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.     

Bohr compactifications and a result of følner

In this paper, we study the Bohr compactification of an arbitrary topological groupT with regard to obtaining relations between relatively dense (or discretely syndetic) subsets ofT, and neighborhoods of the identity in the Bohr compactification. The methods utilized are those algebraic techniques which have been recently applied to topological dynamics (see [2]). For an abelian group, we show that cls (A ^?1 AAa ^?1), forA relatively dense anda∈A, is usually a neighborhood of the identity, thus generalizing a result of Følner [4]. Moreover, an analogous result is proved in the non-abelian case under additional assumptions. Finally, we utilize these results to obtain a generalization of a result of Cotlar-Ricabarra [1] concerning maximal almost periodicity in abelian topological groups.     

Conjecture of Li and Praeger concerning the isomorphisms of Cayley graphs of A5

LetG be a finite group, andS a subset ofG \ |1| withS =S ^?1. We useX = Cay(G,S) to denote the Cayley graph ofG with respect toS. We callS a Cl-subset ofG, if for any isomorphism Cay(G,S) ≈ Cay(G,T) there is an α∈ Aut(G) such thatS ^α =T. Assume that m is a positive integer.G is called anm-Cl-group if every subsetS ofG withS =S ^?1 and | S | ≤m is Cl. In this paper we prove that the alternating groupA ₅ is a 4-Cl-group, which was a conjecture posed by Li and Praeger.     

On normal control processes

Control processes of the form \(\dot x - A(t) x = B(t) u(t)\) , which are normal with respect to the unit ballB _{p′, r′} of the control spaceL ^p′([τ, T]),l _m ^r ′ are characterized in terms ofH(t)=X(T)X ^?1(t),B(t),X(t) any fundamental matrix solution of \(\dot x - A(t)x = 0\) , and directly in terms ofA, B, when bothA andB are independent oft.     

Conditional expectations and operator decompositions

For aC ^*-algebraA with a conditional expectation Φ:A → A onto a subalgebraB we have the linear decompositionA=B⊕H whereH=ker(Φ). Since Φ preserves adjoints, it is also clear that a similar decomposition holds for the selfadjoint parts:A ^s =B ^s ⊕H ^s (we useV ^s ={aεV;a ^*=a} for any subspaceV of A). Apply now the exponential function to each of the three termsA ^s ,B ^s , andH ^s . The results are: the setG ⁺ of positive invertible elements ofA, the setB ⁺ of positive invertible elements ofB, and the setC={e^h;h ^*=h, Φ(h)=0}, respectively. We consider here the question of lifting the decompositionA ^s =B ^s ⊕H ^s to the exponential sets. Concretely, is every element ofG ⁺ the product of elements ofB ⁺ andC, respectively, just as any selfadjoint element ofA is the sum of selfadjoint elements ofB andH? The answer is yes in the following sense: Eacha ε G ⁺ is the positive part of a productbe of elementsb ε B ⁺ and c εC, and bothb andc are uniquely determined and depend analytically ona. This can be rephrased as follows: The map (6, c) →(bc) ⁺ is an analytic diffeomorphism fromB ⁺ x C ontoG ⁺, where for any invertiblex ε A we denote with x⁺ the positive square root ofxx ^*. This result can be expressed equivalently as: The map (b, c) →bcb is a diffeomorphism between the same spaces. Notice that combining the polar decomposition with these results we can write every invertibleg ε A asg=bcu, whereb ε B ⁺,c ε C, andu is unitary. This decomposition is unique and the factorsb, c, u depend analytically ofg. In the case of matrix algebras with Φ=trace/dimension, the factorization corresponds tog=| det(g)|cu withc > 0,det(c)=1, andu unitary. This paper extends some results proved by G. Corach and the authors in [2]. Also, Theorem 2 states that the reductive homogeneous space resulting from a conditional expectation satisfies the regularity hypothesis introduced by L. Mata-Lorenzo and L. Recht in [5], Definition 11.1. The situation considered here is the ”general context” for regularity indicated in the introduction of the last mentioned paper.     

Multiobjective programming and penalty functions

This paper generalizes the penalty function method of Zang-will for scalar problems to vector problems. The vector penalty function takes the form $$g(x,\lambda ) = f(x) + \lambda ^{ - 1} P(x)e,$$ wheree ?R ^m, with each component equal to unity;f:R ⁿ →R ^m, represents them objective functions {f ⁱ} defined onX \( \subseteq \) R ⁿ; λ ∈R ¹, λ>0;P:R ⁿ →R ¹ X \( \subseteq \) Z \( \subseteq \) R ⁿ,P(x)≦0, ∨x ∈R ⁿ,P(x) = 0 ?x ∈X. The paper studies properties of {E (Z, λ _r )} for a sequence of positive {λ _r } converging to 0 in relationship toE(X), whereE(Z, λ _r ) is the efficient set ofZ with respect tog(·, λ_r) andE(X) is the efficient set ofX with respect tof. It is seen that some of Zangwill's results do not hold for the vector problem. In addition, some new results are given.     

Lower semicontinuous perturbations of maximal monotone differential inclusions

We prove existence of solutions to 1 $$\dot x \in - Ax + F\left( {t,x} \right),x\left( a \right) = x^0 ,$$ whereA is a maximal monotone operator inR ⁿ andF is a multifunction measurable in (t, x) and l.s.c. inx, satisfying a sublinear growth condition.     

A limit theorem for matrix-solutions of Hamiltonian systems

The following limit theorem on Hamiltonian systems (resp. corresponding Riccati matrix equations) is shown: Given(N, N)-matrices,A, B, C andn ∈ {1,…, N} with the following properties:A and kemelB(x) are constant, rank(I, A, …, A ^n?1) B(x)≠N,B(x)∈C _n(R), andB(x)(A ^T)^j-1 C(x)∈C _n-j(R) forj=1, …, n. Then \(\mathop {\lim }\limits_{x \to x_0 } \eta _1^T \left( x \right)V\left( x \right)U^{ - 1} \left( x \right)\eta _2 \left( x \right) = d_1^T \left( {x_0 } \right)U\left( {x_0 } \right)d_2 \) forx ₀∈R, whenever the matricesU(x), V(x) are a conjoined basis of the differential systemU′=AU + BV, V′=CU?A ^TV, and whenever η_i(x)∈R ^N satisfy η_i(x ₀)=U(x ₀)d _i ∈ imageU(x ₀) η′_i-Aη_ni(x) ∈ imageB(x),B(x)(η′_i(x)-Aη_i(x)) ∈C _n-1 R fori=1,2.     

The Dirichlet problem with a volume constraint

For B the open unit disk in R², let W₁(B) denote the Sobolev space of vector functions x: B→R³ such that x and its first partial derivatives are square integrable. For any y∈W₁(B), S(y) is the set of all x in W₁(B) for which x-y∈W₁₀(B), the closure in W₁(B) of C ₀ ^∞ (B). Assume that for all x ∈ S(y) the area functional A(x)>0. For a given constant K, we show that there is an x_o∈S(y) minimizing the “Dirichlet Integral” $$D(x) = \iint_B {(|x_u |^2 } + |x_v |^2 )dudv$$ in the subset of all x ∈ S(y) for which the oriented volume enclosed by y and x, V(y,x)=K. x_o is analytic on B and is a solution to the differential equation Δx=2H(x_u∧x_v) for some constant H.     

Sur le theoreme de Stone-Weierstrass en Algebre commutative

We show that the sufficient conditions given by Cahen, Grazzini and Haouat for a version of the Stone-Weierstrass theorem in commutative algebra are the widest. More precisely, letA be a Noetherian ring andI a proper ideal ofA such thatA is Hausdorff with respect to theI-adic topology. Note the completion ofA andC(Â,Â) the ring of continuous functions from to with uniform convergence topology. The subset of polynomial functions is dense inC(Â,Â) if and only if the radical ofI is a maximal idealm ofA and the local ringA _m is a one-dimensional analytically irreducible domain with finite residue field.     

Domains with growth conditions for the quasihyperbolic metric

For a given growth functionH, we say that a domainD ?R ⁿ is anH-domain if δ_D x≤δ_D(x ₀)H(k _D(x,x ₀)),x ∈D, where δ_D(x)=d(x?D) andk _D denotes the quasihyperbolic distance. We show that ifH satisfiesH(0)=1, |H'|≤H, andH"≤H, then there exists an extremalH-domain. Using this fact, we investigate some fundamental properties ofH-domains.     

Stability theorems for projections of convex sets

LetH ^n?1 denote the set of all (n ? 1)-dimensional linear subspaces of euclideann-dimensional spaceE ⁿ (n≧3), and letJ andK be two compact convex subsets ofE ⁿ. It is well-known thatJ andK are translation equivalent (or homothetic) if for allH ∈H ^n?1 the respective orthogonal projections, sayJ _H, K_H, are translation equivalent (or homothetic). This fact gives rise to the following stability problem: Ifε≧0 is given, and if for everyH ∈H ^n?1 a translate (or homothetic copy) ofK _H is within Hausdorff distanceε ofJ _H, how close isJ to a nearest translate (or homothetic copy) ofK? In the case of translations it is shown that under the above assumptions there is always a translate ofK within Hausdorff distance (1 + 2√2)ε ofJ. Similar results for homothetic transformations are proved and applications regarding the stability of characterizations of centrally symmetric sets and spheres are given. Furthermore, it is shown that these results hold even ifH ^n?1 is replaced by a rather small (but explicitly specified) subset ofH ^n?1.     

Counterexamples related to high-frequency oscillation of Poisson's kernel

Let $$L_\varepsilon = \frac{\partial }{{\partial x^i }}a^{ij} \left( {\frac{x}{\varepsilon }} \right)\frac{\partial }{{\partial x^j }},L_0 = q^{ij} \frac{{\partial ^2 }}{{\partial x_i \partial x_j }},$$ wherea is a smooth periodic matrix andL ₀ is the homogenized operator corresponding to the family (L _ε). LetD be a nice domain, and letP _ε (x, y), P ₀ (x, y) be the Poisson kernels associated withL _ε andL ₀. We show that in generalP _ε (x, ·) does not converge strongly toP ₀ (x, ·) inL ^p , by exhibiting two counterexamples. This result has the following implication in the theory of boundary control of distributed systems: if , withz given,u _ε (x) = ∫ P _ε (x, y)g(y) andu ₀ (x) = ∫P ₀ (x,y)g(y), then, in general, .     

On an estimate of Calderón-Zygmund operators by dyadic positive operators

Given a general dyadic grid D and a sparse family of cubes S = {Q _j ^k ∈ D, define a dyadic positive operator A _D,S by $${A_{D,S}}f(x) = \sum\limits_{j,k} {{f_{Q_j^k}}{\chi _{Q_j^k}}} (x)$$ . Given a Banach function space X(? ⁿ ) and the maximal Calderón-Zygmund operator ${T_\natural }$ , we show that $${\left\| {{T_\natural}f} \right\|_X} \leqslant c(T,n)\mathop {\sup }\limits_{D,S} {\left\| {{A_{D,S}}|f|} \right\|_X}$$ This result is applied to weighted inequalities. In particular, it implies (i) the “twoweight conjecture” by D. Cruz-Uribe and C. Pérez in full generality; (ii) a simplification of the proof of the “A ₂ conjecture”; (iii) an extension of certain mixed A _p ?A _r estimates to general Calderón-Zygmund operators; (iv) an extension of sharp A ₁ estimates (known for T ) to the maximal Calderón-Zygmund operator $\natural $ .     

Analytic factorizations and completely bounded maps

We prove an analytic factorization theorem in the setting of the recently developed theory of operator spaces. We especially obtain the following result: LetA be aC ^*-algebra andH be a Hilbert space. Let π be an element ofH ^∞ (CB(A, B(H))), i.e. a bounded analytic function valued in the space of completely bounded maps fromA intoB(H). Then there exist a Hilbert spaceK, a representation π:A→B(K), ?₁ ∈₁ H ^∞ (B(H,K)) and ∈₂ H ^∞ (B(K,H)) such that ‖ε₁‖∞‖∈₂‖∞ ≤ ‖∈‖∞ and: $\forall z \in D, \forall a \in A, \varphi (z)(a) = \varphi _2 (z)\pi (a)\varphi _1 (z).$ We also prove an analogous result for completely bounded multilinear maps. The last part of the paper is devoted to a new proof of Pisier's theorem about gamma-norms.     

On Marcus-Wang conjecture

LetA _m ,B _m ,m=1, ...,p, be linear operators on ann-dimensional unitary space \(V.L = \sum\limits_{m = 1}^p {A_m \otimes B_m } \) is a linear operator on ?² V, the tensor product space with the customarily induced inner product. The numerical range ofL is defined as $$W\tfrac{1}{2}(L) = \left\{ {(L)x \otimes y,x \otimes y):x,y o.n.} \right\}$$ where “o.n.” means “orthonormal”. In [1], M.Marcus and B.Y. Wang conjecture: There exists no non-zero operatorL of minimum length less thann for whichW ₂ ¹ (L)=0. In this paper, we prove that this conjecture is true.     

Isometries on complex matric Heisenberg groups

Complex matric Heisenberg groupsH=H _C (p,q,r) endowed with suitable left-invariant Hermitian metrics have a large connected groupK ₀ of isometries fixing the identity ofH. In fact, forpr=1,K ₀ is the conformal symplectic groupCSp(q), while forpr>1 it is the factor groupU(p)×U(q)×U(r)/T, whereU(n) is the unitary group ofC ⁿ andT a normal subgroup of the product isomorphic toU(1).     

Convergence rates in the law of large numbers for random variables on partially ordered sets

Let (A, ≤) be a partially ordered set, {X _α} a collection of i. i. d. random variables, indexed byA. Let \(S_\alpha = \sum _{\beta \leqslant \alpha } X_\beta \) , |α|=card {β∈A, β∈α}. We study the convergence rates ofS _α/|α|. We derive for a large class of partially ordered sets theorems, like the following one: For suitabler, t with 1/2< <r/t≤1:E|X| ^t M (|X| ^t/r )<∞ andEX=μ if and only if $$S_\alpha = \sum _{\beta \leqslant \alpha } X_\beta $$ for all ε>0, where \(M(x) = \sum _{j< x} d(j)\) withd(j)=card {α∈A, |α|=j}.     

Dirichlet problems for the quasilinear second order subelliptic equations

In this paper, we study the Dirichlet problems for the following quasilinear second order sub-elliptic equation, $$\left\{ {\begin{array}{*{20}c} {\sum\limits_{i,j = 1}^m {X_i^* (A_{i,j} (x,u)X_j u) + \sum\limits_{j = 1}^m {B_j (x,u)X_j u + C(x,u) = 0in\Omega ,} } } \\ {u = \varphi on\partial \Omega ,} \\ \end{array} } \right.$$ whereX={X ₁, ...,X _m } is a system of real smooth vector fields which satisfies the Hörmander's condition,A _i,j ,B _j ,C ∈C ^∞( $\bar \Omega$ ×R) and (A _i,j (x,z)) is a positive definite matrix. We have proved the existence and the maximal regularity of solutions in the “non-isotropic” Hölder space associated with the system of vector fieldsX.