Simple algebras of Weyl type, II

Over a field of any characteristic, for a commutative associative algebra , and for a commutative subalgebra of , the vector space which consists of polynomials of elements in with coefficients in and which is regarded as operators on forms naturally an associative algebra. It is proved that, as an associative algebra, is simple if and only if is -simple. Suppose is -simple. Then, (a) is a free left -module; (b) as a Lie algebra, the subquotient is simple (except for one case), where is the center of . The structure of this subquotient is explicitly described. This extends the results obtained by Su and Zhao.

     

-Weyl's theorem for operator matrices

If is a upper triangular matrix on the Hilbert space , then -Weyl's theorem for and need not imply -Weyl's theorem for , even when . In this note we explore how -Weyl's theorem and -Browder's theorem survive for operator matrices on the Hilbert space.

     

On the irreducibility of the iterates of

Let be a field and suppose that is irreducible in . We discuss the following question: under what conditions are all iterates of irreducible over ?

     

LCM-splitting sets in some ring extensions

Let be a saturated multiplicative set of an integral domain . Call an lcm splitting set if and are principal ideals for every and . We show that if is an -stable overring of (that is, if whenever and is principal, it follows that and if is an lcm splitting set of , then the saturation of in is an lcm splitting set in . Consequently, if is Noetherian and is a (nonzero) prime element, then is also a prime element of the integral closure of . Also, if is Noetherian, is generated by prime elements of and if the integral closure of is a UFD, then so is the integral closure of .

     

An automatic adjoint theorem and its applications

In this paper, we prove the following automatic adjoint theorem: For any sequence spaces and , if has the signed-weak gliding hump property and is an infinite matrix which transforms into , then the transpose matrix of transforms into , and for any and , . That is, the adjoint operator of automatically exists and is just the transpose matrix of . From the theorem we obtain a class of infinite matrix topological algebras , and prove also a -multiplier convergence theorem of Orlicz-Pettis type. The theorem improves substantially the famous Stiles' Orlicz-Pettis theorem.

     

Normal bases for Hopf-Galois algebras

Let be a Hopf algebra over a commutative ring such that is a finitely generated, projective module over , let be a right -comodule algebra, and let be the subalgebra of -coinvariant elements of . If is a Galois extension of and is a local subalgebra of the center of , then is a cleft right -comodule algebra or, equivalently, there is a normal basis for over .

     

An application of Ramsey's Theorem to the Banach Contraction Principle

One of the most fundamental fixed-point theorems is Banach's Contraction Principle, of which the following conjecture is a generalization.

Generalized Banach Contraction Conjecture (GBCC). Let be a self-map of a complete metric space , and let . Let be a positive integer. Assume that for each pair , . Then has a fixed point.

Unlike Banach's original theorem (the case ), the above hypothesis does not compel to be continuous. In this paper we use Ramsey's Theorem from combinatorics to establish the GBCC for arbitrary in the case when is assumed to be continuous, and also derive a result which enables us to prove the GBCC when without the assumption of continuity; it is known that the case includes instances where is not continuous.

     

Abelian point stabilizers in transitive permutation groups

In this note we prove that if the point stabilizer in a transitive permutation group of degree is abelian, then the exponent of is less than . This extends an earlier result of Andrea Lucchini, who proved this in the case where is cyclic.

     

A new invariant of stable equivalences of Morita type

It was proved in an earlier paper by the author that the Hochschild cohomology algebras of self-injective algebras are invariant under stable equivalences of Morita type. In this note we show that the orbit algebra of a self-injective algebra (considered as an --bimodule) is also invariant under stable equivalences of Morita type, where the orbit algebra is the algebra of all stable --bimodule morphisms from the non-negative Auslander-Reiten translations of to .

     

Schur orthogonality relations and invariant sesquilinear forms

Important connections between the representation theory of a compact group and are summarized by the Schur orthogonality relations. The first part of this work is to generalize these relations to all finite-dimensional representations of a connected semisimple Lie group The second part establishes a general framework in the case of unitary representations of a separable locally compact group. The key step is to identify the matrix coefficient space with a dense subset of the Hilbert-Schmidt endomorphisms on .

     

The sumset phenomenon

Answering a problem posed by Keisler and Leth, we prove a theorem in non-standard analysis to reveal a phenomenon about sumsets, which says that if two sets and are large in terms of ``measure', then the sum is not small in terms of ``order-topology'. The theorem has several corollaries about sumset phenomenon in the standard world; these are described in sections 2-4. One of these is a new result in additive number theory; it says that if two sets and of non-negative integers have positive upper or upper Banach density, then is piecewise syndetic.

     

A question of B. Plotkin about the semigroup of endomorphisms of a free group

Let be a free group of finite rank , let be the semigroup of endomorphisms of , and let be the group of automorphisms of .

Theorem. If is an automorphism of , then there is an such that for all .

     

A global pinching theorem for surfaces with constant mean curvature in

Let be a compact immersed surface in the unit sphere with constant mean curvature . Denote by the linear map from into , , where is the linear map associated to the second fundamental form and is the identity map. Let denote the square of the length of . We prove that if , then is either totally umbilical or an -torus, where is a constant depending only on the mean curvature .

     

Existence of multiwavelets in

For a -regular Multiresolution Analysis of multiplicity with arbitrary dilation matrix for a general lattice in , we give necessary and sufficient conditions in terms of the mask and the symbol of the vector scaling function in order that an associated wavelet basis exists. We also show that if where is the absolute value of the determinant of , then these conditions are always met, and therefore an associated wavelet basis of -regular functions always exists. This extends known results to the case of multiwavelets in several variables with an arbitrary dilation matrix for a lattice .

     

On stable equivalences of Morita type for finite dimensional algebras

In this paper, we assume that algebras are finite dimensional algebras with 1 over a fixed field and modules over an algebra are finitely generated left unitary modules. Let and be two algebras (where is a splitting field for and ) with no semisimple summands. If two bimodules and induce a stable equivalence of Morita type between and , and if maps any simple -module to a simple -module, then is a Morita equivalence. This conclusion generalizes Linckelmann's result for selfinjective algebras. Our proof here is based on the construction of almost split sequences.

     

-identities on associative algebras

Let be an algebra over a field and a finite group of automorphisms and anti-automorphisms of . We prove that if satisfies an essential -polynomial identity of degree , then the -codimensions of are exponentially bounded and satisfies a polynomial identity whose degree is bounded by an explicit function of . As a consequence we show that if is an algebra with involution satisfying a -polynomial identity of degree , then the -codimensions of are exponentially bounded; this gives a new proof of a theorem of Amitsur stating that in this case must satisfy a polynomial identity and we can now give an upper bound on the degree of this identity.

     

Spectral synthesis for

Let be a locally compact group, the Fourier algebra of and the von Neumann algebra generated by the left regular representation of . We introduce the notion of -spectral set and -Ditkin set when is an -invariant linear subspace of , thus providing a unified approach to both spectral and Ditkin sets and their local variants. Among other things, we prove results on unions of -spectral sets and -Ditkin sets, and an injection theorem for -spectral sets.

     

Set-theoretic complete intersections on binomials

Let be an affine toric variety of codimension over a field of any characteristic. We completely characterize the affine toric varieties that are set-theoretic complete intersections on binomials. In particular we prove that in the characteristic zero case, is a set-theoretic complete intersection on binomials if and only if is a complete intersection. Moreover, if are binomials such that , then . While in the positive characteristic case, is a set-theoretic complete intersection on binomials if and only if is completely -glued.

These results improve and complete all known results on these topics.

     

Proof of the prime power conjecture for projective planes of order

Let be an abelian collineation group of order of a projective plane of order . We show that must be a prime power, and that the -rank of is at least if for an odd prime .

     

Minimal index of a -crossed product by a finite group

Let be the -crossed product of a simple unital -algebra by a finite group . In this paper we show that the canonical conditional expectation from to has the minimal index if is simple. It is also proved that if is an outer action, then the canonical one is the unique conditional expectation of index-finite type from to , while there are infinitely many conditional expectations when a nontrivial subgroup of acts innerly on .