Commutative group algebras of -summable abelian groups

In this note we study the commutative modular and semisimple group rings of -summable abelian -groups, which group class was introduced by R. Linton and Ch. Megibben. It is proved that is -summable if and only if is -summable, provided is an abelian group and is a commutative ring with 1 of prime characteristic , having a trivial nilradical. If is a -summable -group and the group algebras and over a field of characteristic are -isomorphic, then is a -summable -group, too. In particular provided is totally projective of a countable length.

Moreover, when is a first kind field with respect to and is -torsion, is -summable if and only if is a direct sum of cyclic groups.

     

On the asymptotic spectrum of Hermitian block Toeplitz matrices with Toeplitz blocks

We study the asymptotic behaviour of the eigenvalues of Hermitian block Toeplitz matrices , with Toeplitz blocks. Such matrices are generated by the Fourier coefficients of an integrable bivariate function , and we study their eigenvalues for large and , relating their behaviour to some properties of as a function; in particular we show that, for any fixed , the first eigenvalues of tend to , while the last tend to , so extending to the block case a well-known result due to Szegö. In the case the 's are positive-definite, we study the asymptotic spectrum of , where is a block Toeplitz preconditioner for the conjugate gradient method, applied to solve the system , obtaining strict estimates, when and are fixed, and exact limit values, when and tend to infinity, for both the condition number and the conjugate gradient convergence factor of the previous matrices. Extensions to the case of a deeper nesting level of the block structure are also discussed.

     

The generalized Berg theorem and BDF-theorem

Let be a separable simple -algebra with finitely many extreme traces. We give a necessary and sufficient condition for an essentially normal element , i.e., is normal ( is the quotient map), having the form for some normal element and We also show that a normal element can be quasi-diagonalized if and only if the Fredholm index for all In the case that is a simple -algebra of real rank zero, with stable rank one and with continuous scale, and has countable rank, we show that a normal element with zero Fredholm index can be written as

where is an (increasing) approximate identity for consisting of projections, is a bounded sequence of numbers and with for any given

     

A sequential property of and a covering property of Hurewicz

has the monotonic sequence selection property if there is for each , and for every sequence where for each is a sequence converging pointwise monotonically to , a sequence such that for each is a term of , and converges pointwise to . We prove a theorem which implies for metric spaces that has the monotonic sequence selection property if, and only if, has a covering property of Hurewicz.

     

Mean growth of Koenigs eigenfunctions

In 1884, G. Koenigs solved Schroeder's functional equation

in the following context: is a given holomorphic function mapping the open unit disk into itself and fixing a point , is holomorphic on , and is a complex scalar. Koenigs showed that if , then Schroeder's equation for has a unique holomorphic solution satisfying

moreover, he showed that the only other solutions are the obvious ones given by constant multiples of powers of . We call the Koenigs eigenfunction of . Motivated by fundamental issues in operator theory and function theory, we seek to understand the growth of integral means of Koenigs eigenfunctions. For , we prove a sufficient condition for the Koenigs eigenfunction of to belong to the Hardy space and show that the condition is necessary when is analytic on the closed disk. For many mappings the condition may be expressed as a relationship between and derivatives of at points on that are fixed by some iterate of . Our work depends upon a formula we establish for the essential spectral radius of any composition operator on the Hardy space .

     

A Cantor-Lebesgue theorem with variable ``coefficients'

If is a lacunary sequence of integers, and if for each , and are trigonometric polynomials of degree then must tend to zero for almost every whenever does. We conjecture that a similar result ought to hold even when the sequence has much slower growth. However, there is a sequence of integers and trigonometric polynomials such that tends to zero everywhere, even though the degree of does not exceed for each . The sequence of trigonometric polynomials tends to zero for almost every , although explicit formulas are developed to show that the sequence of corresponding conjugate functions does not. Among trigonometric polynomials of degree with largest Fourier coefficient equal to , the smallest one ``at' is while the smallest one ``near' is unknown.

     

On Bernstein type theorems concerning the growth of derivatives of entire functions

A subspace of which is invariant under all left translation operators is called admissible if is a Banach space satisfying the following properties:

(i) If then there exists a subsequence such that almost everywhere.

(ii) The group is a bounded strongly continuous group. In this case, let

Typical admissible spaces are and all spaces for More generally, all of the Peetre interpolation spaces of two admissible spaces are also admissible.

A function is called subexponential if for every With these definitions our main result goes as follows: . If is an entire function of exponential type such that its restriction to the real axis, denoted by , is subexponential and belongs to some admissible space then the derivative is also in Moreover,
for each real

This result yields as consequences and in a systematic way many new and old Bernstein type inequalities.

     

Mean-boundedness and Littlewood-Paley for separation-preserving operators

Suppose that is a -finite measure space, , and is a bounded, invertible, separation-preserving linear operator such that the linear modulus of is mean-bounded. We show that has a spectral representation formally resembling that for a unitary operator, but involving a family of projections in which has weaker properties than those associated with a countably additive Borel spectral measure. This spectral decomposition for is shown to produce a strongly countably spectral measure on the ``dyadic sigma-algebra' of , and to furnish with abstract analogues of the classical Littlewood-Paley and Vector-Valued M. Riesz Theorems for .

     

Grothendieck operators on tensor products

We prove that for Banach spaces and operators , the tensor product is a Grothendieck operator, provided is a Grothendieck operator and is compact.

     

Unramified cohomology and Witt groups of anisotropic Pfister quadrics

The unramified Witt group of an anisotropic conic over a field , with , defined by the form is known to be a quotient of the Witt group of and isomorphic to . We compute the unramified cohomology group , where is the three dimensional anisotropic quadric defined by the quadratic form over . We use these computations to study the unramified Witt group of .

     

Isomorphisms of row and column finite matrix rings

This paper investigates the ring-theoretic similarities and the categorical dissimilarities between the ring of row finite matrices and the ring of row and column finite matrices. For example, we prove that two rings and are Morita equivalent if and only if the rings and are isomorphic. This resembles the result of V. P. Camillo (1984) for . We also show that the Picard groups of and are isomorphic, even though the rings and are never Morita equivalent.

     

Chern characters associated with almost commuting algebras

A trace formula related to -almost commuting subalgebras and is established. By means of this formula, homomorphisms from to and from to are established. An index map from to is also given.

     

Smooth exhaustion functions in convex domains

We show that in every bounded convex domain in there exists a smooth convex exhaustion function such that the product of all eigenvalues of the matrix is . Moreover, if the domain is strictly convex, then can be chosen so that every eigenvalue is .

     

Cohomological dimension and approximate limits

Approximate (inverse) systems of compacta have been useful in the study of covering dimension, dim, and cohomological dimension over an abelian group , . Such systems are more general than (classical) inverse systems. They have limits and structurally have similar properties. In particular, the limit of an approximate system of compacta satisfies the important property of being an approximate resolution. We shall prove herein that if is an abelian group, a compactum is the limit of an approximate system of compacta , , and for each , then .

     

On Kestenband–Ebert partitions

We study Kestenband–Ebert partitions from a group-theoretic point of view. © 1997 John Wiley & Sons, Inc. J Combin Designs 5: 367–375, 1997     

A note on -hyponormal operators

Let be a -hyponormal operator on a Hilbert space with polar decomposition and let for and We study order and spectral properties of In particular we refine recent Furuta's result on -hyponormal operators.

     

Six mutually orthogonal Latin squares of order 48

A direct construction of six mutually orthogonal Latin squares of order 48 is given. © 1997 John Wiley & Sons, Inc. J Combin Designs 5:463–466, 1997     

The quantum analog of a symmetric pair: a construction in type

Let be the ideal in the enveloping algebra of generated by the maximal compact subalgebra of . In this paper we construct an analog of in the quantized enveloping algebra corresponding to a type diagram at generic . We find generators for and explicit bases for .

     

A remark on commuting operator exponentials

In a previous paper the author proved that for square matrices with algebraic entries expexpexpexp if and only if . This result is extended here to bounded operators on an arbitrary Banach space.

     

Integration of the intertwining operator for -harmonic polynomials associated to reflection groups

Let be the intertwining operator with respect to the reflection invariant measure on the unit sphere in Dunkl's theory on spherical -harmonics associated with reflection groups. Although a closed form of is unknown in general, we prove that

where is the unit ball of and is a constant. The result is used to show that the expansion of a continuous function as Fourier series in -harmonics with respect to is uniformly Cesáro summable on the sphere if , provided that the intertwining operator is positive.