Über die Wiener-Eigenschaft bei projektiven Limites lokalkompakter Gruppen

Leptin posed in [1] the problem to determine the class [W] of locally compact groups G characterized by the following property: Every proper closed two-sided idealJ in the Banach-*-algebraL ¹(G) is annihilated by some nondegenerate continuous *-representation ofL ¹(G) in a Hilbert space. Our main result: A locally compact group G, which is representable as a projective limit of a system of factor groups G/k, k compact normal subgroups, lies in [W] if and only if all the G/k are in [W].     

Complemented subspaces ofL ∞ (G), ideals ofL 1 (G) and amenability

LetG be a locally compact group andX a weak *-closed translation invariant subspace ofL (G). It is shown that the following conditions are equivalent: (i)X has a closedG-invariant complement inL (G); (ii)X has a closedL ¹ (G)-invariant complement inL (G); (iii) the annihilatorX ofX inL ¹ (G) has bounded approximate units. The following result of Lau and Losert is then deduced: ifG is amenable andX complemented, thenX has a closedG-invariant complement. This implies for amenableG thatX is complemented if and only if the idealX has bounded approximate units. This duality unifies and generalizes results of Gilbert, Liu, van Rooij, Wang, Rosenthal and Reiter.     

Representation of measurable positive definite generalized Toeplitz Kernels inR

We define the signature of a bounded operatorA onL ² S ² and prove thatA is smooth for the action ofSO(3) onL ² S ² if and only if its signature is smooth and any finite application of certain differential operators to it yields the signature of a bounded operator. Moreover, we show that the formal Fourier multipliers with bounded and smoothly variable coefficients are well defined bounded operators which areSO(3)-smooth.     

On the abstract Cauchy problem for the operator\frac{{d^2 }}{{dt^2 }} - A.

LetX be a Banach space, A a closed operator with dense domainD(A) and non-void resolvent set; topologized by the semi-norm system. We prove that the Cauchy problem is well posed in the sense of distributions for the operator d²/dt²–A if and only if A restricted toD generates a locally — equicontinuous cosine function of class C inL _S D . This is an extension of Ushijima's smoothness result of distribution semi-groups [10].     

Centrally large subgroups of finite p-groups

Let S be a finite p-group. We say that an abelian subgroup A of S is a large abelian subgroup of S if |A||A^*| for every abelian subgroup A^* of S. We say that a subgroup Q of S is a centrally large subgroup, or CL-subgroup, of S if |Q||Z(Q)||Q^*||Z(Q^*)| for every subgroup Q^* of S. The study of large abelian subgroups and variations on them began in 1964 with Thompson's second normal p-complement theorem [J.G. Thompson, Normal p-complements for finite groups, J. Algebra 1 (1964) 43–46]. Centrally large subgroups possess some similar properties. In 1989, A. Chermak and A. Delgado [A. Chermak, A. Delgado, A measuring argument for finite groups, Proc. Amer. Math. Soc. 107 (1989) 907–914] studied several families of subgroups that include centrally large subgroups as a special case. In this paper, we extend their work to prove some further properties of centrally large subgroups. The proof uses an analogue for finite p-groups of an application of Borel's Fixed Point Theorem for algebraic groups.     

Classification Of Locally 2-Connected Compact Metric Spaces

The aim of this paper is to prove that, for compact metric spaces which do not contain infinite complete graphs, the (strong) property of being locally 2-dimensional is guaranteed just by a (weak) local connectivity condition. Specifically, we prove that a locally 2-connected, compact metric space M either contains an infinite complete graph or is surface like in the following sense: There exists a unique surface S such that S and M contain the same finite graphs. Moreover, M is embeddable in S, that is, M is homeomorphic to a subset of S.     

Analytic theory of subnormal operators

LetS be a pure subnormal operator and (·) be its mosaic introduced in [13]. In this paper, the author proves that rank (z)<+ if [S ^*,S] is compact and rank (z)=–g(z) whereg(·) is the Pincus principal function if [S ^*,S] is in the trace class. Also a result is given about the form of pure and irreducible subnormal operators with finite rank self-commutators.This work is supported in part by a NSF grant.     

Periodic groups saturated with <Emphasis Type="Italic">L</Emphasis><Subscript>3</Subscript>(2<Superscript>m</Superscript>)

Let be a set of finite groups. A group G is saturated with groups from if every finite subgroup of G is contained in a subgroup isomorphic to some member of . It is proved that a periodic group G saturated with groups from the set {L₃(2^m)|m = 1, 2, …} is isomorphic to L₃(Q), for a locally finite field Q of characteristic 2; in particular, it is locally finite. __________ Translated from Algebra i Logika, Vol. 46, No. 5, pp. 606–626, September–October, 2007.     

Embeddings of Rearrangement Invariant Spaces that are not Strictly Singular

We give partial answers to the following conjecture: the natural embedding of a rearrangement invariant space E into L ₁([0,1]) is strictly singular if and only if G does not embed into E continuously, where G is the closure of the simple functions in the Orlicz space L with (x) = exp(x²)-1.     

The center of the 3-dimensional and 4-dimensional Sklyanin algebras

LetA=A(E, ) denote either the 3-dimensional or 4-dimensional Sklyanin algebra associated to an elliptic curveE and a point E. Assume that the base field is algebraically closed, and that its characteristic does not divide the dimension ofA. It is known thatA is a finite module over its center if and only if is of finite order. Generators and defining relations for the centerZ(A) are given. IfS=Proj(Z(A)) andA is the sheaf ofO _S -algebras defined byA(S _(f))=A[f ^–1]₀ then the centerL ofA is described. For example, for the 3-dimensional Sklyanin algebra we obtain a new proof of M. Artin's result thatSpec L². However, for the 4-dimensional Sklyanin algebra there is not such a simple result: althoughSpec L is rational and normal, it is singular. We describe its singular locus, which is also the non-Azumaya locus ofA.     

Bounded tightness for weak topologies

In every Hausdorff locally convex space for which there exists a strictly finer topology than its weak topology but with the same bounded sets (like for instance, all infinite dimensional Banach spaces, the space of distributions or the space of analytic functions in an open set , etc.) there is a set A such that 0 is in the weak closure of A but 0 is not in the weak closure of any bounded subset B of A. A consequence of this is that a Banach space X is finite dimensional if, and only if, the following property [P] holds: for each set and each x in the weak closure of A there is a bounded set such that x belongs to the weak closure of B. More generally, a complete locally convex space X satisfies property [P] if, and only if, either X is finite dimensional or linearly topologically isomorphic to . Received: 11 June 2003     

Properties of Standard n-Ideals of a Lattice

An n-ideal of a lattice L is a convex sublattice containing a fixed element n L and it is called standard if it is a standard element of the lattice of n-ideals I_n(L). In this paper we have shown that, for a neutral element n of a lattice L, the principal n-ideal a_n of a lattice L is a standard n-ideal if and only if a n is standard and a n is dual standard. We have also shown that if n is a neutral element and (n] and [n) are relatively complemented, then every homomorphism n-kernels of L is a standard n-ideal and every standard n-ideal is the n-kernel of precisely one congruence relation. Finally, we have shown that, for a relatively complemented lattice L with 0 and 1, C(L) is a Boolean algebra if and only if every standard n-ideal of L is a principal n-ideal.AMS Subject Classification (2000) 06B10 06B99 06C15     

Pointwise bounds for L 2 eigenfunctions on locally symmetric spaces

We prove pointwise bounds for L ² eigenfunctions of the Laplace-Beltrami operator on locally symmetric spaces with -rank one if the corresponding eigenvalues lie below the continuous part of the L ² spectrum. Furthermore, we use these bounds in order to obtain some results concerning the L ^p spectrum.     

On finiteness conditions for Rees matrix semigroups

Let T=[S; I; J; P] be a Rees matrix semigroup where S is a semigroup, I and J are index sets, and P is a J × I matrix with entries from S, and let U be the ideal generated by all the entries of P. If U has finite index in S, then we prove that T is periodic (locally finite) if and only if S is periodic (locally finite). Moreover, residual finiteness and having solvable word problem are investigated.     

A note on Schatten‐class membership of Hankel operators with antiholomorphic symbols on generalized Fock‐spaces

In this paper we investigate Hankel operators with anti‐holomorphic L²‐symbols on generalized Fock spaces A_m² in one complex dimension. The investigation of the mentioned operators was started in [4] and [3]. Here, we show that a Hankel operator with anti‐holomorphic L²‐symbol is in the Schatten‐class S_p if and only if the symbol is a polynomial with degree N satisfying 2N < m and p > . The result has been proved independently before in the recent work [2], which also considers the case of several complex variables. However, in addition to providing a different proof for the result the present work shows that the methodology developed in [4] and [3] can be adopted in order to work to characterize Schatten‐class membership. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Parallel Computation of Invariant Measures

Let S:[0,1][0,1] be a nonsingular transformation and let P:L ¹(0,1)L ¹(0,1) be the corresponding Frobenius–Perron operator. In this paper we propose a parallel algorithm for computing a fixed density of P, using Ulam's method and a modified Monte Carlo approach. Numerical results are also presented.     

On Strongly Inert Subalgebras of an Infinite-Dimensional Lie Algebra

We study infinite-dimensional Lie algebras L over an arbitrary field that contain a subalgebra A such that dim(A + [A, L])/A < . We prove that if an algebra L is locally finite, then the subalgebra A is contained in a certain ideal I of the Lie algebra L such that dimI/A <. We show that the condition of local finiteness of L is essential in this statement.     

On Infinite Cycles I

We adapt the cycle space of a finite graph to locally finite infinite graphs, using as infinite cycles the homeomorphic images of the unit circle S¹ in the graph compactified by its ends. We prove that this cycle space consists of precisely the sets of edges that meet every finite cut evenly, and that the spanning trees whose fundamental cycles generate this cycle space are precisely the end-faithful spanning trees. We also generalize Eulers theorem by showing that a locally finite connected graph with ends contains a closed topological curve traversing every edge exactly once if and only if its entire edge set lies in this cycle space.To the memory of C. St. J. A. Nash-Williams     

Nonabelian K-theory: The nilpotent class of K1 and general stability

A functorial filtration GL _n =S^–1L _n S⁰L _n S ⁱ L _n E _n of the general linear group GL _{n, n} 3, is defined and it is shown for any algebra A, which is a direct limit of module finite algebras, that S^–1 L _n (A)/S⁰L _n (A) is abelian, that S⁰L _n (A) S¹L _n (A) is a descending central series, and that S ⁱ L _n (A) = E _n(A) whenever i the Bass-Serre dimension of A. In particular, the K-functors k ₁ S ⁱ L _n =S ⁱ L _n /E _n are nilpotent for all i 0 over algebras of finite Bass-Serre dimension. Furthermore, without dimension assumptions, the canonical homomorphism S ⁱ L _n (A)/S ⁱ⁺¹ L _n (A)S ⁱ L _{n+ 1}(A)/S ⁱ⁺¹ L _{n + 1} (A) is injective whenever n i + 3, so that one has stability results without stability conditions, and if A is commutative then S⁰L _n (A) agrees with the special linear group SL _n (A), so that the functor S⁰L _n generalizes the functor SL _n to noncommutative rings. Applying the above to subgroups H of GL _n (A), which are normalized by E _n(A), one obtains that each is contained in a sandwich GL _n (A, ) H E _n(A, ) for a unique two-sided ideal of A and there is a descending S⁰L _n (A)-central series GL _n (A, ) S⁰L _n (A, ) S¹L _n (A, ) S ⁱ L _n (A, ) E _n(A, ) such that S ⁱ L _n (A, )=E _n(A, ) whenever i Bass-Serre dimension of A.Dedicated to Alexander Grothendieck on his sixtieth birthday     

Automorphism groups of finite distributive lattices with a given sublattice of fixed points

For a latticeL, aL is a fixed point ofL if and only iff(a)=a for every automorphismf ofL. Let Aut (L) andS (L) denote the group of automorphisms ofL and the sublattice of fixed points ofL, respectively. It is shown that ifG is a finite group other thanZ ₂ ² ,Z ₂ ³ ,Z ₂ ⁴ ,Z ₃ ² or the quaternion group of order 8 andL is a finite automorphism free distributive lattice that is not near Boolean then there is a finite distributive latticeL such that Aut (L)G andS(L)L.The support of the National Research Council of Canada is gratefully acknowledged.