An asymptotic formula for the ranks of homotopy groups

Let X be a finite simply connected CW complex of dimension n. The loop space homology H_∗(ΩX;Q) is the universal enveloping algebra of a graded Lie algebra LX isomorphic with π∗−1(X)⊗Q. Let QX⊂LX be a minimal generating subspace, and set .Theorem: If dimLX=∞ and , then

     

A completion theorem for pro-G-spectra

We give a very general completion theorem for pro-spectra. We show that, if G is a compact Lie group, M[∗] is a pro-G-spectrum, and F is a family of (closed) subgroups of G, then the mapping pro-spectrum F(EF₊,M[∗]) is the F-adic completion of M[∗], in the sense that the map M[∗]→F(EF₊,M[∗]) is the universal map into an algebraically F-adically complete pro-spectrum. Here, F(EF₊,M[∗]) denotes the pro-G-spectrum , where runs over the finite subcomplexes of EF₊.     

Extension of Fourier algebra homomorphisms to duals of algebras of uniformly continuous functionals

For a locally compact group G, let XG be one of the following introverted subspaces of VN(G): , the C^∗-algebra of uniformly continuous functionals on A(G); , the space of weakly almost periodic functionals on A(G); or , the C^∗-algebra generated by the left regular representation on the measure algebra of G. We discuss the extension of homomorphisms of (reduced) Fourier-Stieltjes algebras on G and H to cb-norm preserving, weak^∗-weak^∗-continuous homomorphisms of into , where (XG,XH) is one of the pairs , , or . When G is amenable, these extensions are characterized in terms of piecewise affine maps.     

Immersions of non-orientable surfaces

Let F be a closed non-orientable surface. We classify all finite order invariants of immersions of F into R³, with values in any Abelian group. We show they are all functions of a universal order 1 invariant which we construct as T⊕P⊕Q, where T is a Z valued invariant reflecting the number of triple points of the immersion, and P,Q are Z/2 valued invariants characterized by the property that for any regularly homotopic immersions , P(i)−P(j)∈Z/2 (respectively, Q(i)−Q(j)∈Z/2) is the number mod 2 of tangency points (respectively, quadruple points) occurring in any generic regular homotopy between i and j.For immersion and diffeomorphism such that i and i○h are regularly homotopic we show:

P(i○h)−P(i)=Q(i○h)−Q(i)=(rank(h_∗−Id)+ε(deth∗∗))mod2     

The Pettis integral for multi-valued functions via single-valued ones

We study the Pettis integral for multi-functions defined on a complete probability space (Ω,Σ,μ) with values into the family cwk(X) of all convex weakly compact non-empty subsets of a separable Banach space X. From the notion of Pettis integrability for such an F studied in the literature one readily infers that if we embed cwk(X) into ?_∞(BX^∗) by means of the mapping defined by j(C)(x^∗)=sup(x^∗(C)), then j○F is integrable with respect to a norming subset of B?_∞^∗(BX^∗). A natural question arises: When is j○F Pettis integrable? In this paper we answer this question by proving that the Pettis integrability of any cwk(X)-valued function F is equivalent to the Pettis integrability of j○F if and only if X has the Schur property that is shown to be equivalent to the fact that cwk(X) is separable when endowed with the Hausdorff distance. We complete the paper with some sufficient conditions (involving stability in Talagrand's sense) that ensure the Pettis integrability of j○F for a given Pettis integrable cwk(X)-valued function F.     

Rieffel deformation of homogeneous spaces

Let G₁⊂G be a closed subgroup of a locally compact group G and let X=G/G₁ be the quotient space of left cosets. Let X=(C₀(X),ΔX) be the corresponding G-C^∗-algebra where G=(C₀(G),Δ). Suppose that Γ is a closed abelian subgroup of G₁ and let Ψ be a 2-cocycle on the dual group . Let GΨ be the Rieffel deformation of G. Using the results of the previous paper of the author we may construct GΨ-C^∗-algebra XΨ - the Rieffel deformation of X. On the other hand we may perform the Rieffel deformation of the subgroup G₁ obtaining the closed quantum subgroup , which in turn, by the results of S. Vaes, leads to the GΨ-C^∗-algebra . In this paper we show that . We also consider the case where Γ⊂G is not a subgroup of G₁, for which we cannot construct the subgroup . Then generically XΨ cannot be identified with a quantum quotient. What may be shown is that it is a GΨ-simple object in the category of GΨ-C^∗-algebras.     

Nonlinear conditions for weighted composition operators between Lipschitz algebras

Let T:Lip₀(X)→Lip₀(Y) be a surjective map between pointed Lipschitz ^∗-algebras, where X and Y are compact metric spaces. On the one hand, we prove that if T satisfies the non-symmetric norm ^∗-multiplicativity condition:

     

Universal factorization property of certain polycyclic groups

Let H be a torsion-free strongly polycyclic (torsion-free virtually polycyclic, resp.) group. Let G be any group with maximal condition. We show that there exists a torsion-free strongly polycyclic (torsion-free virtually polycyclic, resp.) group and an epimorphism such that for any homomorphism ?:G→H, it factors through , i.e., there exists a homomorphism such that . We show that this factorization property cannot be extended to any finitely generated group G. As an application of factorization, we give necessary and sufficient conditions for N(f,g)=R(f,g) to hold for maps f,g:X→Y between closed orientable n-manifolds where π₁(X) has the maximal condition, Y is an infra-solvmanifold, N(f,g) and R(f,g) denote the Nielsen and Reidemeister coincidence numbers, respectively.     

Chen-Ruan cohomology of some moduli spaces of parabolic vector bundles

Let (X,D) be an ?-pointed compact Riemann surface of genus at least two. For each point x∈D, fix parabolic weights such that . Fix a holomorphic line bundle ξ over X of degree one. Let PMξ denote the moduli space of stable parabolic vector bundles, of rank two and determinant ξ, with parabolic structure over D and parabolic weights . The group of order two line bundles over X acts on PMξ by the rule E_∗⊗L?E_∗⊗L. We compute the Chen-Ruan cohomology ring of the corresponding orbifold.     

On Fourier algebra homomorphisms

Let G be a locally compact group and let B(G) be the dual space of C∗(G), the group C∗ algebra of G. The Fourier algebra A(G) is the closed ideal of B(G) generated by elements with compact support. The Fourier algebras have a natural operator space structure as preduals of von Neumann algebras. Given a completely bounded algebra homomorphism we show that it can be described, in terms of a piecewise affine map with Y in the coset ring of H, as follows

     

Dimension free estimates for discrete Riesz transforms on products of abelian groups

Let G be a lca group with a fixed g₀∈G, spanning an infinite subgroup. Let τ_j, acting on L²(Gⁿ), be translation by g_o in the jth coordinate; the discrete derivatives ∂_j=I−τ_j define a discrete Laplacian and discrete Riesz transforms . We get dimension-free estimates

     

The extension of the Krein-Šmulian theorem for Orlicz sequence spaces and convex sets

If X is a Banach space and C⊂X∗∗ a convex subset, for x∗∗∈X∗∗ and A⊂X∗∗ let be the distance from x∗∗ to C and . In this paper we prove that if φ is an Orlicz function, I an infinite set and X=?φ(I) the corresponding Orlicz space, equipped with either the Luxemburg or the Orlicz norm, then for every w^∗-compact subset K⊂X∗∗ we have if and only if φ satisfies the Δ₂-condition at 0. We also prove that for every Banach space X, every nonempty convex subset C⊂X and every w^∗-compact subset K⊂X∗∗ then and, if K∩C is w^∗-dense in K, then .     

A bilinear version of Orlicz-Pettis theorem

Given three Banach spaces X, Y and Z and a bounded bilinear map , a sequence x=n(xn)⊆X is called B-absolutely summable if is finite for any y∈Y. Connections of this space with are presented. A sequence x=n(xn)⊆X is called B-unconditionally summable if is finite for any y∈Y and z^∗∈Z^∗ and for any M⊆N there exists xM∈X for which _∑n∈M〈B(xn,y),z^∗〉=〈B(xM,y),z^∗〉 for all y∈Y and z^∗∈Z^∗. A bilinear version of Orlicz-Pettis theorem is given in this setting and some applications are presented.     

A Morita type equivalence for dual operator algebras

We generalize the main theorem of Rieffel for Morita equivalence of W^∗-algebras to the case of unital dual operator algebras: two unital dual operator algebras A,B have completely isometric normal representations α,β such that α(A)=[M^∗β(B)M]^−w∗ and β(B)=[Mα(A)M^∗]^−w∗ for a ternary ring of operators M (i.e. a linear space M such that MM^∗M⊂M) if and only if there exists an equivalence functor which “extends” to a ∗-functor implementing an equivalence between the categories and . By we denote the category of normal representations of A and by the category with the same objects as and Δ(A)-module maps as morphisms (Δ(A)=A∩A^∗). We prove that this functor is equivalent to a functor “generated” by a B,A bimodule, and that it is normal and completely isometric.     

Localization and duality in topology and modular representation theory

We develop a duality theory for localizations in the context of ring spectra in algebraic topology. We apply this to prove a theorem in the modular representation theory of finite groups.Let G be a finite group and k be an algebraically closed field of characteristic p. If p is a homogeneous nonmaximal prime ideal in H^∗(G,k), then there is an idempotent module κ_p which picks out the layer of the stable module category corresponding to p, and which was used by Benson, Carlson and Rickard [D.J. Benson, J.F. Carlson, J. Rickard, Thick subcategories of the stable module category, Fund. Math. 153 (1997) 59-80] in their development of varieties for infinitely generated kG-modules. Our main theorem states that the Tate cohomology is a shift of the injective hull of H^∗(G,k)/p as a graded H^∗(G,k)-module. Since κ_p can be constructed using a version of the stable Koszul complex, this can be viewed as a statement of localized Gorenstein duality in modular representation theory. Various consequences of this theorem are given, including the statement that the stable endomorphism ring of the module κ_p is the p-completion of cohomology , and the statement that κ_p is a pure injective kG-module.In the course of proving the theorem, we further develop the framework introduced by Dwyer, Greenlees and Iyengar [W.G. Dwyer, J.P.C. Greenlees, S. Iyengar, Duality in algebra and topology, Adv. Math. 200 (2006) 357-402] for translating between the unbounded derived categories and . We also construct a functor to the full stable module category, which extends the usual functor and which preserves Tate cohomology. The main theorem is formulated and proved in , and then translated to and finally to .The main theorem in can be viewed as stating that a version of Gorenstein duality holds after localizing at a prime ideal in H^∗(BG;k). This version of the theorem holds more generally for a compact Lie group satisfying a mild orientation condition. This duality lies behind the local cohomology spectral sequence of Greenlees and Lyubeznik for localizations of H^∗(BG;k).In a companion paper [D.J. Benson, Idempotent kG-modules with injective cohomology, J. Pure Appl. Algebra 212 (7) (2008) 1744-1746], a more recent and shorter proof of the main theorem is given. The more recent proof seems less natural, and does not say anything about localization of the Gorenstein condition for compact Lie groups.