Subnormal Subgroups of Group Ring Units

Let be an arbitrary group. If satisfies , , then the units , generate a nonabelian free subgroup of units. As an application we show that if is contained in an almost subnormal subgroup of units in then either contains a nonabelian free subgroup or all finite subgroups of are normal. This was known before to be true for finite groups only.

     

Every complete doubling metric space carries a doubling measure

We prove that a complete metric space carries a doubling measure if and only if is doubling and that more precisely the infima of the homogeneity exponents of the doubling measures on and of the homogeneity exponents of are equal. We also show that a closed subset of carries a measure of homogeneity exponent . These results are based on the case of compact due to Volberg and Konyagin.

     

Properties of subgenerators of regularized semigroups

We introduce two operations , in the set of subgenerators of a given - regularized semigroup and prove that is a complete partially ordered lattice with respect to , and the operator inclusion . Also presented are some other properties and examples for

     

On uniqueness of invariant means

The following results on uniqueness of invariant means are shown:

(i) Let be a connected almost simple algebraic group defined over . Assume that , the group of the real points in , is not compact. Let be a prime, and let be the compact -adic Lie group of the -points in . Then the normalized Haar measure on is the unique invariant mean on .

(ii) Let be a semisimple Lie group with finite centre and without compact factors, and let be a lattice in . Then integration against the -invariant probability measure on the homogeneous space is the unique -invariant mean on .

     

A remark on Gelfand-Kirillov dimension

Let be a finitely generated non-PI Ore domain and the quotient division algebra of . If is the center of , then .

     

A pointwise spectrum and representation of operators

For a self-adjoint operator commuting with an increasing family of projections we study the multifunction an open set of the topology containing , where is the spectrum of on . Let be the measure of maximal spectral type. We study the condition that is essentially a singleton, is not a singleton. We show that if is the density topology and if satisfies the density theorem, in particular if it is absolutely continuous with respect to the Lebesgue measure, then this condition is equivalent to the fact that is a Borel function of . If is the usual topology then the condition is equivalent to the fact that is approched in norm by step functions , where the set of intervals covers the set where is a singleton.

     

Inradius and integral means for green's functions and conformal mappings

Let be a convex planar domain of finite inradius . Fix the point and suppose the disk centered at and radius is contained in . Under these assumptions we prove that the symmetric decreasing rearrangement in of the Green's function , for fixed , is dominated by the corresponding quantity for the strip of width . From this, sharp integral mean inequalities for the Green's function and the conformal map from the disk to the domain follow. The proof is geometric, relying on comparison estimates for the hyperbolic metric of with that of the strip and a careful analysis of geodesics.

     

Subaveraging estimate for

Let be a smooth real hypersurface of and a compact submanifold of . We generalize a result of A. Boggess and R. Dwilewicz giving, under some geometric conditions on and , an estimate of the submeanvalue on of any function on a neighbourhood of , by the norm of on a neighbourhood of in .

     

Automorphic-differential identities and actions of pointed coalgebras on rings

In this paper, we prove the following two results which generalize the theorem concerning automorphic-differential endomorphisms asserted by J. Bergen. Let be a ring, its left Martindale quotient ring and a right ideal of having no nonzero left annihilator. (1) Let be a pointed coalgebra which measures such that the group-like elements of act as automorphisms of . If is prime and for , then . Furthermore, if the action of extends to and if such that , then . (2) Let be an endomorphism of given as a sum of composition maps of left multiplications, right multiplications, automorphisms and skew-derivations. If is semiprime and , then .

     

Products of similar matrices

Let and be matrices of determinant over a field , with or . We show that if is not a scalar matrix, then is a product of matrices similar to . Analogously, we conjecture that if and are elements of a semisimple algebraic group over a field of characteristic zero, and if there is no normal subgroup of containing but not , then is a product of conjugates of . The conjecture is verified for orthogonal groups and symplectic groups, and for all semisimple groups over local fields. Thus, in a connected, semisimple Lie group with finite center, the only conjugation-invariant subsemigroups are the normal subgroups.

     

A representation theorem for Schauder bases in Hilbert space

A sequence of vectors in a separable Hilbert space is said to be a Schauder basis for if every element has a unique norm-convergent expansion

If, in addition, there exist positive constants and such that

then we call a Riesz basis. In the first half of this paper, we show that every Schauder basis for can be obtained from an orthonormal basis by means of a (possibly unbounded) one-to-one positive self adjoint operator. In the second half, we use this result to extend and clarify a remarkable theorem due to Duffin and Eachus characterizing the class of Riesz bases in Hilbert space.

     

On the Deleted Product Criterion For Embeddability in

For a space let . Let act on and on by exchanging factors and antipodes respectively. We present a new short proof of the following theorem by Weber: For an -polyhedron and , if there exists an equivariant map , then is embeddable in . We also prove this theorem for a peanian continuum and . We prove that the theorem is not true for the 3-adic solenoid and .

     

Some results on finite Drinfeld modules

Let be a global function field, a degree one prime divisor of and let be the Dedekind domain of functions in regular outside . Let be the Hilbert class field of , the integral closure of in . Let be a rank one normalized Drinfeld -module and let be a prime ideal in . We explicitly determine the finite -module structure of . In particular, if , is an odd prime number and is the Carlitz -module, then the finite -module is always cyclic.

     

A remark on normal derivations

Given a Hilbert space , let be operators on . Anderson has proved that if is normal and , then for all operators . Using this inequality, Du Hong-Ke has recently shown that if (instead) , then for all operators . In this note we improve the Du Hong-Ke inequality to for all operators . Indeed, we prove the equivalence of Du Hong-Ke and Anderson inequalities, and show that the Du Hong-Ke inequality holds for unitarily invariant norms.

     

A note on Kamenev type theorems for second order matrix differential systems

Some oscillation criteria are given for the second order matrix differential system , where and are real continuous matrix functions with symmetric, . These results improve oscillation criteria recently discovered by Erbe, Kong and Ruan by using a generalized Riccati transformation , where is the identity matrix, is a given function on and .

     

Fusion in the character table

Suppose that is a Sylow -subgroup of a finite -solvable group . If , then the number of -conjugates of in can be read off from the character table of .

     

On the suspension order of

It is shown that the suspension order of the -fold cartesian product of real projective -space is less than or equal to the suspension order of the -fold symmetric product of and greater than or equal to , where and satisfy and . In particular has suspension order , and for fixed the suspension orders of the spaces are unbounded while their stable suspension orders are constant and equal to .

     

Enveloping algebras of lie color algebras: primeness versus graded-primeness

Let be a finite abelian group and let be a, possibly restricted, -graded Lie color algebra. Then the enveloping algebra is also -graded, and we consider the question of whether being graded-prime implies that it is prime. The first section of this paper is devoted to the special case of Lie superalgebras over a field of characteristic . Specifically, we show that if and if has a unique minimal graded-prime ideal, then this ideal is necessarily prime. As will be apparent, the latter result follows quickly from the existence of an anti-automorphism of whose square is the automorphism of the enveloping algebra associated with its -grading. The second section, which is independent of the first, studies more general Lie color algebras and shows that if is graded-prime and if most homogeneous components of are infinite dimensional over , then is prime. Here we use -methods to study the grading on the extended centroid of . In particular, if is generated by the infinite support of , then we prove that is homogeneous.