Isomorphic Copy Research Articles

Bidual renormings of Banach spaces

Given a separable Banach space X with no isomorphic copies of ℓ 1 and a separable subspace Y of its bidual, we provide a sufficient condition on Y to ensure that X admits an equivalent norm such that the restriction to Y of the corresponding bidual norm is midpoint locally uniformly rotund. This result applies to the separable subspaces of the bidual of a Banach space with a shrinking unconditional Schauder basis and to the bidual of the James space.

Σ-Definability of countable structures over real numbers, complex numbers, and quaternions

We study Σ-definability of countable models over hereditarily finite {ie193-01} superstructures over the field ℝ of reals, the field ℂ of complex numbers, and over the skew field ℍ of quaternions. In particular, it is shown that each at most countable structure of a finite signature, which is Σ-definable over {ie193-02} with at most countable equivalence classes and without parameters, has a computable isomorphic copy. Moreover, if we lift the requirement on the cardinalities of the classes in a definition then such a model can have an arbitrary hyperarithmetical complexity, but it will be hyperarithmetical in any case. Also it is proved that any countable structure Σ-definable over {ie193-03}, possibly with parameters, has a computable isomorphic copy and that being Σ-definable over {ie193-04} is equivalent to being Σ-definable over {ie193-05}.

$A$-identities for the Grassmann algebra: The conjecture of Henke and Regev

Let K be an algebraically closed field of characteristic 0, and let E be the infinite dimensional Grassmann (or exterior) algebra over K. Denote by P n the vector space of the multilinear polynomials of degree n in x 1 , ..., x n in the free associative algebra K(X). The symmetric group S n acts on the left-hand side on P n , thus turning it into an S n -module. This fact, although simple, plays an important role in the theory of PI algebras since one may study the identities satisfied by a given algebra by applying methods from the representation theory of the symmetric group. The S n -modules P n and KS n are canonically isomorphic. Letting An be the alternating group in S n , one may study KA n and its isomorphic copy in P n with the corresponding action of An. Henke and Regev described the A n -codimensions of the Grassmann algebra E, and conjectured a finite generating set of the An-identities for E. Here we answer their conjecture in the affirmative.

On the Adjoint of a Strongly Continuous Semigroup

Using some techniques from vector integration, we prove the weak measurability of the adjoint of strongly continuous semigroups which factor through Banach spaces without isomorphic copy ofl1; we also prove the strong continuity away from zero of the adjoint if the semigroup factors through Grothendieck spaces. These results are used, in particular, to characterize the space of strong continuity of{T**(t)}t≥0, which, in addition, is also characterized for abstractL- andM-spaces. As a corollary, it is proven that abstractL-spaces with no copy ofl1are finite-dimensional.

More on decompositions of edge-colored complete graphs

Let G be a family of graphs whose edges are colored with elements from a set R of r colors. We assume no two vertices of G are joined by more than one edge of color i for any i ∈ R , for each G ∈ G . K n ( r ) will denote the complete graph with r edges joining any pair of distinct vertices, one of each of the r colors. We describe necessary and asymptotically sufficient conditions on n for the existence of a family D of subgraphs of K n ( r ) , each of which is an isomorphic copy of some graph in G , so that each edge of K n ( r ) appears in exactly one of the subgraphs in D .

On the number of generators needed for free profinite products of finite groups

We provide lower estimates on the minimal number of generators of the profinite completion of free products of finite groups.In particular, we show that if C 1, ..., C n are finite cyclic groups then there exists a finite group G which is generated by isomorphic copies of C 1, ..., C n and the minimal number of generators of G is n.

Interpolation Banach lattices containing no isomorphic copies of [formula omitted

This article is concerned with the question of when the Banach lattices generated by the interpolation method of constants contain no isomorphic copies of c 0 . For this we study the Köthe dual spaces of Banach lattices obtained by the methods of constants and means. We find an explicit formula for the Köthe dual of Banach lattices generated by the means method with a mild restriction on the parameters. We apply these results to describe a large class of Banach lattices, generated by the method of constants, containing no subspaces isomorphic to c 0 . Moreover, we present applications to the strict singularity of certain inclusion maps between Banach sequence lattices.

Errata and addendum: tensor products and Dunford–Pettis sets

AbstractGhenciu and Lewis introduced the notion of a strong Dunford–Pettis set and used this notion to study the presence or absence of isomorphic copies of c0 in Banach spaces. The authors asserted that they could obtain a fundamental result of J. Elton without resorting to Ramsey theory. While the stated theorems are correct, unfortunately there is a flaw in the proof of the first theorem in the paper which also affects subsequent corollaries and theorems. The difficulty is discussed, and Elton's results are employed to establish a Schauder basis proposition which leads to a quick proof of the theorem in question. Additional results where questions arise are discussed on an individual basis.

Amalgams, blocks, weights, fusion systems and finite simple groups

We associate an iterated amalgam of finite groups to a certain class of fusion systems on finite p-groups ( p a prime), in such a way that the p-group of the fusion system is a maximal finite p-subgroup of the resulting group, unique up to conjugacy, and, furthermore, the conjugation action of the resulting (usually infinite) group on p-subgroups induces the original fusion system on the p-group. 1 In view of earlier work of Puig and of Broto, Castellana, Grodal, Levi and Oliver [C. Broto, N. Castellana, J. Grodal, R. Levi, R. Oliver, Subgroup families controlling p-local finite groups, Proc. London Math. Soc. (3) 91 (2005) 325–354], the fusion systems we deal with include all saturated fusion systems. 1 Since this paper was written, we have been informed that I. Leary and R. Stancu are currently writing a paper with a different construction to realise a fusion system on a finite p-group via a group. Their work and ours are independent of each other. The resulting amalgam has free normal subgroups of finite index, and we examine the images of the group by its maximal free normal subgroups of finite index; these images all contain (isomorphic copies of) the original p-group (and are generated by the (images of the) finite groups used in the amalgamation). If there is no non-trivial normal p-subgroup of the fusion system (equivalently, if the iterated amalgam constructed has no non-trivial normal p-subgroup), then the generalised Fitting subgroup of each of these homomorphic images is a direct product of non-Abelian simple groups, each of order divisible by p (and if there is no proper non-trivial strongly closed p-subgroup in the fusion system, then the generalised Fitting subgroup of any of these finite groups is characteristically simple). We note that the (finite-dimensional) representation theory of this amalgam is (almost by construction) p-locally determined. In the case of the fusion system associated to a p-block of a finite group, we suggest strong links between block-theoretic invariants of the above finite epimorphic images of the associated amalgam and of the original block. We believe that the results of this paper offer the prospect of at least the beginnings of a structural explanation for some of the current numerical conjectures in block theory (e.g. [J.L. Alperin, Weights for finite groups, in: Proc. Sympos. Pure Math., vol. 47, Amer. Math. Soc., Providence, 1987, pp. 369–379; E.C. Dade, Counting characters in blocks, I, Invent. Math. 109 (1) (1992) 187–210]).

Turing degrees of nonabelian groups

For a countable structure $\mathcal {A}$, the (Turing) degree spectrum of $\mathcal {A}$ is the set of all Turing degrees of its isomorphic copies. If the degree spectrum of $\mathcal {A}$ has the least degree $\mathbf {d}$, then we say that $\mathbf {d}$ is the (Turing) degree of the isomorphism type of $\mathcal {A}$. So far, degrees of the isomorphism types have been studied for abelian and metabelian groups. Here, we focus on highly nonabelian groups. We show that there are various centerless groups whose isomorphism types have arbitrary Turing degrees. We also show that there are various centerless groups whose isomorphism types do not have Turing degrees.

Order-weakly compact operators from vector-valued function spaces to Banach spaces

Let 𝐸 be an ideal of 𝐿⁰ over a 𝜎 -finite measure space (Ω,Σ,𝜇) , and let 𝐸^{∼} stand for the order dual of 𝐸 . For a real Banach space (𝑋,|⋅|_{_{𝑋}}) let 𝐸(𝑋) be a subspace of the space 𝐿⁰(𝑋) of 𝜇 -equivalence classes of strongly Σ -measurable functions 𝑓:Ω⟶𝑋 and consisting of all those 𝑓∈𝐿⁰(𝑋) for which the scalar function |𝑓(⋅)|_{_{𝑋}} belongs to 𝐸 . For a real Banach space (𝑌,|⋅|_{_{𝑌}}) a linear operator 𝑇:𝐸(𝑋)⟶𝑌 is said to be order-weakly compact whenever for each 𝑢∈𝐸⁺ the set T ( { f ∈ E ( X ) : ‖ f ( ⋅ ) ‖ X ≤ u } ) \,T(\{f\in E(X):\; \|f(\cdot )\|_{_X}\le u\})\, is relatively weakly compact in 𝑌 . In this paper we examine order-weakly compact operators 𝑇:𝐸(𝑋)⟶𝑌 . We give a characterization of an order-weakly compact operator 𝑇 in terms of the continuity of the conjugate operator of 𝑇 with respect to some weak topologies. It is shown that if (𝐸,|⋅|_{_{𝐸}}) is an order continuous Banach function space, 𝑋 is a Banach space containing no isomorphic copy of 𝑙¹ and 𝑌 is a weakly sequentially complete Banach space, then every continuous linear operator 𝑇:𝐸(𝑋)⟶𝑌 is order-weakly compact. Moreover, it is proved that if (𝐸,|⋅|_{_{𝐸}}) is a Banach function space, then for every Banach space 𝑌 any continuous linear operator 𝑇:𝐸(𝑋)⟶𝑌 is order-weakly compact iff the norm |⋅|_{_{𝐸}} is order continuous and 𝑋 is reflexive. In particular, for every Banach space 𝑌 any continuous linear operator 𝑇:𝐿¹(𝑋)⟶𝑌 is order-weakly compact iff 𝑋 is reflexive.

Isomorphisms into section spaces of Banach bundles

Let X be a compact Hausdorff space, E a (real) Banach space, and let C(X, E) be the space of continuous functions from X to E. The monograph of Cembranos and Mendoza provides a primer on when C (X, E) might contain isomorphic copies of the classical spaces c 0, l 1, and l ∞. In this paper, we investigate what isomorphisms of classical spaces might occur if the continuous vector-valued functions on X take their values in spaces which may vary over X.

Strong planning under uncertainty in domains with numerous but identical elements (a generic approach)

The typical AI problem is that of making a plan of the actions to be performed by a robot so that the robot could get into a set of final situations, if it started with a certain initial situation. The planning problem is known to be generally very complex. Even within the case of ‘well-balanced’ actions, strong planning under uncertainty about the effects of actions, or games such as ‘Robot against Nature’, is EXPTIME-complete. As a result, AI planners are very sensitive to the number of the variables involved in making a plan, the inherent symmetry of the problem, and the nature of the logical formalisms being used. This paper shows that linear logic provides a convenient and adequate tool for representing strong and weak planning problems in non-deterministic domains. A particular focus of this paper is on planning problems with an unbounded number of functionally identical objects. We show that for such problems linear logic is especially effective and leads to a dramatic contraction of the search space from exponential to polynomial in size. We employ the ability of linear logic to reason about multisets, which in this instance are created by identifying several distinct objects as being functionally equivalent for the problem at hand (think of a number of balls, each of which must be moved to some new location — the balls are distinct, but are functionally equivalent for the problem). In linear logic terms, we establish a clear syntactic condition that allows us to show that solving a generic planning problem where there is only one generic object, directly implies a solution to the original real planning problem over several real objects, the isomorphic copies of the generic object. Moreover, this correspondence also guarantees to produce a real solution that works in polynomial time.

Turing degrees of isomorphism types of algebraic objects

The Turing degree spectrum of a countable structure 𝒜 is the set of all Turing degrees of isomorphic copies of 𝒜. The Turing degree of the isomorphism type of 𝒜, is the least Turing degree in its degree spectrum. We show that there are structures with isomorphism types of arbitrary Turing degrees in each of the following classes: countable fields, rings, and torsion-free Abelian groups of any finite rank. We also show that there are structures in each of these classes the isomorphism types of which do not have Turing degrees. The case of torsion-free Abelian groups of finite rank settles a question left open by Knight, Downey and Jockusch [Downey, Complexity, logic, and recursion theory, Lecture Notes in Pure and Applied Mathematics 187 (ed. A. Sorbi; Marcel Dekker, New York, 1997) 157–205].

Computability of Homogeneous Models

In the last five years there have been a number of results about the computable content of the prime, saturated, or homogeneous models of a complete decidable (CD) theory T in the spirit of Vaught's "Denumerable models of complete theories" combined with computability methods for (Turing) degrees d ≤ 0′. First we recast older results by Goncharov, Peretyat'kin, and Millar in a more modern framework which we then apply. Then we survey recent results by Lange, "The degree spectra of homogeneous models," which generalize the older results and which include positive results on when a certain homogeneous model $\cal A$ of T has an isomorphic copy of a given Turing degree. We then survey Lange's "A characterization of the 0-basis homogeneous bounding degrees" for negative results about when $\cal A$ does not have such copies, generalizing negative results by Goncharov, Peretyat'kin, and Millar. Finally, we explain recent results by Csima, Harizanov, Hirschfeldt, and Soare, "Bounding homogeneous models," about degrees d that are homogeneous bounding and explain their relation to the PA degrees (the degrees of complete extensions of Peano Arithmetic).

On Extremal k-Graphs Without Repeated Copies of 2-Intersecting Edges

The problem of determining extremal hypergraphs containing at most r isomorphic copies of some element of a given hypergraph family was first studied by Boros et al. in 2001. There are not many hypergraph families for which exact results are known concerning the size of the corresponding extremal hypergraphs, except for those equivalent to the classical Turán numbers. In this paper, we determine the size of extremal k-uniform hypergraphs containing at most one pair of 2-intersecting edges for $k\in\{3,4\}$. We give a complete solution when $k=3$ and an almost complete solution (with eleven exceptions) when $k=4$.

Copies of c0(Г) and ℓ∞(Г)/ c0(Г) in quotients of Banach spaces with applications to Orlicz and Marcinkiewicz spaces

Let X be a Banach space, let Y be its subspace, and let Г be an infinite set. We study the consequences of the assumption that an operator T embeds ℓ 221E;(Г) into X isomorphically with T( c 0(Г)) ⊂ Y. Under additional assumptions on T we prove the existence of isomorphic copies of c 0(Г ℵ0) in X/Y, and complemented copies ℓ ∞(Г) in X/Y. In concrete cases we obtain a new information about the structure of X/Y. In particular, L∞[O,1]/ C[O,1] contains a complemented copy of ℓ ∞/ c 0, and some natural (lattice) quotients of real Orlicz and Marcinkiewicz spaces contain lattice-isometric and positively I-complemented copies of(real) ℓ ∞/ c 0.

On Subgroups of the Nottingham Group of Positive Hausdorff Dimension

By a Theorem of Camina, the Nottingham group 𝒩(𝔽p) contains an isomorphic copy of every finitely generated pro-p group. In this article we obtain several restrictions on the structure of subgroups of 𝒩(𝔽p) which have positive Hausdorff dimension and show that such groups, if finitely generated, cannot be linear over a local field. This implies that just-infinite subgroups of 𝒩(𝔽p) studied in Fesenko (1999) and Barnea and Klopsch (2003) are not linear over any profinite ring.

On a problem of Namioka on norm-attaining functionals

Isaac Namioka conjectured that every nonreflexive Banach space can be renormed is such a way that, in the new norm, the set of norm attaining functionals has an empty interior in the norm topology. We prove the rightness of this conjecture for spaces containing an isomorphic copy of l1. As a consequence, we prove also that the same result holds for a wide class of Banach spaces containing, for example, the weakly compactly generated ones.

Complete arcs in [formula omitted]: The spectrum of the sizes and the classification of the smallest complete arcs

In this paper it has been verified, by an exhaustive computer search, that in PG ( 2 , 25 ) the smallest size of a complete arc is 12 and that complete 19-arcs and 20-arcs do not exist. Therefore, the spectrum of the sizes of the complete arcs in PG ( 2 , 25 ) is completely determined. The classification of the smallest complete arcs is also given: the number of non-equivalent complete 12 -arcs is 606 and for each of them the automorphism group has been found and some geometrical properties have been studied. The exhaustive search has been feasible because projective equivalence properties have been exploited to prune the search tree and to avoid generating too many isomorphic copies of the same arc.