Malliavin's theorem for weak synthesis on nonabelian groups

Malliavin's celebrated theorem on the failure of spectral synthesis for the Fourier algebra A(G) on nondiscrete abelian groups was strengthened to give failure of weak synthesis by Parthasarathy and Varma. We extend this to nonabelian groups by proving that weak synthesis holds for A(G) if and only if G is discrete. We give the injection theorem and the inverse projection theorem for weak X-spectral synthesis, as well as a condition for the union of two weak X-spectral sets to be weak X-spectral for an A(G)-submodule X of VN(G). Relations between weak X-synthesis in A(G) and A(G×G) and the Varopoulos algebra V(G) are explored. The concept of operator synthesis was introduced by Arveson. We extend several recent investigations on operator synthesis by defining and studying, for a V^∞(G)-submodule M of B(L²(G)), sets of weak M-operator synthesis. Relations between X-Ditkin sets and M-operator Ditkin sets and between weak X-spectral synthesis and weak M-operator synthesis are explored.     

The combinatorial derivation and its inverse mapping

Let G be a group and P _G be the Boolean algebra of all subsets of G. A mapping Δ: P _G → P _G defined by Δ(A) = {g ∈ G: gA ∩ A is infinite} is called the combinatorial derivation. The mapping Δ can be considered as an analogue of the topological derivation d: P _X → P _X , A ? A ^d , where X is a topological space and A ^d is the set of all limit points of A. We study the behaviour of subsets of G under action of Δ and its inverse mapping ?. For example, we show that if G is infinite and I is an ideal in P _G such that Δ(A) ∈ I and ?(A) ? I for each A ∈ I then I = P _G .     

Orbits and invariants of visible group actions

We consider actions G?×?X?→?X of the affine, algebraic group G on the irreducible, affine, variety X. If [k[X] ^G ]?=?[k[X]] ^G we call the action visible. Here [A] denotes the quotient field of the integral domain A. If the action is not visible we construct a G-invariant, birational morphism φ: Z?→?X such that G?×?Z?→?Z is a visible action. We use this to obtain visible open subsets U of X. We also discuss visibility in the presence of other desirable properties: What if G?×?X?→?X is stable? What if there is a semi-invariant f ∈ k[X] such that G?×?X _f ?→?X _f is visible? What if X is locally factorial? What if G is reductive?     

The recognition of the class of indecomposable digraphs under low hemimorphy

Given a digraph G=(V,A), the subdigraph of G induced by a subset X of V is denoted by G[X]. With each digraph G=(V,A) is associated its dual G^?=(V,A^?) defined as follows: for any x,y∈V, (x,y)∈A^? if (y,x)∈A. Two digraphs G and H are hemimorphic if G is isomorphic to H or to H^?. Given k>0, the digraphs G=(V,A) and H=(V,B) are k-hemimorphic if for every X⊆V, with |X|≤k, G[X] and H[X] are hemimorphic. A class C of digraphs is k-recognizable if every digraph k-hemimorphic to a digraph of C belongs to C. In another vein, given a digraph G=(V,A), a subset X of V is an interval of G provided that for a,b∈X and x∈V−X, (a,x)∈A if and only if (b,x)∈A, and similarly for (x,a) and (x,b). For example, 0?, {x}, where x∈V, and V are intervals called trivial. A digraph is indecomposable if all its intervals are trivial. We characterize the indecomposable digraphs which are 3-hemimorphic to a non-indecomposable digraph. It follows that the class of indecomposable digraphs is 4-recognizable.     

Proper actions, fixed-point algebras and naturality in nonabelian duality

Suppose a locally compact group G acts freely and properly on a locally compact Hausdorff space X, and let γ be the induced action on C₀(X). We consider a category in which the objects are C^∗-dynamical systems (A,G,α) for which there is an equivariant homomorphism of (C₀(X),γ) into the multiplier algebra M(A). Rieffel has shown that such systems are proper and saturated, and hence have a generalized fixed-point algebra Aα which is Morita equivalent to A_×α,rG. We show that the assignment (A,α)?Aα is functorial, and that Rieffel's Morita equivalence is natural in a suitable sense. We then use our results to prove a categorical version of Landstad duality which characterizes crossed products by coactions, and to prove that Mansfield imprimitivity for crossed products by homogeneous spaces is natural.     

Classifying Cubic Semisymmetric Graphs of Order 18 p n

A graph X is said to be G-semisymmetric if it is regular and there exists a subgroup G of A := Aut (X) acting transitively on its edge set but not on its vertex set. In the case of G = A, we call X a semisymmetric graph. Let p be a prime. It was shown by Folkman (J Comb Theory 3:215–232, 1967) that a regular edge-transitive graph of order 2p or 2p ² is necessarily vertex-transitive. The smallest semisymmetric graph is the Folkman graph. In this study, we classify all connected cubic semisymmetric graphs of order 18p ⁿ , where p is a prime and \({n \geq 1}\) .     

On universality of countable powers of absolute retracts

We construct an absolute retract X of arbitrarily high Borel complexity such that the countable power X ^ω is not universal for the Borelian class A ₁ of sigma-compact spaces, and the product X ^ω x ∑, where ∑ is the radial interior of the Hilbert cube, is not universal for the Borelian class A ₂ of absolute G _δσ-spaces.     

Classifying Hilbert bundles

A Hilbert bundle (p, B, X) is a type of fibre space p:B → X such that each fibre p^?1(x) is a Hilbert space. However, p^?1(x) may vary in dimension as x varies in X. We generalize the classical homotopy classification theory of vector bundles to a “homotopy” classification of certain Hilbert bundles. An (m, n)-bundle over the pair (X, A) is a Hilbert bundle (p, B, X) such that the dimension of p^?1(x) is m for x in A and n otherwise. The main result here is that if A is a compact set lying in the “edge” of the metric space X (e.g. if X is a topological manifold and A is a compact subset of the boundary of X), then the problem of classifying (m, n)-bundles over (X, A) reduces to a problem in the classical theory of vector bundles. In particular, we show there is a one-to-one correspondence between the members of the orbit set, [A, G_m($\text{C}$ⁿ)]/[X, U(n)] ¦ A, and the isomorphism classes of (m, n)-bundles over (X, A) which are trivial over X, A.     

Torsion units in integral group rings of metacyclic groups

It is proved that if G is a split extension of a cyclic p-group by a cyclic p′-group with faithful action then any torsion unit of augmentation one of $\text{Z}$G is rationally conjugate to a group element. It is also proved that if G is a split extension of an abelian group A by an abelian group X with (|A|, |X|) = 1 then any torsion unit of $\text{Z}$G of augmentation one and order relatively prime to |A| is rationally conjugate to an element of X.     

Torsion units in integral group rings of some metabelian groups,II

We prove that any torsion unit of the integral group ring ZG is rationally conjugate to a trivial unit if G = A ⋊ X with both A and X abelian, |Xz.sfnc; < p for every prime p dividing |A| provided either |X| is prime or A ic cyclic.     

Relative uniform completeness and order-convex representations of archimedean ?-groups and f-rings

Results of Henriksen and Johnson, for archimedean f-rings with identity, and of Aron and Hager, for archimedean ?-groups with unit, relating uniform completeness to order-convexity of a representation in a D(X) (the lattice of almost real continuous functions on the space X) are extended to situations without identity or unit. For an archimedean ?-group, G, we show: if G admits any representation G?D(X) in which G is order-convex, then G is divisible and relatively uniformly complete. A converse to this would seem to require some sort of canonical representation of G, which seems not to exist in the ?-group case. But for a reduced archimedean f-ring, A, there is the Johnson representation A?D(XA), and we show: A is divisible, relatively uniformly complete and square-dominated if and only if A is order-convex in D(XA) and square-root-closed. Also, we expand on the situation with unit, where we have the Yosida representation, G?D(YG): if G is divisible, relatively uniformly complete, and the unit is a near unit, then G is order-convex in D(YG).     

Reduced Weyl asymptotics for pseudodifferential operators on bounded domains I. The finite group case

Let G⊂O(n) be a compact group of isometries acting on n-dimensional Euclidean space Rn, and X a bounded domain in Rn which is transformed into itself under the action of G. Consider a symmetric, classical pseudodifferential operator A₀ in L²(Rn) with G-invariant Weyl symbol, and assume that it is semi-bounded from below. We show that the spectrum of the Friedrichs extension A of the operator is discrete, and derive asymptotics for the number Nχ(λ) of eigenvalues of A less or equal λ and with eigenfunctions in the χ-isotypic component of L²(X) as λ→∞, giving also an estimate for the remainder term in case that G is a finite group. In particular, we show that the multiplicity of each unitary irreducible representation in L²(X) is asymptotically proportional to its dimension.     

Exponential sets and their geometric motions

Let us call an “exponential set” in a C*-algebraA any set consisting of the exponentialse ^X of all the self-adjoint elementsX of a subspaceH ofA. For example, ifH = A the resulting exponential setG ⁺ consists of all the positive invertible elements ofA, and all other exponential sets are contained in G⁺. An exponential setC ? G⁺ inherits the geometric structure of the space G⁺ when the defining subspaceH has suitable properties. Here we investigate reasonable conditions onH that permit, for example, reduction of the canonical connection of G⁺ toC. As a consequence, in these cases the setC has a rich family of motions that are “rigid” for the geometry of G⁺. In particular we find thatC itself operates on C by the actionL _g a = (g^?1)*ag^? of the groupG of all invertible elements ofA in G⁺, and that the subgroup generated byC is transitive. Similarly, in several cases the productscu withc ε C andu unitary form a closed Lie subgroup ofG that acts onC, withC contained in it. This is the case forH, the space of elements of trace zero, when there is a trace. The conditions onH are all additions to the following basic situation:H is the kernel of a (bounded linear) projection Φ:A → A. For example, ifH is closed under triple brackets [X, [Y, Z]] then parallel transport in G⁺ along geodesics inC through 1 ∈C preserves vectors tangent toC. Similarly, if the symmetric part of [e ^X Ye ^?X ,Z] is inH for allX, Y, Z ∈H ^s thenC is “geodesically convex” in the sense that geodesics tangent toC stay inC. The most interesting cases correspond to a conditional expectation. Two additional conditions produce the groups described in the first paragraph: the case of a Z₂-graded C*-algebra with Φ the projection on the elements of degree 0 (which is automatically a conditional expectation) and the case of a conditional expectation such that the anti-symmetric part ofe ^X Ye ^?Y is in the range of Φ wheneverX, Y are self-adjoint and Φ(X)= Φ(Y) = 0. This is verified for example in the case of central traces.     

X-quasinormal subgroups

Considering two subgroups A and B of a group G and ? ≠ X ? G, we say that A is X-permutable with B if AB ^x = B ^x A for some element x ∈ X. We use this concept to give new characterizations of the classes of solvable, supersolvable, and nilpotent finite groups.     

On the Laczkovich-Komjáth property of sigma-ideals

Komjáth in 1984 proved that, for each sequence (An) of analytic subsets of a Polish space X, if lim supn∈HAn is uncountable for every H∈ω[N] then _?n∈GAn is uncountable for some G∈ω[N]. This fact, by our definition, means that the σ-ideal [X]^?ω has property (LK). We prove that every σ-ideal generated by X/E has property (LK), for an equivalence relation E⊂X² of type Fσ with uncountably many equivalence classes. We also show the parametric version of this result. Finally, the invariance of property (LK) with respect to various operations is studied.     

Note: A conjecture on partitions of groups

We conjecture that every infinite group G can be partitioned into countably many cells \(G = \bigcup\limits_{n \in \omega } {A_n }\) such that cov(A _n A _n ^?1 ) = |G| for each n ∈ ω Here cov(A) = min{|X|: X} ? G, G = X A}. We confirm this conjecture for each group of regular cardinality and for some groups (in particular, Abelian) of an arbitrary cardinality.     

Reduced Weyl asymptotics for pseudodifferential operators on bounded domains II. The compact group case

Let G⊂O(n) be a compact group of isometries acting on n-dimensional Euclidean space Rn, and X a bounded domain in Rn which is transformed into itself under the action of G. Consider a symmetric, classical pseudodifferential operator A₀ in L²(Rn) that commutes with the regular representation of G, and assume that it is elliptic on X. We show that the spectrum of the Friedrichs extension A of the operator is discrete, and using the method of the stationary phase, we derive asymptotics for the number Nχ(λ) of eigenvalues of A equal or less than λ and with eigenfunctions in the χ-isotypic component of L²(X) as λ→∞, giving also an estimate for the remainder term for singular group actions. Since the considered critical set is a singular variety, we recur to partial desingularization in order to apply the stationary phase theorem.     

Hyperspaces of compact sets in metric linear spaces

We write 2^x for the hyperspace of all non-empty compact sets in a complete metric linear space X topologized by the Hausdorff metric. Using the notation $\text{F}$(X) = {A ϵ 2^X: A is finite}, l^f₂ = {x} = (x_i) ϵ l₂: x_i = 0 for almost all i}, and l^σ₂ = {x = (x _i) ϵ l₂:σ^∞_i=1 (ix_i)² < ∞}, we have the following theorem:A family $\text{G}$ ⊂$\text{F}$(X) is homeomorphic to l^f₂ if $\text{G}$ is σ-fd-compact, the closure $\text{G}$ of $\text{G}$ in 2^x is not locally compact and if whenever A, B ∈ $\text{G}$, λ ∈ [0, 1] and C ⊂ λA + (1 - λ)B with card C⩽ max{card A, card B} then C ϵ $\text{G}$.Moreover, for any G_δ-AR-set $\text{G}$$\text{G}$ of $\text{G}$ with $\text{G}$$\text{G}$ ⊃ $\text{G}$ we have ($\text{G}$$\text{G}$, $\text{G}$)≅(l₂, l^ƒ₂).Similar conditions for hyperspaces to be homeomorphic to l^σ₂ are also established.     

First Alexandroff Decomposition Theorem for Topological Lattice Group Valued Measures

Let (X, F) be an Alexandroff space, let A(F) be the Boolean subalgebra of 2 ^X generated by F, let G be a Hausdorff commutative topological lattice group and let rba_F(A(F), G) denote the set of all order bounded F-inner regular finitely additive set functions from A(F) into G. Using some special properties of the elements of rba_F(A(F), G), we extend to this setting the first decomposition theorem of Alexandroff.     

On IP-graphs of association schemes and applications to group theory

Let G be a group acting transitively on a set X such that all subdegrees are finite. Isaacs and Praeger (1993) [5] studied the common divisor graph of (G,X). For a group G and its subgroup A, based on the results in Isaacs and Praeger (1993) [5], Kaplan (1997) [6] proved that if A is stable in G and the common divisor graph of (A,G) has two components, then G has a nice structure. Motivated by the notion of the common divisor graph of (G,X), Camina (2008) [3] introduced the concept of the IP-graph of a naturally valenced association scheme. The common divisor graph of (G,X) is the IP-graph of the association scheme arising from the action of G on X. Xu (2009) [8] studied the properties of the IP-graph of an arbitrary naturally valenced association scheme, and generalized the main results in Isaacs and Praeger (1993) [5] and Camina (2008) [3]. In this paper we first prove that if the IP-graph of a naturally valenced association scheme (X,S) is stable and has two components (not including the trivial component whose only vertex is 1), then S has a closed subset T such that the thin residue O?(T) and the quotient scheme (X/O?(T),S//O?(T)) have very nice properties. Then for an association scheme (X,S) and a closed subset T of S such that S//T is an association scheme on X/T, we study the relations between the closed subsets of S and those of S//T. Applying these results to schurian schemes and common divisor graphs of groups, we obtain the results of Kaplan [6] as direct consequences.