Self-commutator approximants

This paper deals with minimizing , where is fixed, self-adjoint and , and where varies such that and , . (Here, , , denotes the von Neumann-Schatten class and its norm.) The upshot of this paper is that , , is minimized if, and for only if, , and that the map , , has a critical point at if and only if (with related results for normal if or ).

     

Commutator inequalities associated with the polar decomposition

Let be a polar decomposition of an complex matrix . Then for every unitarily invariant norm , it is shown that

where denotes the operator norm. This is a quantitative version of the well-known result that is normal if and only if . Related inequalities involving self-commutators are also obtained.

     

Entire pluricomplex Green functions and Lelong numbers of projective currents

Let be a positive closed current of bidimension (1,1) and unit mass on the complex projective space . We prove that the set of points where has Lelong number larger than is contained in a complex line if , and for some complex line if . We also prove that in dimension 2 and if , then for some conic .

     

Poincaré duality algebras and rings of coinvariants

Let be a faithful representation of a finite group over the field . Via the group acts on and hence on the algebra of homogenous polynomial functions on the vector space . R. Kane (1994) formulated the following result based on the work of R. Steinberg (1964): If the field has characteristic 0, then is a Poincaré duality algebra if and only if is a pseudoreflection group. The purpose of this note is to extend this result to the case (i.e. the order of is relatively prime to the characteristic of ).

     

On Korenblum's maximum principle

Let be the Bergman space over the open unit disk in the complex plane. Korenblum's maximum principle states that there is an absolute constant , such that whenever ( ) in the annulus , then . In this paper we prove that Korenblum's maximum principle holds with .

     

Equicompact sets of operators defined on Banach spaces

Let and be Banach spaces. We say that a set denotes the space of all compact operators from into ) is equicompact if there exists a null sequence in such that for all and all . It is easy to show that collectively compactness and equicompactness are dual concepts in the following sense: is equicompact iff is collectively compact. We study some properties of equicompact sets and, among other results, we prove: 1) a set is equicompact iff each bounded sequence in has a subsequence such that is a converging sequence uniformly for ; 2) if does not have finite cotype and is a maximal equicompact set, then, given and a finite set in , there is an operator such that for and all .

     

Topological entropy and AF subalgebras of graph -algebras

Let be the canonical AF subalgebra of a graph -algebra associated with a locally finite directed graph . For Brown and Voiculescu's topological entropy of the canonical completely positive map on , is known to hold for a finite graph , where is the loop entropy of Gurevic and is the block entropy of Salama. For an irreducible infinite graph , the inequality has recently been known. It is shown in this paper that

where is the graph with the direction of the edges reversed. Some irreducible infinite graphs 1)$"> with are also examined.

     

On a conjecture about MRA Riesz wavelet bases

Let be a compactly supported refinable function in such that the shifts of are stable and for a -periodic trigonometric polynomial . A wavelet function can be derived from by . If is an orthogonal refinable function, then it is well known that generates an orthonormal wavelet basis in . Recently, it has been shown in the literature that if is a -spline or pseudo-spline refinable function, then always generates a Riesz wavelet basis in . It was an open problem whether can always generate a Riesz wavelet basis in for any compactly supported refinable function in with stable shifts. In this paper, we settle this problem by proving that for a family of arbitrarily smooth refinable functions with stable shifts, the derived wavelet function does not generate a Riesz wavelet basis in . Our proof is based on some necessary and sufficient conditions on the -periodic functions and in such that the wavelet function , defined by , generates a Riesz wavelet basis in .

     

Bounds for the index of the centre in capable groups

A group is called capable if it is isomorphic to for some group . Let be a capable group. I. M. Isaacs (2001) showed that if is finite, then the index of the centre is bounded above by some function of . We show that if , then with some constant and this bound is essentially best possible. We complete a result of Isaacs, showing that if is a cyclic group, then .

     

A dual graph construction for higher-rank graphs, and -theory for finite 2-graphs

Given a -graph and an element of , we define the dual -graph, . We show that when is row-finite and has no sources, the -algebras and coincide. We use this isomorphism to apply Robertson and Steger's results to calculate the -theory of when is finite and strongly connected and satisfies the aperiodicity condition.

     

Lifts of - and -morphisms to -morphisms

Let be the Hochschild complex of cochains on and let be the space of multivector fields on . In this paper we prove that given any -structure (i.e. Gerstenhaber algebra up to homotopy structure) on , and any -morphism (i.e. morphism of a commutative, associative algebra up to homotopy) between and , there exists a -morphism between and that restricts to . We also show that any -morphism (i.e. morphism of a Lie algebra up to homotopy), in particular the one constructed by Kontsevich, can be deformed into a -morphism, using Tamarkin's method for any -structure on . We also show that any two of such -morphisms are homotopic.

     

Eigenvalues of the Laplacian acting on -forms and metric conformal deformations

Let be a compact connected orientable Riemannian manifold of dimension and let be the -th positive eigenvalue of the Laplacian acting on differential forms of degree on . We prove that the metric can be conformally deformed to a metric , having the same volume as , with arbitrarily large for all .

Note that for the other values of , that is and , one can deduce from the literature that, 0$">, the -th eigenvalue is uniformly bounded on any conformal class of metrics of fixed volume on .

For , we show that, for any positive integer , there exists a metric conformal to such that, , , that is, the first eigenforms of are all exact forms.

     

A perturbed elementary operator and range-kernel orthogonality

Let denote the algebra of operators on a Hilbert . If and are commuting normal operators, and and are commuting quasi-nilpotents such that , then define and by , , and . It is proved that and , where is some scalar and is the quasi-nilpotent part of the operator .

     

On Schwarz type inequalities

We show Schwarz type inequalities and consider their converses. A continuous function is said to be semi-operator monotone on if is operator monotone on . Let be a bounded linear operator on a complex Hilbert space and be the polar decomposition of . Let and for . (1) If a non-zero function is semi-operator monotone on , then for , where . (2) If are semi-operator monotone on , then for . Also, we show converses of these inequalities, which imply that semi-operator monotonicity is necessary.

     

The Banach algebra generated by a contraction

Let be a contraction on a Banach space and the Banach algebra generated by . Let be the unitary spectrum (i.e., the intersection of with the unit circle) of . We prove the following theorem of Katznelson-Tzafriri type: If is at most countable, then the Gelfand transform of vanishes on if and only if

     

Characterizations of elements with compact support in the dual spaces of -modules of

For a locally compact group and , let be the Figà-Talamanca-Herz algebra and let be its dual Banach space. For a Banach -module of , we denote the norm closure of the subspace of the elements in with compact support by . We prove that an element of is in if and only if for any 0$">, there exists a compact subset of such that for all with and . In particular, we have that an element of is in if and only if for any 0$">, there exists a compact subset of such that for all with . If has an orthogonal complement in , we characterize by the following condition: is in if and only if for any 0$"> and any compact subset of , there exists some with and such that \Vert u\Vert - \epsilon $">. Some results of Flory (1971) and Miao (1999) can be obtained from our main theorems by taking and as some -subalgebras of .

     

A note on periodic points of order preserving subhomogeneous maps

Let be the standard closed positive cone in and let be the set of integers for which there exists a continuous, order preserving, subhomogeneous map , which has a periodic point with period . It has been shown by Akian, Gaubert, Lemmens, and Nussbaum that is contained in the set consisting of those for which there exist integers and such that , , and for some . This note shows that for all .

     

On mixing and completely mixing properties of positive -contractions of finite von Neumann algebras

Akcoglu and Suchaston proved the following result: Let be a positive contraction. Assume that for the sequence converges weakly in . Then either or there exists a positive function , such that . In the paper we prove an extension of this result in a finite von Neumann algebra setting, and as a consequence we obtain that if a positive contraction of a noncommutative -space has no nonzero positive invariant element, then its mixing property implies the completely mixing property.

     

The amenability constant of the Fourier algebra

For a locally compact group , let denote its Fourier algebra and its dual object, i.e., the collection of equivalence classes of unitary representations of . We show that the amenability constant of is less than or equal to and that it is equal to one if and only if is abelian.

     

Multiplication and division by inner functions in the space of Bloch functions

A subspace of the Hardy space is said to have the -property if whenever and is an inner function with . We let denote the space of Bloch functions and the little Bloch space. Anderson proved in 1979 that the space does not have the -property. However, the question of whether or not ( ) has the -property was open. We prove that for every the space does not have the -property.

We also prove that if is any infinite Blaschke product with positive zeros and is a Bloch function with , as , then the product is not a Bloch function.