Isospectrality for Spherical Space Forms

Consider the set S(q, Γ) of isometry classes of q-dimensional spherical space forms whose fundamental groups are isomorphic to a fixed group Γ. We define a certain group ${\cal A}(q,\Gamma)$ of transformations on the finite set ${\cal S}(q,\Gamma)$ , prove that any two elements in the same ${\cal A}(q,\Gamma)$ -orbit are strongly isospectral, and study some consequences. Then a number of the results are carried over to riemannian quotients of oriented real Grassmann manifolds. Some of these results were first obtained by Ikeda, mostly for the special case (Γ cyclic) of lens spaces, and by Gilkey and Ikeda for the case where every Sylow subgroup of Γ is cyclic.     

Integrals,conditional expectations,and martingales of multivalued functions

Let (Ω, $\text{A}$, μ) be a finite measure space and $\text{X}$ a real separable Banach space. Measurability and integrability are defined for multivalued functions on Ω with values in the family of nonempty closed subsets of $\text{X}$. To present a theory of integrals, conditional expectations, and martingales of multivalued functions, several types of spaces of integrably bounded multivalued functions are formulated as complete metric spaces including the space L¹(Ω; $\text{X}$) isometrically. For multivalued functions in these spaces, multivalued conditional expectations are introduced, and the properties possessed by the usual conditional expectation are obtained for the multivalued conditional expectation with some modifications. Multivalued martingales are also defined, and their convergence theorems are established in several ways.     

Injective metrizability and the duality theory of cubings

Following his discovery that finite metric spaces have injective envelopes naturally admitting a polyhedral structure, Isbell, in his pioneering work on injective metric spaces, attempted a characterization of cellular complexes admitting the structure of an injective metric space. A bit later, Mai and Tang confirmed Isbell’s conjecture that a simplicial complex is injectively metrizable if and only if it is collapsible. Considerable advances in the understanding, classification and applications of injective envelopes have since been made by Dress, Huber, Sturmfels and collaborators, and most recently by Lang. Unfortunately a combination theory for injective polyhedra is still unavailable.Here we expose a connection to the duality theory of cubings –simply connected non-positively curved cubical complexes –which provides a more principled and accessible approach to Mai and Tang’s result, providing one with a powerful tool for systematic construction of locally-compact injective metric spaces:Main Theorem. Any complete pointed Gromov–Hausdorff limit of locally-finite piecewise-${?}_{\infty }$ cubings is injective. □This result may be construed as a combination theorem for the simplest injective polytopes, ${?}_{\infty }$-parallelopipeds, where the condition for retaining injectivity is the combinatorial non-positive curvature condition on the complex. Thus it represents a first step towards a more comprehensive combination theory for injective spaces.In addition to setting the earlier work on injectively metrizable complexes within its proper context of non-positively curved geometry, this paper is meant to provide the reader with a systematic review of the results – otherwise scattered throughout the geometric group theory literature – on the duality theory and the geometry of cubings, which make this connection possible.     

Cartesian-closed coreflective subcategories of uniform spaces generated by classes of metric spaces

The main result, in Theorem 3, is that in the category Unif of Hausdorff uniform spaces and uniformly continuous maps, the coreflective hulls of the following classes are cartesian-closed: all metric spaces having no infinite uniform partition, all connected metric spaces, all bounded metric spaces, and all injective metric spaces.Furthermore, Theorems 1 and 4 imply that if $\text{C}$ is any coreflective, cartesian-closed subcategory of Unif in which enough function space structures are finer than the uniformity of uniform convergence (as in the above examples), then either (1) $\text{C}$ is a subclass of the locally fine spaces, or (2) $\text{C}$ contains all injective metric spaces and $\text{C}$ is a subclass of the coreflective hull of all uniform spaces having no infinite uniform partition.     

On representation theory and near-vector spaces

Topics relating to representation theory and near-vector spaces are explored. The general form of finite dimensional near-vector spaces of finite fields is given. We show that a near-vector space can be associated with a particular representation and in the case where F is a finite field, properties of these near-vector spaces are given.     

Compact holomorphic mappings on Banach spaces and the approximation property

Let $\text{H}$(E) be the space of complex valued holomorphic functions on a complex Banach space E. The approximation property for $\text{H}$(E), endowed with various natural locally convex topologies, is studied. For example, $\text{H}$(E) with the compact-open topology has the approximation property if and only if E has the approximation property. In order to characterize when $\text{H}$(E) has the approximation property for topologies other than the compact-open, the notion of a compact holomorphic map between Banach spaces in introduced and studied.     

Combinatorics of the Toric Hilbert Scheme

The toric Hilbert scheme is a parameter space for all ideals with the same multigraded Hilbert function as a given toric ideal. Unlike the classical Hilbert scheme, it is unknown whether toric Hilbert schemes are connected. We construct a graph on all the monomial ideals on the scheme, called the flip graph, and prove that the toric Hilbert scheme is connected if and only if the flip graph is connected. These graphs are used to exhibit curves in P ⁴ whose associated toric Hilbert schemes have arbitrary dimension. We show that the flip graph maps into the Baues graph of all triangulations of the point configuration defining the toric ideal. Inspired by the recent discovery of a disconnected Baues graph, we close with results that suggest the existence of a disconnected flip graph and hence a disconnected toric Hilbert scheme. Received May 15, 2000, and in revised form March 8, 2001. Online publication January 7, 2002.     

Bilinear forms over a finite field,with applications to coding theory

Let Ω be the set of bilinear forms on a pair of finite-dimensional vector spaces over GF(q). If two bilinear forms are associated according to their q-distance (i.e., the rank of their difference), then Ω becomes an association scheme. The characters of the adjacency algebra of Ω, which yield the MacWilliams transform on q-distance enumerators, are expressed in terms of generalized Krawtchouk polynomials. The main emphasis is put on subsets of Ω and their q-distance structure. Certain q-ary codes are attached to a given $\text{X ? Ω}$; the Hamming distance enumerators of these codes depend only on the q-distance enumerator of X. Interesting examples are provided by Singleton systems $\text{X ? Ω}$, which are defined as t-designs of index 1 in a suitable semilattice (for a given integer t). The q-distance enumerator of a Singleton system is explicitly determined from the parameters. Finally, a construction of Singleton systems is given for all values of the parameters.     

Non-commutative crepant resolutions of Hibi rings with small class group

In this paper, we study splitting (or toric) non-commutative crepant resolutions (=?NCCRs) of some toric rings. In particular, we consider Hibi rings, which are toric rings arising from partially ordered sets, and show that Gorenstein Hibi rings with class group ${\mathbb{Z}}^2$ have a splitting NCCR.In the appendix, we also discuss Gorenstein toric rings with class group $\mathbb{Z}$, in which case the existence of splitting NCCRs is already known. We especially observe the mutations of modules giving splitting NCCRs for the three dimensional case, and show the connectedness of the exchange graph.     

On convergence rates in approximation theory for operator semigroups

We create a new, functional calculus, approach to approximation formulas for _C0$C_0$-semigroups on Banach spaces restricted to the domains of fractional powers of their generators. This approach allows us to equip the approximation formulas with rates which appear to be optimal in a natural sense. In the case of analytic semigroups, we improve our general results obtaining better convergence rates which are optimal in that case too. The setting of analytic semigroups includes also the case of convergence on the whole space. As an illustration of our approach, we deduce optimal convergence rates in classical approximation formulas for _C0$C_0$-semigroups restricted to the domains of fractional powers of their generators.     

Polynomials in operator space theory

The aim of this article is to start a metric theory of homogeneous polynomials in the category of operator spaces. For this purpose we take advantage of the basic fact that the space P^m(E)$P^m(E)$ of all m-homogeneous polynomials on a vector space E can be identified with the algebraic dual of the m  -th symmetric tensor product ⊗^m,sE${\otimes }^{m,s}E$. Given an operator space V, we study several different types of completely bounded polynomials on V   which form the operator space duals of ⊗^m,sV${\otimes }^{m,s}V$ endowed with related operator structures. Of special interest are what we call Haagerup, Kronecker, and Schur polynomials – polynomials associated with different types of matrix products.     

Generalised weighted composition operators on Maeda–Ogasawara spaces

Every Archimedean Riesz space can be embedded as an order dense subspace of some $C^{\infty }\left(X\right)$, the Riesz space of all extended continuous functions on a Stonean space $X$, called its Maeda–Ogasawara space. Furthermore, it is a fact that every Riesz homomorphism between spaces of ordinary continuous functions on compact Hausdorff spaces is a weighted composition operator. We prove that a generalised statement holds for Maeda–Ogasawara spaces and refine these results in case the homomorphism preserves order limits.     

A1-homotopy groups,excision, and solvable quotients

We study some properties of ${\mathbb{A}}^1$-homotopy groups: geometric interpretations of connectivity, excision results, and a re-interpretation of quotients by free actions of connected solvable groups in terms of covering spaces in the sense of ${\mathbb{A}}^1$-homotopy theory. These concepts and results are well suited to the study of certain quotients via geometric invariant theory. As a case study in the geometry of solvable group quotients, we investigate ${\mathbb{A}}^1$-homotopy groups of smooth toric varieties. We give simple combinatorial conditions (in terms of fans) guaranteeing vanishing of low degree ${\mathbb{A}}^1$-homotopy groups of smooth (proper) toric varieties. Finally, in certain cases, we can actually compute the “next” non-vanishing ${\mathbb{A}}^1$-homotopy group (beyond ${\pi }_1^{{\mathbb{A}}^1}$) of a smooth toric variety. From this point of view, ${\mathbb{A}}^1$-homotopy theory, even with its exquisite sensitivity to algebro-geometric structure, is almost “as tractable” (in low degrees) as ordinary homotopy for large classes of interesting varieties.     

Banach spaces of distributions having two module structures

After a discussion of a space of test functions and the corresponding space of distributions, a family of Banach spaces (B, ∥ ∥_B) in standard situation is described. These are spaces of distributions having a pointwise module structure and also a module structure with respect to convolution. The main results concern relations between the different spaces associated to B established by means of well-known methods from the theory of Banach modules, among them B₀ and $\text{B}\text{?}$, the closure of the test functions in B and the weak relative completion of B, respectively. The latter is shown to be always a dual Banach space. The main diagram, given in Theorem 4.7, gives full information concerning inclusions between these spaces, showing also a complete symmetry. A great number of corresponding formulas is established. How they can be applied is indicated by selected examples, in particular by certain Segal algebras and the A_p-algebras of Herz. Various further applications are to be given elsewhere.     

A class of extension properties that are not simply generated

Let $\text{P}$ be a closed-hereditary topological property preserved by products. Call a space $\text{P}$-regular if it is homeomorphic to a subspace of a product of spaces with $\text{P}$. Suppose that each $\text{P}$-regular space possesses a $\text{P}$-regular compactification. It is well-known that each $\text{P}$-regular space X is densely embedded in a unique space γ_scPX with $\text{P}$ such that if f: X → Y is continuous and Y has $\text{P}$, then f extends continuously to γ_scPX. Call $\text{P}$-pseudocompact if γ_scPX is compact.Associated with $\text{P}$ is another topological property $\text{P}$^#, possessing all the properties hypothesized for $\text{P}$ above, defined as follows: a $\text{P}$-regular space X has $\text{P}$^# if each $\text{P}$-pseudocompact closed subspace of X is compact. It is known that the $\text{P}$-pseudocompact spaces coincide with the $\text{P}$^#-pseudocompact spaces, and that $\text{P}$^# is the largest closed-hereditary, productive property for which this is the case. In this paper we prove that if $\text{P}$ is not the property of being compact and $\text{P}$-regular, then $\text{P}$^# is not simply generated; in other words, there does not exist a space E such that the spaces with $\text{P}$^# are precisely those spaces homeomorphic to closed subspaces of powers of E.