Multiplication operators on sobolev disk algebra

In this paper,we study the algebra consisting of analytic functions in the Sobolev space W~(2,2) (D) (D is the unit disk),called the Sobolev disk algebra,explore the properties of the multiplication operators M_f on it and give the characterization of the corn- mutant algebra A′(M_f) of M_f.We show that A′(M_f) is commutative if and only if M_f~* is a Cowen-Douglas operator of index 1.     

Characterizations of *-Cancellation Ideals of an Integral Domain

Let D be an integral domain and * a star-operation on D. For a nonzero ideal I of D, let I ^{* _f} = ?{J* | (0) ≠ J ? I is finitely generated} and I ^{* _w} = ? _{P∈* f -Max(D)} ID _P . A nonzero ideal I of D is called a *-cancellation ideal if (IA)* = (IB)* for nonzero ideals A and B of D implies A* =B*. Let X be an indeterminate over D and N _* = {f ∈ D[X] | (c(f))* =D}. We show that I is a * _w -cancellation ideal if and only if I is * _f -locally principal, if and only if ID[X] _{N *} is a cancellation ideal. As a corollary, we have that each nonzero ideal of D is a * _w -cancellation ideal if and only if D _P is a principal ideal domain for all P ∈ * _f -Max(D), if and only if D[X] _{N *} is an almost Dedekind domain. We also show that if I is a * _w -cancellation ideal of D, then I ^{* _w}  = I ^{* _f}  = I _t , and I is * _w -invertible if and only if I ^{* _w}  = J _v for a nonzero finitely generated ideal J of D.     

A Criterion for the Triviality of G(D) and Its Applications to the Multiplicative Structure of D

Let D be an F-central division algebra of index n. Here we present a criterion for the triviality of the group G(D) = D*/Nrd _D/F (D*)D′ and thus generalizing various related results published recently. To be more precise, it is shown that G(D) = 1 if and only if SK ₁(D) = 1 and F ^*2 = F ^*2n . Using this, we investigate the role of some particular subgroups of D* in the algebraic structure of D. In this direction, it is proved that a division algebra D of prime index is a symbol algebra if and only if D* contains a non-abelian nilpotent subgroup. More applications of this criterion including the computation of G(D) and the structure of maximal subgroups of D* are also investigated     

Approximate Amenability of Certain Semigroup Algebras

In the present paper, it is shown that a left cancellative semigroup S (not necessarily with identity) is left amenable whenever the Banach algebra ℓ¹(S) is approximately amenable. It is also proved that if S is a Brandt semigroup over a group G with an index set I, then ℓ¹(S) is approximately amenable if and only if G is amenable. Moreover ℓ¹(S) is amenable if and only if G is amenable and I is finite. For a left cancellative foundation semigroup S with an identity such that for every M_a(S)-measurable subset B of S and s ∈ S the set sB is M_a(S)-measurable, it is proved that if the measure algebra M_a(S) is approximately amenable, then S is left amenable. Concrete examples are given to show that the converse is negative.     

Dihedral manifold approximate fibrations over the circle

Consider the cyclic group C ₂ of order two acting by complex-conjugation on the unit circle S ¹. The main result is that a finitely dominated manifold W of dimension > 4 admits a cocompact, free, discontinuous action by the infinite dihedral group D _∞ if and only if W is the infinite cyclic cover of a free C ₂-manifold M such that M admits a C ₂-equivariant manifold approximate fibration to S ¹. The novelty in this setting is the existence of codimension-one, invariant submanifolds of M and W. Along the way, we develop an equivariant sucking principle for orthogonal actions of finite groups on Euclidean space.     

Finite Blaschke product and the multiplication operators on Sobolev disk algebra

Let R(D) be the algebra generated in Sobolev space W22(D) by the rational functions with poles outside the unit disk D. In this paper the multiplication operators Mg on R(D) is studied and it is proved that Mg ～ Mzn if and only if g is an n-Blaschke product. Furthermore, if g is an n-Blaschke product, then Mg has uncountably many Banach reducing subspaces if and only if n > 1.     

On abelian principal bundles

Let T be a complex torus and E _T a holomorphic principal T-bundle over a connected complex manifold M. We prove that the total space of E _T admits a K?hler structure if and only if M admits a K?hler structure and E _T admits a flat holomorphic connection whose monodromy preserves a K?hler form on T. If E _T admits a K?hler structure, then is isomorphic to . Received: 2 September 2005     

Maximal subgroups of <Emphasis Type="Italic">GL</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>(<Emphasis Type="Italic">D</Emphasis>) with finite conjugacy classes

Let D be an infinite division ring. A famous result due to Herstein says that every non-central element of D has infinitely many conjugates and so, if D ^* is an FC-group, then D is a field. Let M be a maximal subgroup of GL _n (D), where n ≥ 1. In this paper, we prove that if M is an FC-group, then it is the multiplicative group of some maximal subfield of M _n (D). Moreover, if M is algebraic over Z(D), then [D : Z(D)] < ∞.     

Projections on weak*-closed subspace of dual Banach algebras

Our first purpose in this paper is to provide necessary conditions for a weak*-closed translation invariant subspace in the semigroup algebra of a locally compact topological foundation semigroup to be completely complemented. We give conditions when a weak*-closed left translation invariant subspace in Ma(S)* of a compact cancellative foundation semigroup S is the range of a weak*-weak* continuous projection on M _a (S)* commuting with translations. Let G be a locally compact group and A be a Banach G-module. Our second purpose in this paper is to study some projections on A* and B(A*) which commutes with translations and convolution.     

On the Volume of a Domain Obtained by a Holomorphic Motion Along a Complex Curve

Let D be a domain obtained by a holomorphic motion of a domain D _p M _p ^n–1 along a complex curve P in a complex space form M ⁿ . We prove that, if n= 2, the volume of D depends only on the geometry of D _p and the intrinsic geometry of P, but not on the extrinsic geometry of P. When M is closed (compact without boundary), then the dependence on P is only through its topology. When n > 2, and for arbitrary domains D _p, a similar result holds only for Frenet motions, but when D _p has certain integral symmetries (and only in this case) this result is still true for any motion .     

Goldie's theorem for semigroups

In this paper, we try to look for a semigroup analogue of Goldie's theorem which characterizes the orders of the full matrix ringM _n (D) over a division ringD. The main result of this paper, Theorem 4.11, does give a characterization of the semigroups which have a classical left quotient semigroupM _n (G ⁰), whereG ⁰ is a group with zero. The author is grateful to the referee for his careful modification of this paper and helpful comments which resulted in several improvements.     

Irreducible Subgroups of GL 1(D) Satisfying Group Identities

ABSTRACT

Let D be a finite dimensional F -central division algebra and G an irreducible subgroup of D*: = GL ₁(D). Here we investigate the structure of D under various group identities on G. In particular, it is shown that when [D:F] = p ², p a prime, then D is cyclic if and only if D* contains a nonabelian subgroup satisfying a group identity.     

On the Unital C*-Algebras Generated by Certain Subnormal Tuples

We consider an important class of subnormal operator m-tuples M _p (p = m,m + 1, . . .) that is associated with a class of reproducing kernel Hilbert spaces H_p{{\mathcal H}_p} (with M _m being the multiplication tuple on the Hardy space of the open unit ball \mathbb B^2m{{\mathbb B}^{2m}} in \mathbb C^m{{\mathbb C}^m} and M _m+1 being the multiplication tuple on the Bergman space of \mathbb B^2m{{\mathbb B}^{2m}}). Given any two C*-algebras A{\mathcal A} and B{\mathcal B} from the collection {C^*(M_p), C^*([(M)\tilde]_p): p 3 m}{\{C^*({M}_p), C^*({\tilde M}_p): p \geq m\}} , where C*(M _p ) is the unital C*-algebra generated by M _p and C^*([(M)\tilde]_p){C^*({\tilde M}_p)} the unital C*-algebra generated by the dual [(M)\tilde]_p{{\tilde M}_p} of M _p , we verify that A{\mathcal A} and B{\mathcal B} are either *-isomorphic or that there is no homotopy equivalence between A{\mathcal A} and B{\mathcal B} . For example, while C*(M _m ) and C*(M _m+1) are well-known to be *-isomorphic, we find that C^*([(M)\tilde]_m){C^*({\tilde M}_m)} and C^*([(M)\tilde]_m+1){C^*({\tilde M}_{m+1})} are not even homotopy equivalent; on the other hand, C*(M _m ) and C^*([(M)\tilde]_m){C^*({\tilde M}_{m})} are indeed *-isomorphic. Our arguments rely on the BDF-theory and K-theory.     

Counterexamples to the Kneser conjecture in dimension four

We construct a connected closed orientable smooth four-manifold whose fundamental group is the free product of two non-trivial groups such that it is not homotopy equivalent toM ₀#M ₁ unlessM ₀ orM ₁ is homeomorphic toS ⁴. LetN be the nucleus of the minimal elliptic Enrique surfaceV ₁(2, 2) and putM=N∪ _∂NN. The fundamental group ofM splits as ℤ/2 * ℤ/2. We prove thatM#k(S ²×S²) is diffeomorphic toM ₀#M ₁ for non-simply connected closed smooth four-manifoldsM ₀ andM ₁ if and only ifk≥8. On the other hand we show thatM is homeomorphic toM ₀#M ₁ for closed topological four-manifoldsM ₀ andM ₁ withπ ₁(M_i)=ℤ/2.     

Arens multiplication on Banach algebras related to locally compact semigroups

Let S be a locally compact semigroup, let ω be a weight function on S, and let M_a (S, ω) be the weighted semigroup algebra of S. Let L^∞₀ (S;M_a (S, ω)) be the C^*‐algebra of allM_a (S, ω)‐measurable functions g on S such that g /ω vanishes at infinity. We introduce and study an Arens multiplication on L^∞₀ (S;M_a (S, ω))^* under which M_a (S, ω) is a closed ideal. We show that the weighted measure algebra M (S, ω) plays an important role in the structure of L^∞₀ (S;M_a (S, ω))^*. We then study Arens regularity of L^∞₀ (S;M_a (S, ω))^* and ist relation with Arens regularity of M_a (S, ω), M (S, ω) and the discrete convolution algebra ℓ¹(S, ω). As the main result, we prove that L^∞₀ (S;M_a (S, ω))^* is Arens regular if and only if S is finite, or S is discrete and Ω is zero cluster. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

*-orderable semigroups

Fix a *-orderable field k. We introduce the class of *-orderable semigroups as those semigroups with involution S for which the semigroup algebra kS endowed with the canonical involution admits a *-ordering. It is shown that this class is a quasivariety that is locally and residually closed. A cancellative nilpotent semigroup with involution is proved to be *-orderable if and only if it has unique extraction of roots. In general this equivalence fails, although every *-orderable semigroup has unique extraction of roots.     

Some remarks on dualisability and endodualisability

The fundamental problem of dualisability and the particular problem of endodualisability are discussed. It is proved tha every finite generating algebra of a quasi-variety generated by a finite dualisable algebra D is also dualisable. The corresponding result for endodualisability is true when D is subdirectly irreducible. Under special conditions, it is also proved that a finite algebra M is endodualisable if and only if any finite power M ⁿ of M is endodualisable. Received January 27, 1999; accepted in final form September 17, 1999.     

On the limit-classifications of even and odd-order formally symmetric differential expressions

In this paper we consider the formally symmetric differential expressionM [.] of any order (odd or even) ≥ 2. We characterise the dimension of the quotient spaceD(T _max)/D(T _min) associated withM[.] in terms of the behaviour of the determinants det [[f _rg_s](∞)] where 1 ≤n ≤ (order of the expression +1); here [fg](∞) = lim [fg](x), where [fg](x) is the sesquilinear form in f andg associated withM. These results generalise the well-known theorem thatM is in the limit-point case at ∞ if and only if [fg](∞) = 0 for everyf, g ε the maximal domain Δ associated withM.     

Fonctions lipschitziennes et espaces de Sobolev fractionnaires

We consider a semigroup of Markovian and symmetric operators to which we associate fractional Sobolev spaces D^α_p (0 < α < 1 and 1 < p < ∞) defined as domains of fractional powers (−A_p)^α/2, where A_p is the generator of the semigroup in L^p. We show under rather general assumptions that Lipschitz continuous functions operate by composition on D^α_p if p ≥ 2. This holds in particular in the case of the Ornstein-Uhlenbeck semigroup on an abstract Wiener space.     

Two-Sided Closed Ideals of Certain Algebras of Unbounded Operators

This paper investigates the two-sided uniformly closed ideals of the maximal Op*-algebra L⁺(D) of (bounded or unbounded) operators on a dense domain D in a HILBERT space. It is assumed that D is a FRECHET space with respect to the graph topology. The set of all non-trivial two-sided closed ideals of L⁺(D) is well-ordered by inclusion and the α-th closed ideal ??_α is generated by the orthogonal projections onto HILBERTian subspaces of D of dimension less then ??_α. An element A in L⁺(D) belongs to the minimal closed ideal ??₀ if and only if the following two equivalent conditions are satisfied: a) A maps bounded subsets of D into relatively compact sets. b) A maps weakly convergent sequences in D into convergent sequences.