Direct Sums of Operator Spaces

It is proved that if X and Y are operator spaces such that everycompletely bounded operator from X into Y is completely compactand Z is a completely complemented subspace of X Y, then thereexists a completely bounded automorphism : X Y X Y with completelybounded inverse such that Z = X₀ Y₀, where X₀ and Y₀ are completelycomplemented subspaces of X and Y, respectively. If X and Yare homogeneous, the existence is proved of such a under aweaker assumption that any operator from X to Y is strictlysingular. An upper estimate is obtained for ||||_cb||^–1||_cbif X and Y are separable homogeneous Hilbertian operator spaces.Also proved is the uniqueness of a ‘completely unconditional’basis in X Y if X and Y satisfy certain conditions.     

An algebraic loop theorem and the decomposition of PD3-pairs

Let (Y, X) denote a three-dimensional Poincaré pair (PD³-pair).By the work of Eckmann, Müller and Linnell we may suppose,up to a homotopy equivalence, that the boundary X is a closed2-manifold. We show that if a component of X fails to be ₁-injectivein Y, then there is an essential simple loop in X which is nullhomotopicin Y. It follows that there is a finite process of attaching2-disks along essential simple loops on X, and filling sphericalcomponents of X, which transforms (Y, X) into a PD³-pair (Y',X') with aspherical incompressible boundary X' and such that₁(Y) = ₁(Y'). The PD³-pair (Y', X') then admits a canonicaldecomposition as a connected sum of a finite number of asphericalPD³-pairs with incompressible boundary, together with a PD³-pairhaving virtually free (possibly finite) fundamental group andboundary a (possibly empty) disjoint union of projective planes.     

A Generalised Skolem-Mahler-lech Theorem for Affine Varieties

The Skolem–Mahler–Lech theorem states that if f(n)is a sequence given by a linear recurrence over a field of characteristic0, then the set of m such that f(m) is equal to 0 is the unionof a finite number of arithmetic progressions in m 0 and afinite set. We prove that if X is a subvariety of an affinevariety Y over a field of characteristic 0 and q is a pointin Y, and is an automorphism of Y, then the set of m such that^m(q) lies in X is a union of a finite number of complete doubly-infinitearithmetic progressions and a finite set. We show that thisis a generalisation of the Skolem–Mahler–Lech theorem.     

Normal Transversality and Uniform Bounds

Let A be a commutative ring. A graded A-algebra U = _n0 U_n isa standard A-algebra if U₀ = A and U = A[U₁] is generated asan A-algebra by the elements of U₁. A graded U-module F = _n0F_nis a standard U-module if F is generated as a U-module by theelements of F₀, that is, F_n = U_nF₀ for all n 0. In particular,F_n = U₁F_n–1 for all n 1. Given I, J, two ideals of A,we consider the following standard algebras: the Rees algebraof I, R(I) = _n0Iⁿtⁿ = A[It] A[t], and the multi-Rees algebraof I and J, R(I, J) = _n0(_p+q=nI^pJ^qu^pv^q) = A[Iu, Jv] A[u, v].Consider the associated graded ring of I, G(I) = R(I) A/I =_n0Iⁿ/Iⁿ⁺¹, and the multi-associated graded ring of I and J,G(I, J) = R(I, J) A/(I+J) = _n0(_p+q=nI^pJ^q/(I+J)I^pJ^q). We canalways consider the tensor product of two standard A-algebrasU = _p0U_p and V = _q0V_q as a standard A-algebra with the naturalgrading U V = _n0(_p+q=nU_p V_q). If M is an A-module, we havethe standard modules: the Rees module of I with respect to M,R(I; M) = _n0IⁿMtⁿ = M[It] M[t] (a standard R(I)-module), andthe multi-Rees module of I and J with respect to M, R(I, J;M) = _n0(_p+q=nI^pJ^qMu^pv^q) = M[Iu, Jv] M[u, v] (a standard R(I,J)-module). Consider the associated graded module of M withrespect to I, G(I; M) = R(I; M) A/I = _n0IⁿM/Iⁿ⁺¹M (a standardG(I)-module), and the multi-associated graded module of M withrespect to I and J, G(I, J; M) = R(I, J; M) A/(I+J) = _n0(_p+q=nI^pJ^qM/(I+J)I^pJ^qM)(a standard G(I, J)-module). If U, V are two standard A-algebras,F is a standard U-module and G is a standard V-module, thenF G = _n0(_p+q=nF_p G_q) is a standard U V-module. Denote by :R(I) R(J; M) R(I, J; M) and :R(I, J; M) R(I+J;M) the natural surjective graded morphisms of standard RI) R(J)-modules. Let :R(I) R(J; M) R(I+J; M) be . Denote by :G(I) G(J; M) G(I, J; M) and :G(I, J; M) G(I+J; M) the tensor productof and by A/(I+J); these are two natural surjective gradedmorphisms of standard G(I) G(J)-modules. Let :G(I) G(J; M) G(I+J; M) be . The first purpose of this paper is to prove the following theorem.     

The l{infty} direct sum of Lp (1 < p < {infty}) is primary

Using Szemeredi's theorem on arithmetic progressions, it isshown that, for 1 < p < , the infinite l direct sum (L^p L^p · · · )_l is a primary Banach space.     

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.     

Linear Automorphisms that are Symplectomorphisms

Let K be the field of real or complex numbers. Let (X K²ⁿ,) be a symplectic vector space and take 0 < k < n,N =. Let L₁,...,L_N X be 2k-dimensionallinear subspaces which are in a sufficiently general position.It is shown that if F : X X is a linear automorphism whichpreserves the form ^k on all subspaces L₁,...,L_N, then F is an_k-symplectomorphism (that is, F^* = _k, where ). In particular, if K = R and k is odd then F mustbe a symplectomorphism. The unitary version of this theoremis proved as well. It is also observed that the set A_l,2r ofall l-dimensional linear subspaces on which the form has rank 2r is linear in the Grassmannian G(l,2n), that is, there isa linear subspace L such that A_l,2r = L G(l, 2n). In particular,the set A_l,2r can be computed effectively. Finally, the notionof symplectic volume is introduced and it is proved that itis another strong invariant.     

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.     

A Remark on Contraction Semigroups on Banach Spaces

Let X be a complex Banach space and let J:XX^* be a duality sectionon X (that is, x,J(x)=||J(x)||||x||=||J(x)||²)=||x||²). Forany unit vector x and any (C⁰) contraction semigroup T={e^tA:t0}, Goldstein proved that if X is a Hilbert space and |T(t)x,j(x)|1 as t, then x is an eigenvector of A corresponding toa purel imaginary eigenvalue. In this article, we prove thata similar result holds if X is a strictly convex complex Banachspace.     

On a theorem of K. Schmidt

Let Y be a locally compact group, Aut(Y) be the group of topologicalautomorphisms of Y and (Y) be the set of continuous positivedefinite functions on Y which have unit value at the identity.A function (Y²) is said to be of product type if there aresuch functions _j (Y) that (u, v) = ₁(u)₂(v). Define the mappingT: Y² Y² by the formula T(u, v) = (A₁ uA₂ v, A₃ u A₄ v), whereA_j Aut(Y), and assume that T is a one-to-one transform. K.Schmidt proved: (i) if both (u, v) and (T(u, v)) are of producttype, then the functions _j are infinitely divisible; (ii) ifY is Abelian, both (u, v) and (T(u, v)) are of product type,and (u, v) 0, then the functions _j are Gaussian. We show thatstatement (i), generally, is not valid, but K. Schmidt's proofholds true if (u, v) 0. We also give another proof of statement(ii). Our proof uses neither the Levy–Khinchin formulafor a continuous infinitely divisible positive definite functionnor (i) on which K. Schmidt's proof is based.     

The Weiss Conjecture for Bounded Analytic Semigroups

New results concerning the so-called Weiss conjecture on admissibleoperators for bounded analytic semigroups are given. Let be a bounded analytic semigroup withgenerator –A on some Banach space X. It is proved thatif A^1/2 is admissible for A, that is, if there is an estimate then any continuous mappingC : D(A) Y valued in a Banach space Y is admissible for A providedthat there is an estimate .for , Re()<0. This holds in particular if is a contractive (analytic) semigroup on Hilbertspace. In the converse direction, it is shown that this mayhappen for a bounded analytic semigroup on Hilbert space thatis not similar to a contractive one. Applications in non-HilbertianBanach spaces are also given.     

Generalizations of Pełczyński’s decomposition method for Banach spaces containing a complemented copy of their squares

Suppose that X and Y are Banach spaces isomorphic to complemented subspaces of each other. In 1996, W. T. Gowers solved the Schroeder-Bernstein Problem for Banach spaces by showing that X is not necessarily isomorphic to Y. However, if X ² is complemented in X with supplement A and Y ² is complemented in Y with supplement B, that is,

Type 2 Semi-Algebras of Continuous Functions

A semi-algebra of continuous functions is a cone A of continuousreal functions on a compact Hausdorff space X such that A containsthe products of its elements. A cone A is said to be of typen if fA implies fⁿ(1 + f)^–1 A. Uniformly closed semi-algebrasof types 0 and 1 have long been characterized in a manner analogousto the Stone–Weierstrass theorem, but, except for thecase when A is generated by a single function, little has beenknown about type 2. Here, progress is reported on two problems.The first is the characterization of those continuous linearfunctionals on C(X) that determine semi-algebras of type 2.The second is the determination of the type of the tensor productof two type 1 semi-algebras. 1991 Mathematics Subject Classification:46J10.     

Polynomial symplectomorphisms

Let be the field of real or complex numbers. Let (X ²ⁿ, )be a symplectic affine space. We study the group of polynomialsymplectomorphisms of X. We show that for an arbitrary k thegroup of polynomial symplectomorphisms acts k-transitively onX. Moreover, if 2 l 2n – 2 then elements of this groupcan be characterized by polynomial automorphisms which preservethe symplectic type of all algebraic l-dimensional subvarietiesof X.     

Numerical Solution of Matrix Polynomial Equations by Newton's Method

Let P(X) = _v=1ⁿ A_vX^v with A_v, X C^m?m (v = 1, ..., n) be a matrixpolynomial. We present a Newton method to solve the equationP(X) = B, and we prove that the algorithm converges quadraticallynear simple solvents. We need the inverse of the Fr?chet-derivativeP' of P. This leads to linear equations for the correctionsH of type In the second part, we turn to the case of scalar coefficients, i.e. A_v = _vI, with_v C (v = 1, ..., n). The derivative P' and the usual algebraicderivative P' are compared and we show that the use of P' leadsto difficulties. In particular, those algorithms based on P'are not self-correcting, while our proposed method is self-correcting.Numerical examples are included. In the Appendix, an existencetheorem is proved by using a modified Newton method.     

Castelnuovo-Mumford Regularity and Analytic Deviation of Ideals

Let (A, m) be a local ring. For convenience we will assume throughoutthis paper that the residue field of A is infinite. Let I be an ideal of A. An ideal J I is called a reduction ofI if JIⁿ = Iⁿ⁺¹ for some integer n. The least number n withthis property is denoted by rJ (I). A reduction of I is saidto be minimal if it does not contain any other reduction ofI. The reduction number r(I) of I is the minimum of r_J(I) forall minimal reductions J of I. A minimal reduction of I usuallyhas better properties than I. It can be viewed as an approximationof I and the reduction number is a measure for how it is differentfrom I. The minimal number of generators of every minimal reductionof I is equal to the dimension of the fibre ring _n0Iⁿ/mIⁿ. Thisinvariant is called the analytic spread of I and denoted byl(I). All these notions have played an important role in idealtheory since their introduction by Northcott and Rees [16].     

Real Abelian Varieties with Many Line Bundles

Let X and Y be affine nonsingular real algebraic varieties.A general problem in real algebraic geometry is to try to decidewhen a continuous map f: X Y can be approximated by regularmaps in the space of c⁰ mappings from X to Y, equipped withthe c⁰ topology. This paper solves this problem when X is theconnected component containing the origin of the real part ofa complex Abelian variety and Y is the standard 2-dimensionalsphere.     

Bounding the number of stable homotopy types of a parametrized family of semi-algebraic sets defined by quadratic inequalities

We prove a nearly optimal bound on the number of stable homotopytypes occurring in a k-parameter semi-algebraic family of setsin R, each defined in terms of m quadratic inequalities. Ourbound is exponential in k and m, but polynomial in . More precisely,we prove the following. Let R be a real closed field and let = {P₁, ... , P_m} R[Y₁, ... ,Y,X₁, ... ,X_k], with deg_Y(P_i) 2, deg_X(P_i) d, 1 i m. Let S R^+k be a semi-algebraic set,defined by a Boolean formula without negations, with atoms ofthe form P 0, P 0, P . Let : R^+k R^k be the projection onthe last k coordinates. Then the number of stable homotopy typesamongst the fibers S_x = ^–1(x) S is bounded by (2^mkd)^O(mk).     

Eigenvectors of Order-Preserving Linear Operators

Suppose that K is a closed, total cone in a real Banach spaceX, that A:XX is a bounded linear operator which maps K intoitself, and that A' denotes the Banach space adjoint of A. Assumethat r, the spectral radius of A, is positive, and that thereexist x₀0 and m1 with A^m(x₀)=r^mx₀ (or, more generally, thatthere exist x₀(–K) and m1 with A^m(x₀)r^mx₀). If, in addition,A satisfies some hypotheses of a type used in mean ergodic theorems,it is proved that there exist uK–{0} and K'–{0}with A(u)=ru, A'()=r and (u)>0. The support boundary of Kis used to discuss the algebraic simplicity of the eigenvaluer. The relation of the support boundary to H. Schaefer's ideasof quasi-interior elements of K and irreducible operators Ais treated, and it is noted that, if dim(X)>1, then thereexists an xK–{0} which is not a quasi-interior point.The motivation for the results is recent work of Toland, whoconsidered the case in which X is a Hilbert space and A is self-adjoint;the theorems in the paper generalize several of Toland's propositions.     

Coarse version of the Banach-Stone theorem

We show that if there exists a Lipschitz homeomorphism T between the nets in the Banach spaces C(X) and C(Y) of continuous real valued functions on compact spaces X and Y, then the spaces X and Y are homeomorphic provided . By l(T) and l(T−1) we denote the Lipschitz constants of the maps T and T−1. This improves the classical result of Jarosz and the recent result of Dutrieux and Kalton where the constant obtained is . We also estimate the distance of the map T from the isometry of the spaces C(X) and C(Y).