(2n, O) has tofactor through a representation of Sp(2n, O/p^l) for some l.     

On the Discrepancy for Cartesian Products

Let B₂ denote the family of all circular discs in the plane.It is proved that the discrepancy for the family {B₁ x B₂ :B₁, B₂ B₂} in R⁴ is O(n^1/4+) for an arbitrarily small constant > 0, that is, it is essentially the same as that for thefamily B₂ itself. The result is established for the combinatorialdiscrepancy, and consequently it holds for the discrepancy withrespect to the Lebesgue measure as well. This answers a questionof Beck and Chen. More generally, we prove an upper bound forthe discrepancy for a family {^k_i=1A_i:A_iA_i, i = 1, 2, ..., k},where each A_i is a family in R^d_i, each of whose sets is describedby a bounded number of polynomial inequalities of bounded degree.The resulting discrepancy bound is determined by the ‘worst’of the families A_i, and it depends on the existence of certaindecompositions into constant-complexity cells for arrangementsof surfaces bounding the sets of A_i. The proof uses Beck's partialcoloring method and decomposition techniques developed for therange-searching problem in computational geometry.     

One-Dimensional p-Adic Subanalytic Sets

In this paper we extend two theorems from [2] on p-adic subanalyticsets, where p is a fixed prime number, Q_p is the field of p-adicnumbers and Z_p is the ring of p-adic integers. One of thesetheorems [2, 3.32] says that each subanalytic subset of Z_p issemialgebraic. This is extended here as follows.     

Ternary forms as invariants of Markoff forms and other SL2 (Z)-bundles

Consider the free group Γ = {A,B} generated by matrices A, B in SL₂(Z). We can construct a ternary form Φ(x,y,z) whose GL₃(Z) equivalence class is invariant, as it depends on Γ and not the choice of generators. If Γ is the commutator of SL₂(Z), then the generating matrices have fixed points corresponding to different fields and inequivalent Markoff forms, but they are all biuniquely determined by Φ = -z²+ y(2x+y+z) to within equivalence. When referred to transformations A, B of the upper half plane, this phenomenon is interpreted in terms of inequivalent homotopy elements which are primitive for the perforated torus.     

One Cubic Diophantine Inequality

Suppose that G(x) is a form, or homogeneous polynomial, of odddegree d in s variables, with real coefficients. Schmidt [15]has shown that there exists a positive integer s₀(d), whichdepends only on the degree d, so that if s s₀(d), then thereis an x Z^s\{0} satisfying the inequality |G(x)|<1. (1) In other words, if there are enough variables, in terms of thedegree only, then there is a nontrivial solution to (1). Lets₀(d) be the minimum integer with the above property. In thecourse of proving this important result, Schmidt did not explicitlygive upper bounds for s₀(d). His methods do indicate how todo so, although not very efficiently. However, in fact muchearlier, Pitman [13] provided explicit bounds in the case whenG is a cubic. We consider a general cubic form F(x) with realcoefficients, in s variables, and look at the inequality |F(x)|<1. (2) Specifically, Pitman showed that if s(1314)²⁵⁶–1, (3) then inequality (2) is non-trivially soluble in integers. Wepresent the following improvement of this bound.     

Bounds for the Singular Values of a Matrix

A lower-bound theorem is developed for the singular values ofa matrix A (and therefore for the eigenvalues of B = A^HA). Itis found that there is always a unique column scaling of A whichproduces the optimum bound. However, sharper bounds still maysometimes be obtained by taking advantage of matrix partitioning.It is shown that the resulting bounds may often (but not always)be better than those obtained by applying Gerschgorin's theoremto B. The equivalent upper-bound theorem is found to be weak.     

How To Make Ext Vanish

We describe a general construction of a module A from a givenmodule B such that Ext(B, A) = 0, and we apply it to answerseveral questions on splitters, cotorsion theories and saturatedrings.     

Tensor Products of C*-Algebras Over Abelian Subalgebras

Suppose that A is a C*-algebra and C is a unital abelian C*-subalgebrawhich is isomorphic to a unital subalgebra of the centre ofM(A), the multiplier algebra of A. Letting = , so that we maywrite C = C(), we call A a C()-algebra (following Blanchard[7]). Suppose that B is another C()-algebra, then we form A_CB, the algebraic tensor product of A with B over C as follows:A B is the algebraic tensor product over C, I_C = {ⁿ_i–1(f_i 1–1f_i)x|f_iC, xAB} is the ideal in AB generated by f1–1f|fC,and A _CB = AB/I_C. Then A_CB is an involutive algebra over C,and we shall be interested in deciding when A_CB is a pre-C*-algebra;that is, when is there a C*-norm on A_C B? There is a C*-semi-norm,which we denote by ||·||_C-min, which is minimal in thesense that it is dominated by any semi-norm whose kernel containsthe kernel of ||·||_C-min. Moreover, if A _C B has a C*-norm,then ||·||_C-min is a C*-norm on A_C B. The problem isto decide when ||·||_C-min is a norm. It was shown byBlanchard [7, Proposition 3.1] that when A and B are continuousfields and C is separable, then ||·||_C-min is a norm.In this paper we show that ||·||_C-min is a norm whenC is a von Neumann algebra, and then we examine some consequences.