π-Complementation in the Unitisation and Multiplication Algebras of a Semiprime Algebra

π-complemented algebras are defined as those (not necessarily associative or unital) algebras such that each annihilator ideal is complemented by other annihilator ideal. For a given semiprime algebra A, we discuss the π-complementation of the unitisation algebra A ¹ of A. Moreover, if in addition the multiplication algebra ?(A) of A is also semiprime, we study the π-complementation in the algebras ?(A) and ?^?(A) (the multiplication ideal of A). In associative setting, we prove that A is π-complemented if and only if ?^?(A) is π-complemented, and that A ¹ π-complemented if and only if ?(A) is π-complemented.     

Quotients of Incidence Algebras and the Euler Characteristic

In this note, we show that, if A ? kQ _A /I _A is a schurian strongly simply connected algebra given by its normed presentation, and Σ is the unique poset whose Hasse quiver coincides with Q _A , then A ? kΣ if and only if I _A has a generating set consisting of exactly χ(Q _A ) elements, where χ(Q _A ) is the Euler characteristic of Q _A . We also prove that a quotient of an incidence algebra A = kΣ/J is strongly simply connected if and only if A is simply connected and kΣ is strongly simply connected.     

Monomorphism Operator and Perpendicular Operator

For a quiver Q, a k-algebra A, and an additive full subcategory 𝒳 of A-mod, the monomorphism category Mon(Q, 𝒳) is introduced. The main result says that if T is an A-module such that there is an exact sequence 0 → T _m  → … → T ₀ → D(A _A ) → 0 with each T _i  ∈ add(T), then Mon(Q, ^⊥ T) =^⊥(kQ ? _k T); and if T is cotilting, then kQ ? _k T is a unique cotilting Λ-module, up to multiplicities of indecomposable direct summands, such that Mon(Q, ^⊥ T) =^⊥(kQ ? _k T).

As applications, the category of the Gorenstein-projective (kQ ? _k A)-modules is characterized as Mon(Q, 𝒢𝒫(A)) if A is Gorenstein; the contravariantly finiteness of Mon(Q, 𝒳) can be described; and a sufficient and necessary condition for Mon(Q, A) being of finite type is given.     

On the first Hochschild cohomology group of an algebra

Let A≡KΔ /I be a factor of a path algebra. We develop a strategy to compute dim H ¹(A), the dimension of the first Hochschild cohomology group of A, using combinatorial data from (Δ,I). That allows us to connect dim H ¹(A) with the rank and p-rank of the fundamental group π₁(Δ,I) of (Δ,I). We get explicit formulae for dim H ¹(A), when every path in Δ parallel to an arrow belongs to I or when I is homogeneous. Received: 12 April 1999 / Revised version: 9 October 2000     

A Markov-Nikolskii type inequality for absolutely monotone polynomials of order <Emphasis Type="Italic">K</Emphasis>

A function Q is called absolutely monotone of order k on an interval I if Q(x) ≥ 0, Q′(x) ≥ 0, …, Q(k)(x) ≥ 0, for all x ε I. An essentially sharp (up to a multiplicative absolute constant) Markov inequality for absolutely monotone polynomials of order k in L _p [−1, 1], p > 0, is established. One may guess that the right Markov factor is cn ²/k, and this indeed turns out to be the case. Similarly sharp results hold in the case of higher derivatives and Markov-Nikolskii type inequalities. There is also a remarkable connection between the right Markov inequality for absolutely monotone polynomials of order k in the supremum norm and essentially sharp bounds for the largest and smallest zeros of Jacobi polynomials. This is discussed in the last section of the paper.     

Linear transformations on symmetric matrices that preserve commutativity

It is shown that if a nonsingular linear transformation T on the space of n-square real symmetric matrices preserves the commutativity, where n ?3, then T(A) = λQAQ^t + Q(A)I_n for all symmetric matricesA, for some scalar λ, orthogonal matrix Q, and linear functional Q.     

Generalized Smash Products

In this paper．we study the ring #(D．B)and obtain two very interesting results. First we prove in Theorem 3 that the category of rational left BU-modules is equivalent to both the category of #-rational left modules and the category of all(B．D)-Hopf modules BM^D．Cai and Chen have proved this result in the case B=D=A．Secondly they have proved that if A has a nonzero left integral then A#A^*rat is a dense subring of Endk(A)．We prove that #(A,A) is a dense subring of Endk(Q),where Q is a certain subspace of #(A．A)under the condition that the antipode is bijective(see Theorem18).This condition is weaker than the condition that A has a nonzero integral.It is well known the antipode is bijective in case A has a nonzero integral．Furthermore if A has nonzero left integral,Q can be chosen to be A(see Corollary 19)and #(A,A)is both left and right primitive.Thus A#A^*rat #(A,A)-Endk(A)．Moreover we prove that the left singular ideal of the ring #(A,A)is zero．A corollary of this is a criterion for A with nonzero left integral to be finite-dimensional,namely the ring #(A,A)has a finite uniform dimension．     

Universal entire functions with gap power series

Let be the family of all compact sets in which have connected complement. For K ε M we denote by A(K) the set of all functions which are continuous on K and holomorphic in its interior.Suppose that {z_n} is any unbounded sequence of complex numbers and let Q be a given sub-sequence of ₀.If Q has density Δ(Q) = 1 then there exists a universal entire function with lacunary power series

1. (1) (z) = ε^∞_{v = 0 v}Z^v, _v = 0 for v Q, which has for all K ε M the following properties simultaneously

2. (2) the sequence {(Z + Z_n)} is dense in A(K)

3. (3) the sequence { (ZZ_n)} is dense in A(K) if 0 K.

Also a converse result is proved: If is an entire function of the form (1) which satisfies (3), then Q must have maximal density Δ_max(Q) = 1.     

Topological *-algebras withC*-enveloping algebras II

UniversalC*-algebrasC*(A) exist for certain topological *-algebras called algebras with aC*-enveloping algebra. A Frechet *-algebraA has aC*-enveloping algebra if and only if every operator representation ofA mapsA into bounded operators. This is proved by showing that every unbounded operator representation π, continuous in the uniform topology, of a topological *-algebraA, which is an inverse limit of Banach *-algebras, is a direct sum of bounded operator representations, thereby factoring through the enveloping pro-C*-algebraE(A) ofA. Given aC*-dynamical system (G,A,α), any topological *-algebraB containingC _c (G,A) as a dense *-subalgebra and contained in the crossed productC*-algebraC*(G,A,α) satisfiesE(B) =C*(G,A,α). IfG = ℝ, ifB is an α-invariant dense Frechet *-subalgebra ofA such thatE(B) =A, and if the action α onB ism-tempered, smooth and by continuous *-automorphisms: then the smooth Schwartz crossed productS(ℝ,B,α) satisfiesE(S(ℝ,B,α)) =C*(ℝ,A,α). WhenG is a Lie group, theC ^∞-elementsC ^∞(A), the analytic elementsC ^ω(A) as well as the entire analytic elementsC ^є(A) carry natural topologies making them algebras with aC*-enveloping algebra. Given a non-unitalC*-algebraA, an inductive system of idealsI _α is constructed satisfyingA =C*-ind limI _α; and the locally convex inductive limit ind limI _α is anm-convex algebra with theC*-enveloping algebraA and containing the Pedersen idealK _a ofA. Given generatorsG with weakly Banach admissible relationsR, we construct universal topological *-algebraA(G, R) and show that it has aC*-enveloping algebra if and only if (G, R) isC*-admissible.     

COMPARATIVE PROBABILITY ON THE C* ALGEBRA OF COMPACT OPERATORS

Abstract

Let A be a von Neumann algebra on a Hilbert space H and let P(A) denote the projections of A. A comparative probability (CP) on A (or more correctly on P(A)) is a preorder ? on P(A) satisfying:

0 ? P ? P ε P(A) with Q ≠ 0 for some Q ε P(A).

If P, Q ε P(A) then either P ? Q or Q ? P.

If P, Q and R are all in P(A) and P⊥R, Q⊥R, then P ? Q ? P + R ? Q + R.

Let τ be any of the usual locally convex topologies on A. We say ? is τ continuous if the interval topology induced on P(A) by ? is weaker than the τ topology on P(A). If μ an additive (completely additive) measure on P(A) then μ induces a uniformly (weakly) continuous CP ?μ on P(A) given by P ?_μ Q if μ(P) ? μ(Q). We show that if A is the C* algebra C(H) of compact operators on an infinite dimensional Hilbert space H, the converse is true under an extra boundedness condition on the CP which is automatically satisfied whenever the identity is present in A = P(C(H)).     

Estimates of Freud-Christoffel functions for some weights with the whole real line as support

Upper and lower bounds for generalized Christoffel functions, called Freud-Christoffel functions, are obtained. These have the form λ_n,p(W,j,x) = infPW_Lp(R)/|P^(j)(X)| where the infimum is taken over all polynomials P(x) of degree at most n − 1. The upper and lower bounds for λ_n,p(W,j,x) are obtained for all 0 < p ∞ and J = 0, 1, 2, 3,… for weights W(x) = exp(−Q(x)), where, among other things, Q(x) is bounded in [− A, A], and Q″ is continuous in β(−A, A) for some A > 0. For p = ∞, the lower bounds give a simple proof of local and global Markov-Bernstein inequalities. For p = 2, the results remove some restrictions on Q in Freud's work. The weights considered include W(x) = exp(− ¦x¦^α/2), α > 0, and W(x) = exp(− exp(¦x¦)), > 0.     

On the inequality μ(mM) ≥ μ(M) over polynomial rings

In [4], Macaulay proved that μ(mI) ≥ Q(μ(I)) ≥ μ(I) for any homogeneous ideal I in k[X ₁, …,X_n ] with minimal generators of a single degree where m is the homogeneous maximal ideal (X ₁, …,X_n ) and Q is the function called the Binomial Expansion base n - 1. The inequelity can be generalised to the case of torsion-fre modules with generators of a single degree. The proof follows the lines of Robbiano [5]. A family of counterexamples to μ(mI) ≥ μ(I) in which I has minimal generators of just two different degrees is given.     

Deformation of certain quadratic algebras and the corresponding quantum semigroups

LetV be a finite-dimensional vector space. Given a decompositionV⊗V=⊕ _i=1,…n I _i , definen quadratic algebrasQ(V, J _(m)) whereJ _(m)=⊕ _i≠m I _i . There is also a quantum semigroupM(V; I ₁, …,I _n ) which acts on all these quadratic algebras. The decomposition determines as well a family of associative subalgebras of End (V ^⊗k ), which we denote byA _k =A _k (I ₁,…,I _n ),k≥2. In the classical case, whenV⊗V decomposes into the symmetric and skewsymmetric tensors,A _k coincides with the image of the representation of the group algebra of the symmetric groupS _k in End(V ^⊗k ). LetI _i,h be deformations of the subspacesI _i . In this paper we give a criteria for flatness of the corresponding deformations of the quadratic algebrasQ(V, J _(m),h ) and the quantum semigroupM(V;I _1,h ,…,I _n,h ). It says that the deformations will be flat if the algebrasA _k (I ₁, …,I _n ) are semisimple and under the deformation their dimension does not change. Usually, the decomposition intoI _i is defined by a given semisimple operatorS onV⊗V, for whichI _i are its eigensubspaces, and the deformationsI _i,h are defined by a deformationS _h ofS. We consider the cases whenS _h is a deformation of Hecke or Birman-Wenzl symmetry, and also the case whenS _h is the Yang-Baxter operator which appears by a representation of the Drinfeld-Jimbo quantum group. Applying the flatness criteria we prove that in all these cases we obtain flat deformations of the quadratic algebras and the corresponding quantum semigroups. Partially supported by a grant from the Israel Science Foundation administered by the Israel Academy of Sciences.     

Idealization of a decomposition theorem

In 1986, Tong [13] proved that a function f : (X,τ)→(Y,φ) is continuous if and only if it is α-continuous and A-continuous. We extend this decomposition of continuity in terms of ideals. First, we introduce the notions of regular-I-closed sets, A _I-sets and A _I -continuous functions in ideal topological spaces and investigate their properties. Then, we show that a function f : (X,τ,I)→(Y, φ) is continuous if and only if it is α-I-continuous and A _I-continuous. This revised version was published online in June 2006 with corrections to the Cover Date.     

Completely integrally closed modules and rings

A ring A is a completely integrally closed right A-module if and only if the maximal right ring of quotients Q _max(A) of A is an injective right A-module and A is a right completely integrally closed subring in Q _max(A). A right Noetherian, right integrally closed ring A is a completely integrally closed right A-module.     

Renormalization group, causality, and nonpower perturbation expansion in QFT

The structure of the QFT expansion is studied in the framework of a new “invariant analytic” version of the perturbative QCD. Here, an invariant coupling constant α(Q ² /Λ ² ) = β ₁ α_s(Q ² )/(4π) becomes a Q ² -analytic invariant function α _an (Q²/Λ ² ) ≡A(x), which, by construction, is free of ghost singularities because it incorporates some nonperturbative structures. In the framework of the “analyticized” perturbation theory, an expansion for an observable F, instead of powers of the analytic invariant charge A(x), may contain specific functions A_n(x)=[aⁿ(x)] _an , the “nth power of a(x) analyticized as a whole.” Functions A _n>2(x) for small Q² ≤Λ ² oscillate, which results in weak loop and scheme dependences. Because of the analyticity requirement, the perturbation series for F(x) becomes an asymptotic expansion à la Erdélyi using a nonpower set {A _n (x)}. The probable ambiguities of the invariant analyticization procedure and the possible inconsistency of some of its versions with the renormalization group structure are also discussed. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 1, pp. 55–66, April, 1999.     

Properly discontinuous actions of subgroups in amenable algebraic groups and its application to affine motions

This note will concern properly discontinuous actions of subgroups in real algebraic groups on contractible manifolds. Let (π,X,ρ) be such an action, where ρ:π→ Diff(X) is a homomorphism. We assume that ? extends to a smooth action of a real algebraic group G containing π. If such π has a nontrivial radical (i.e., unique maximal normal solvable subgroup), then we can apply the method of Seifert construction [14],[17] to yield that the quotient π\X supports the structure of an injective Seifert fibering with typical (resp. exceptional) fiber diffeomorphic to a solv (resp. infrasolv)-manifold (when π acts freely). When G is an amenable algebraic group, we can say about a uniqueness property for such actions. Namely, let (π_i, X_i, ρ_i) be actions as above (i= 1,2). Then, given an isomorphism f of π₁ onto ?₂, there is a diffeomorphism h: X₁→X₂ such that h(ρ₁(r)x)=ρ₂(f(r)h(x).As an application, we try to decide the structure of affine motions of some euclidean space $\text{R}$ⁿ. First we verify the conjecture of [17, 4 5], i.e., a compact complete affinely flat manifold admits a maximal toral action if its fundamental group has a nontrivial center. Second, a compact complete affinity flat manifold whose fundamental group is virtually polycyclic supports the structure of an infrasolvmanifold. This structure varies depending on its solvable kernel (if it is abelian or nilpotent, it must be a euclidean space form or an infranilmanifold respectively). If a group of the affine group A(n) acts properly discontinuously and with compact quotient of $\text{R}$ⁿ, then it is called an affine crystallographic group. Finally, we can say so far as to a uniqueness property that two virtually polycyclic affine crystallographic groups are conjugate inside Diff($\text{R}$ⁿ) if they are isomorphic (cf.[8]).     

Canonical form with respect to semiscalar equivalence for a matrix pencil with nonsingular first matrix

Polynomial n × n matrices A(x) and B(x) over a field \mathbbF \mathbb{F} are called semiscalar equivalent if there exist a nonsingular n × n matrix P over \mathbbF \mathbb{F} and an invertible n × n matrix Q(x) over \mathbbF \mathbb{F} [x] such that A(x) = PB(x)Q(x). We give a canonical form with respect to semiscalar equivalence for a matrix pencil A(x) = A _0x - A ₁, where A ₀ and A ₁ are n × n matrices over \mathbbF \mathbb{F} , and A ₀ is nonsingular.     

On groups of central type,non-degenerate and bijective cohomology classes

A finite group G is of central type (in the non-classical sense) if it admits a non-degenerate cohomology class [c] ∈ H ²(G, ℂ*) (G acts trivially on ℂ*). Groups of central type play a fundamental role in the classification of semisimple triangular complex Hopf algebras and can be determined by their representation-theoretical properties. Suppose that a finite group Q acts on an abelian group A so that there exists a bijective 1-cocycle π ∈ Z ¹(Q,Ǎ), where Ǎ = Hom(A, ℂ*) is endowed with the diagonal Q-action. Under this assumption, Etingof and Gelaki gave an explicit formula for a non-degenerate 2-cocycle in Z ²(G, ℂ*), where G:= A × Q. Hence, the semidirect product G is of central type. In this paper, we present a more general correspondence between bijective and non-degenerate cohomology classes. In particular, given a bijective class [π] ∈ H ¹(Q,Ǎ) as above, we construct non-degenerate classes [cπ] ∈ H ²(G,ℂ*) for certain extensions 1 → A → G → Q → 1 which are not necessarily split. We thus strictly extend the above family of central type groups.     

Evolution Semigroups and Product Formulas for Nonautonomous Cauchy Problems

In this paper, we study nonautonomous Cauchy problems (NCP) {^{u̇(t) = A(t)u(t)}_{u(s) = x ∈ X} for a family of linear operators (A(t))_t∈I on some Banach space X by means of evolution semigroups. In particular, we characterize “stability” in the so called “hyperbolic case” on the level of evolution semigroups and derive a product formula for the solutions of (NCP). Moreover, in Section 4 we connect the “hyperbolic” and the “parabolic” case by showing, that integrals ∫^t_s A(τ) dτ always define generators. This yields another product formula.