Smooth noncompact operators from <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">K</Emphasis>), <Emphasis Type="Italic">K</Emphasis> scattered

Let X be a Banach space, K be a scattered compact and T: B _C(K) → X be a Fréchet smooth operator whose derivative is uniformly continuous. We introduce the smooth biconjugate T**: B _C(K)** → X** and prove that if T is noncompact, then the derivative of T** at some point is a noncompact linear operator. Using this we conclude, among other things, that either is compact or that ℓ₁ is a complemented subspace of X*. We also give some relevant examples of smooth functions and operators, in particular, a C ^1,u -smooth noncompact operator from B _{c O} which does not fix any (affine) basic sequence. P. Hájek was supported by grants A100190502, Institutional Research Plan AV0Z10190503.     

<Emphasis Type="Italic">C</Emphasis><Superscript>*</Superscript>-Subalgebras Generated by a Single Operator in <Emphasis Type="Italic">B</Emphasis>(<Emphasis Type="Italic">H</Emphasis>)

In this paper, we characterize a C ^*-subalgebra C ^*(x) of B(H), generated by a single operator x. We show that if x is polar-decomposed by aq, where a is the partial isometry part and q is the positive operator part of x, then C ^*(x) is *-isomorphic to the groupoid crossed product algebra A_q×_a\mathbbG_a\mathcal{A}_{q}\times_{\alpha }\mathbb{G}_{a} , where A_q=C^*(q)\mathcal{A}_{q}=C^{*}(q) and \mathbbG_a\mathbb{G}_{a} is the graph groupoid induced by a partial isometry part a of x.     

Best approximation by ridge functions in <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-spaces

We study the approximation of the classes of functions by the manifold R _n formed by all possible linear combinations of n ridge functions of the form r(a · x)): It is proved that, for any 1 ≤ q ≤ p ≤ ∞, the deviation of the Sobolev class W ^r _p from the set R _n of ridge functions in the space L _q (B ^d ) satisfies the sharp order n ^-r/(d-1).     

On the Theory of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>)-Banach Lattices

Given 1≤ p,q < ∞, let BL_pL_q be the class of all Banach lattices X such that X is isometrically lattice isomorphic to a band in some L_p(L_q)-Banach lattice. We show that the range of a positive contractive projection on any BL_pL_q-Banach lattice is itself in BL_pL_q. It is a consequence of this theorem and previous results that BL_pL_q is first-order axiomatizable in the language of Banach lattices. By studying the pavings of arbitrary BL_pL_q-Banach lattices by finite dimensional sublattices that are themselves in this class, we give an explicit set of axioms for BL_pL_q. We also consider the class of all sublattices of L_p(L_q)-Banach lattices; for this class (when p/q is not an integer) we give a set of axioms that are similar to Krivine’s well-known axioms for the subspaces of L_p-Banach spaces (when p/2 is not an integer). We also extend this result to the limiting case q = ∞.     

Representations of algebraic groups of type <Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> with small weight multiplicities

Lower estimates for the maximal weight multiplicities in irreducible representations of the algebraic groups of type C _n in characteristic p ≤ 7 are found. If n ≥ 8 and p ≠ 2 , then for an irreducible representation either such a multiplicity is at least n− 4 − [n]₄,where [n]₄ is the residue of n modulo 4, or all the weight multiplicities are equal to 1.For p = 2, the situation is more complicated, and for every n and l there exists a class of representations with the maximal weight multiplicity equal to 2 ^l . For symplectic groups in characteristic p > 7 and spinor groups similar results were obtained earlier. Bibliography: 15 titles.     

Maximal subgroups of <Emphasis Type="Italic">GL</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>(<Emphasis Type="Italic">D</Emphasis>) with finite conjugacy classes

Let D be an infinite division ring. A famous result due to Herstein says that every non-central element of D has infinitely many conjugates and so, if D ^* is an FC-group, then D is a field. Let M be a maximal subgroup of GL _n (D), where n ≥ 1. In this paper, we prove that if M is an FC-group, then it is the multiplicative group of some maximal subfield of M _n (D). Moreover, if M is algebraic over Z(D), then [D : Z(D)] < ∞.     

An Estimate of Growth Bound of Positive <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>-Semigroup on <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> Space and its Applications

Let be a positive C ₀-semigroup on L ^p (Ω), with infinitesimal generator A. In this paper, it is proved that if there exists a such that and , where A ^* is the adjoint of A, then the growth bound of T(t) is upper bounded by b when p = 1, and by when 1 lt; p lt; α and c D(A), where . This is an operator version of a classical stability result on Z-matrix. As application examples, some new results on the asymptotic behaviours of population system and neutron transport system are obtained. Submitted: March 1, 2001?Revised: August 28, 2002     

The <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-Approximation Order of Surface Spline Interpolation for 1 \leq <Emphasis Type="Italic">p</Emphasis> \leq 2

We show that the L_p-approximation order of surface spline interpolation equals m+1/p for p in the range 1 \leq p \leq 2, where m is an integer parameter which specifies the surface spline. Previously it was known that this order was bounded below by m + &frac; and above by m+1/p. With h denoting the fill-distance between the interpolation points and the domain , we show specifically that the L_p()-norm of the error between f and its surface spline interpolant is O(h^{m + 1/p}) provided that f belongs to an appropriate Sobolev or Besov space and that \subset R^d is open, bounded, and has the C^2m-regularity property. We also show that the boundary effects (which cause the rate of convergence to be significantly worse than O(h^2m)) are confined to a boundary layer whose width is no larger than a constant multiple of h |log h|. Finally, we state numerical evidence which supports the conjecture that the L_p-approximation order of surface spline interpolation is m + 1/p for 2 < p \leq \infty.     

Rationality of the vertex operator algebra <Emphasis Type="Italic">V</Emphasis><Subscript><Emphasis Type="Italic">L</Emphasis></Subscript>+ for a positive definite even lattice <Emphasis Type="Italic">L</Emphasis>

The lattice vertex operator algebra V_L associated to a positive definite even lattice L has an automorphism of order 2 lifted from –1-isometry of L. The fixed point set V_L+ of V_L for the automorphism is naturally a vertex operator algebra. We prove that any ₀-graded weak V_L⁺-module is completely reducible.Supported by JSPS Research Fellowships for Young Scientists.     

A space <Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(<Emphasis Type="Italic">X</Emphasis>) is dominated by irrationals if and only if it is<Emphasis Type="Italic"> K</Emphasis>-analytic

Summary We prove that, for any Tychonoff X, the space C_p(X) is K-analytic if and only if it has a compact cover {K_p: p } such that K_p subset K_q whenever p,q and p q. Applying this result we show that if C_p(X) is K-analytic then C_p(X) is K-analytic as well. We also establish that a space C_p(X) is K-analytic and Baire if and only if X is countable and discrete.     

Estimation of the norm of the derivative of a monotone rational function in the spaces <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

We show that the derivative of an arbitrary rational function R of degree n that increases on the segment [−1, 1] satisfies the following equality for all 0 < ε < 1 and p, q > 1:

<Emphasis Type="Italic">K</Emphasis>-homology of <Emphasis Type="Italic">C</Emphasis><Superscript><Emphasis Type="Italic">*</Emphasis></Superscript>-categories and symmetric spectra representing <Emphasis Type="Italic">K</Emphasis>-homology

We define K-homology groups K _* () for small C ^* -categories in terms of Hilbert modules over the C ^* -category . We also define a functor A ^f from the category of small C ^* -categories into the category of C ^* -algebras and show that there is a natural isomorphism . In addition, we give an easy construction of a functor from the category of C ^* -algebras into the category of symmetric spectra which represents K-homology, i.e. we show that the functor comes with a natural isomorphism for C ^* -algebras A. It then follows that the composition A ^f provides a functor that can be used in the Davis-Lück approach for constructing the Baum-Connes map.     

Congruences Concerning Values of the Modular Functions <Emphasis Type="Italic">j</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

The j-function j(z) = q⁻¹+ 744 + 196884q + ⋅s plays an important role in many problems. In [7], Zagier, presented an interesting series of functions obtained from the j-function: j_m(ζ) = (j(ζ) – 744)∨T₀(m), where T₀(m) is the usual m′th normalized weight 0 Hecke operator. In [3], Bruinier et al. show how this series of functions can be used to describe all meromorphic modular forms on SL₂(ℤ). In this note we use these functions and basic notions about modular forms to determine previously unidentified congruence relations between the coefficients of Eisenstein series and the j-function. 2000 Mathematics Subject Classification: Primary–11B50, 11F03, 11F30 The author thanks the National Science Foundation for their generous support.     

On Transference of Multipliers on Matrix Weighted <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-Spaces

We consider a periodic matrix weight W defined on ℝ ^d and taking values in the N×N positive-definite matrices. For such weights, we prove transference results between multiplier operators on L _p (ℝ ^d ;W) and L_p(\mathbb T^d;W)L_{p}(\mathbb {T}^{d};W), 1<p<∞, respectively. As a specific application, we study transference results for homogeneous multipliers of degree zero.     

Compact composition operators on BMOA(<Emphasis Type="Italic">B</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>)

The paper gives some characterization theorems for the compact composition operator on some function spaces over the unit ball B _n in ℂ ⁿ . Especially, it gives a characterization for compact composition operators on BMOA(B _n ), which generalizes a result proved by Bourdon, Cima and Matheson for the case n = 1.     

The -conjecture and equivariant <Emphasis Type="Italic">e</Emphasis><Subscript><Emphasis Type="Italic">C</Emphasis></Subscript>-invariants

Let X be a smooth closed oriented non-spin 4-manifold with even intersection form kE₈nH (n1). The -conjecture states that n is greater than or equal to |k|. In this paper we give a proof of the -conjecture. The strategy of this paper is to use the finite dimensional approximation of the map induced from the Seiberg-Witten equations and equivariant e_C-invariants as in the paper of M. Furuta and Y. Kametani.Mathematics Subject Classification (1991): 57R55This work was supported by Korea Research Foundation Grant (KRF–2002–003–C00011).     

On the <Emphasis Type="Italic">K</Emphasis>-Theory of Graph <Emphasis Type="Italic">C</Emphasis><Superscript>*</Superscript>-Algebras

We classify graph C ^*-algebras, namely, Cuntz-Krieger algebras associated to the Bass-Hashimoto edge incidence operator of a finite graph, up to strict isomorphism. This is done by a purely graph theoretical calculation of the K-theory of the C ^*-algebras and the method also provides an independent proof of the classification up to Morita equivalence and stable equivalence of such algebras, without using the boundary operator algebra. A direct relation is given between the K ₁-group of the algebra and the cycle space of the graph. We thank Jakub Byszewski for his input in Sect. 2.8. The position of the unit in K ₀( _Ч) was guessed based on some example calculations by Jannis Visser in his SCI 291 Science Laboratory at Utrecht University College.     

A Class of <Emphasis Type="Italic">C</Emphasis>*-algebras with non-stable <Emphasis Type="Italic">K</Emphasis><Subscript>1</Subscript>-group property

In this paper, we give a class of C*-algebras with non-stable K ₁-group property which include the example non-simple tracial topological rank zero and stable rank two C*-algebra given by Lin and Osaka.     

Free <Emphasis Type="Bold">C</Emphasis><Subscript>+</Subscript>-actions on <Emphasis Type="Bold">C</Emphasis><Superscript>3</Superscript> are translations

Let X be a smooth contractible three-dimensional affine algebraic variety with a free algebraic C₊-action on it such that S=X//C₊ is smooth. We prove that X is isomorphic to S×C and the action is induced by a translation on the second factor. As a consequence we show that any free algebraic C₊-action on C³ is a translation in a suitable coordinate system.