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1.
Let G be a non-discrete locally compact abelian group, and let M(G) be the convolution algebra of regular bounded Borel measures on G. Let Γ denote the dual group of G. Then the interior of the ?ilov boundary of M(G) is exactly Γ. The proof uses generalized Riesz products for the compact metrizable case and standard liftings from that case.  相似文献   

2.
There exists a compact groupG havingM 4/3(G)≠M 4(G). This answers in the negative (the dual reformulation of) a question of Eymard (Séminaire Bourbaki, 1969/70).  相似文献   

3.
For a nondiscrete σ-compact locally compact Hausdorff group G, L(G) is a commutative Banach algebra under pointwise multiplication which has many nonzero proper closed invariant ideals; there is at least a continuum of maximal invariant ideals {Nα} such that Nα1 + Nα2 = L(G) whenever α1α2. It follows from the construction of these ideals that when G is also amenable as a discrete group, then LIM?TLIM contains at least a continuum of mutually singular elements each of which is singular to any element of TLIM. The supports of left-invariant means are in the maximal ideal space of L(G); the structure of these supports leads to the notion of stationary and transitive maximal ideals. To prove that both these types of maximal ideals are dense among all maximal ideals, one shows that the intersection of all nonzero closed invariant ideals is zero. This is the case even though the intersection of any sequence of closed invariant ideals is not zero and the intersection of all the maximal invariant ideals is not zero.  相似文献   

4.
Let G be a locally compact group and let p∈(1,∞). Let be any of the Banach spaces Cδ,p(G), PFp(G), Mp(G), APp(G), WAPp(G), UCp(G), PMp(G), of convolution operators on Lp(G). It is shown that PFp(G)′ can be isometrically embedded into UCp(G)′. The structure of maximal regular ideals of (and of MAp(G)″, Bp(G)″, Wp(G)″) is studied. Among other things it is shown that every maximal regular left (right, two sided) ideal in is either dense or is the annihilator of a unique element in the spectrum of Ap(G). Minimal ideals of is also studied. It is shown that a left ideal M in is minimal if and only if , where Ψ is either a right annihilator of or is a topologically x-invariant element (for some xG). Some results on minimal right ideals are also given.  相似文献   

5.
Let H(Δ) denote the Banach algebra of bounded analytic functions on the open unit disc, let M denote its maximal ideal space, and let ? denote its Shilov boundary. D. J. Newman has shown that a homomorphism ? in M will be in ? if and only if ? is unimodular on all Blaschke products. We answer a question of K. Hoffman by showing that ? will be in ? if and only if ? is unimodular on every Blaschke product whose zero set is an interpolating sequence. Our method is based on a construction due to L. Carleson, originally developed for the proof of the Corona theorem.  相似文献   

6.
Motivated by the recent interest in the examination of unital completely positive maps and their effects in C*-theory, we revisit an older result concerning the existence of the ?ilov ideal. The direct proof of Hamana’s Theorem for the existence of an injective envelope for a unital operator subspace X of some ${\mathcal{B}(H)}$ that we provide implies that the ?ilov ideal is the intersection of C*(X) with any maximal boundary operator subsystem in ${\mathcal{B}(H)}$ . As an immediate consequence we deduce that the ?ilov ideal is the biggest boundary operator subsystem for X in C*(X). The new proof of the existence of the ?ilov ideal that we give does not use the existence of maximal dilations, provided by Drits- chel and McCullough, and so it is independent of the one given by Arveson. As a consequence, the ?ilov ideal can be seen as the set that contains the abnormalities in a C*-cover ${(C, \iota)}$ of X for all the extensions of the identity map ${{\rm id}_{\iota(X)}}$ . The interpretation of our results in terms of ucp maps characterizes the maximal boundary subsystems of X in ${\mathcal{B}(H)}$ as kernels of X-projections that induce completely minimal X-seminorms; equivalently, X-minimal projections with range being an injective envelope, that we view from now on as the ?ilov boundary for X.  相似文献   

7.
Let G denote a locally compact abelian group and H a separable Hilbert space. Let L p (G, H), 1 ≤ p < ∞, be the space of H-valued measurable functions which are in the usual L p space. Motivated by the work of Helgason [1], Figa-Talamanca [11] and Bachelis [2, 3], we have defined the derived space of the Banach space L p (G, H) and have studied its properties. Similar to the scalar case, we prove that if G is a noncompact, locally compact abelian group, then L p 0 (G, H) = {0} holds for 1 ≤ p < 2. Let G be a compact abelian group and Γ be its dual group. Let S p (G, H) be the L 1(G) Banach module of functions in L p (G, H) having unconditionally convergent Fourier series in L p -norm. We show that S p (G, H) coincides with the derived space L p 0 (G, H), as in the scalar valued case. We also show that if G is compact and abelian, then L p 0 (G, H) = L 2(G, H) holds for 1 ≤ p ≤ 2. Thus, if FL p (G, H), 1 ≤ p < 2 and F has an unconditionally convergent Fourier series in L p -norm, then FL 2(G, H). Let Ω be the set of all functions on Γ taking only the values 1, ?1 and Ω* be the set of all complex-valued functions on Γ having absolute value 1. As an application of the derived space L p 0 (G, H), we prove the following main result of this paper. Let G be a compact abelian group and F be an H-valued function on the dual group Γ such that $$ \sum \omega (\gamma )F(\gamma )\gamma $$ is a Fourier-Stieltjes series of some measure µ ∈ M(G, H) for every scalar function ω such that |ω(γ)| = 1. Then Fl 2(Γ, H).  相似文献   

8.
For a group G, let M(G) denote the near-ring of functions on G. We characterize all maximal subnear-rings of M(G) and show that for many classes of groups, E(G), the near-ring generated by the semigroup, End(G) of G, is never maximal as a subnear-ring of M 0 (G). Received: 25 April 2008  相似文献   

9.
Let M and L be (nonlinear) operators in a reflexive Banach space B for which Rg(M + L) = B and ¦(Mx ? My) + α(Lx ? Ly)¦ ? | mx ? My | for all α > 0 and pairs x, y in D(M) ∩ D(L). Then there is a unique solution of the Cauchy problem (Mu(t))′ + Lu(t) = 0, Mu(0) = v0. When M and L are realizations of elliptic partial differential operators in space variables, this gives existence and uniqueness of generalized solutions of boundary value problems for nonlinear partial differential equations of mixed parabolic-Sobolev type.  相似文献   

10.
This paper solves for torsion free and torsion abelian groups G the problem of presenting nth powers Δn(G) of the augmentation ideal Δ(G) of an integral group ring , in terms of the standard additive generators of Δn(G). A concrete basis for Δn(G) is obtained when G itself has a basis and is torsion. The results are applied to describe the homology of the sequence Δn(N)G?Δn(G)?Δn(G/N).  相似文献   

11.
Let Γ be the fundamental group of a finite connected graph G. Let M be an abelian group. A distribution on the boundary ∂Δ of the universal covering tree Δ is an M-valued measure defined on clopen sets. If M has no χ(G)-torsion, then the group of Γ-invariant distributions on ∂Δ is isomorphic to H1(G,M).  相似文献   

12.
The first Zagreb index M1(G) and the second Zagreb index M2(G) of a (molecular) graph G are defined as M1(G)=∑uV(G)(d(u))2 and M2(G)=∑uvE(G)d(u)d(v), where d(u) denotes the degree of a vertex u in G. The AutoGraphiX system [M. Aouchiche, J.M. Bonnefoy, A. Fidahoussen, G. Caporossi, P. Hansen, L. Hiesse, J. Lacheré, A. Monhait, Variable neighborhood search for extremal graphs. 14. The AutoGraphiX 2 system, in: L. Liberti, N. Maculan (Eds.), Global Optimization: From Theory to Implementation, Springer, 2005; G. Caporossi, P. Hansen, Variable neighborhood search for extremal graphs: 1 The AutoGraphiX system, Discrete Math. 212 (2000) 29-44; G. Caporossi, P. Hansen, Variable neighborhood search for extremal graphs. 5. Three ways to automate finding conjectures, Discrete Math. 276 (2004) 81-94] conjectured that M1/nM2/m (where n=|V(G)| and m=|E(G)|) for simple connected graphs. Hansen and Vuki?evi? [P. Hansen, D. Vuki?evi?, Comparing the Zagreb indices, Croat. Chem. Acta 80 (2007) 165-168] proved that it is true for chemical graphs and it does not hold for all graphs. Vuki?evi? and Graovac [D. Vuki?evi?, A. Graovac, Comparing Zagreb M1 and M2 indices for acyclic molecules, MATCH Commun. Math. Comput. Chem. 57 (2007) 587-590] proved that it is also true for trees. In this paper, we show that M1/nM2/m holds for graphs with Δ(G)−δ(G)≤2 and characterize the extremal graphs, the proof of which implies the result in [P. Hansen, D. Vuki?evi?, Comparing the Zagreb indices, Croat. Chem. Acta 80 (2007) 165-168]. We also obtain the result that M1/nM2/m holds for graphs with Δ(G)−δ(G)≤3 and δ(G)≠2.  相似文献   

13.
After the change of variables Δi = γi ? δi and xi,i + 1 = δi ? δi + 1 we show that the invariant polynomials μG(n)q(, Δi, ; , xi,i+1,) characterizing U(n) tensor operators 〈p, q,…, q, 0,…, 0〉 become an integral linear combination of Schur functions Sλ(γ ? δ) in the symbol γ ? δ, where γ ? δ denotes the difference of the two sets of variables {γ1 ,…, γn} and {δ1 ,…, δn}. We obtain a similar result for the yet more general bisymmetric polynomials mμG(n)q(γ1 ,…, γn; δ1 ,…, δm). Making use of properties of skew Schur functions Sλρ and Sλ(γ ? δ) we put together an umbral calculus for mμG(n)q(γ; δ). That is, working entirely with polynomials, we uniquely determine mμG(n)q(γ; δ) from mμG(n)q ? 1(γ; δ) and combinatorial rules involving Ferrers diagrams (i.e., partitions), provided that n ≥ (μ + 1)q. (This restriction does not interfere with writing the general case of mμG(n)q(γ; δ) as a linear combination of Sλ(γ ? δ).) As an application we deduce “conjugation” symmetry for nμG(n)q(γ; δ) from “transposition” symmetry by showing that these two symmetries are equivalent.  相似文献   

14.
This paper continues the study of spectral synthesis and the topologies τ and τr on the ideal space of a Banach algebra, concentrating on the class of Banach *-algebras, and in particular on L1-group algebras. It is shown that if a group G is a finite extension of an abelian group then τr is Hausdorff on the ideal space of L1(G) if and only if L1(G) has spectral synthesis, which in turn is equivalent to G being compact. The result is applied to nilpotent groups, [FD]-groups, and Moore groups. An example is given of a non-compact, non-abelian group G for which L1(G) has spectral synthesis. It is also shown that if G is a non-discrete group then τr is not Hausdorff on the ideal lattice of the Fourier algebra A(G).  相似文献   

15.
A complete characterization of those compact Hausdorff spaces is given such that for every n, each normal element in the algebra C(X)?Mn of continuous functions from X to Mn can be continuously diagonalized. The conditions are that X be a sub-Stonean space with dim X ? 2 and carries no nontrivial G-bundles over any closed subset, for G a symmetric group or the circle group. In particular, diagonalization is assured on every totally disconnected sub-Stonean space, but also on connected spaces of the form β(Y)/Y, where Y is a simply-connected (noncompact) graph.  相似文献   

16.
Let G=(V,E) be a graph with V={1,2,…,n}. Define S(G) as the set of all n×n real-valued symmetric matrices A=[aij] with aij≠0,ij if and only if ijE. By M(G) we denote the largest possible nullity of any matrix AS(G). The path cover number of a graph G, denoted P(G), is the minimum number of vertex disjoint paths occurring as induced subgraphs of G which cover all the vertices of G.There has been some success with relating the path cover number of a graph to its maximum nullity. Johnson and Duarte [5], have shown that for a tree T,M(T)=P(T). Barioli et al. [2], show that for a unicyclic graph G,M(G)=P(G) or M(G)=P(G)-1. Notice that both families of graphs are outerplanar. We show that for any outerplanar graph G,M(G)?P(G). Further we show that for any partial 2-path G,M(G)=P(G).  相似文献   

17.
This paper introduces the use of conjugate transforms in the study of τT semigroups of probability distribution functions. If Δ+ denotes the space of one-dimensional distribution functions concentrated on [0, ∞) and T is a t-norm, i.e., a suitable binary operation on [0, 1], then the operation τT is defined for F, G in Δ+by τT(F, G)(x) = supu+v = xT(F(u), G(v)) for all x. The pair (Δ+, τT) is then a semigroup. For any Archimedean t-norm T, a conjugate transform CT is defined on (Δ+, τT). These transforms are shown to play a role similar to that played by the Laplace transform on the convolution semigroup. Thus a theory of “characteristic functions” for τT semigroups is developed. In addition to establishing their basic algebraic properties, we also use conjugate transforms to study the algebraic questions of the cancellation law, infinitely divisible elements, and solutions of equations in τT semigroups.  相似文献   

18.
A bicyclic graph is a connected graph in which the number of edges equals the number of vertices plus one. Let Δ(G) and ρ(G) denote the maximum degree and the spectral radius of a graph G, respectively. Let B(n) be the set of bicyclic graphs on n vertices, and B(n,Δ)={GB(n)∣Δ(G)=Δ}. When Δ≥(n+3)/2 we characterize the graph which alone maximizes the spectral radius among all the graphs in B(n,Δ). It is also proved that for two graphs G1 and G2 in B(n), if Δ(G1)>Δ(G2) and Δ(G1)≥⌈7n/9⌉+9, then ρ(G1)>ρ(G2).  相似文献   

19.
For locally compact groups G and H, let BM(G, H) denote the Banach space of bounded bilinear forms on C0(G) × C0(H). Using a consequence of the fundamental inequality of A. Grothendieck. a multiplication and an adjoint operation are introduced on BM(G, H) which generalize the convolution structure of M(G × H) and which make BM(G, H) into a KG2-Banach 1-algebra, where KG is Grothendieck's universal constant. Various topics relating to the ideal structure of BM(G, H) and the lifting of unitary representations of G × H to 1-representations of BM(G, H) are investigated.  相似文献   

20.
Let denote the maximum average degree (over all subgraphs) of G and let χi(G) denote the injective chromatic number of G. We prove that if , then χi(G)≤Δ(G)+1; and if , then χi(G)=Δ(G). Suppose that G is a planar graph with girth g(G) and Δ(G)≥4. We prove that if g(G)≥9, then χi(G)≤Δ(G)+1; similarly, if g(G)≥13, then χi(G)=Δ(G).  相似文献   

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