Norm estimates of almost Mathieu operators

We estimate the norm of the almost Mathieu operator , regarded as an element in the rotation C^*-algebra . In the process, we prove for every λ∈R and the inequality

     

Meet-irreducible ideals and representations of limit algebras

In this paper we give criteria for an ideal of a TAF algebra to be meet-irreducible. We show that is meet-irreducible if and only if the C∗-envelope of is primitive. In that case, admits a faithful nest representation which extends to a ∗-representation of the C∗-envelope for . We also characterize the meet-irreducible ideals as the kernels of bounded nest representations; this settles the question of whether the n-primitive and meet-irreducible ideals coincide.     

Solution to a conjecture by Hofmeier-Wittstock

For a locally compact (non-compact) group , consider as a left module over the algebra , endowed with the first Arens product. We answer in the affirmative a question raised by Hofmeier-Wittstock (Math. Ann. 308 (1) (1997) 141) concerning the automatic boundedness of the corresponding module homomorphisms on . In fact, we prove that an even stronger assertion holds true. Furthermore, we show that the theorem, first obtained by Ghahramani-McClure (Canad. Math. Bull. 35 (2) (1992) 180), on the automatic w∗-continuity of the (bounded) -module homomorphisms on , can be sharpened in a similar fashion. To this end, a general factorization theorem for bounded families in is proved from which we shall derive both results.     

Ideals of operators and the metric approximation property

We prove that a Banach space X has the metric approximation property if and only if , the space of all finite rank operators, is an ideal in , the space of all bounded operators, for every Banach space Y. Moreover, X has the shrinking metric approximation property if and only if is an ideal in for every Banach space Y.Similar results are obtained for u-ideals and the corresponding unconditional metric approximation properties.     

Non-linear numerical radius isometries on atomic nest algebras and diagonal algebras

A nonlinear map φ between operator algebras is said to be a numerical radius isometry if w(φ(T−S))=w(T−S) for all T, S in its domain algebra, where w(T) stands for the numerical radius of T. Let and be two atomic nests on complex Hilbert spaces H and K, respectively. Denote the nest algebra associated with and the diagonal algebra. We give a thorough classification of weakly continuous numerical radius isometries from onto and a thorough classification of numerical radius isometries from onto .     

On some subalgebras of von Neumann algebras with analyticity

Let (M,α,G) be a covariant system on a locally compact Abelian group G with the totally ordered dual group which admits the positive semigroup . Let H^∞(α) be the associated analytic subalgebra of M; i.e. . Let be the analytic crossed product determined by a covariant system . We give the necessary and sufficient condition that an analytic subalgebra H^∞(α) is isomorphic to an analytic crossed product related to Landstad's theorem. We also investigate the structure of σ-weakly closed subalgebra of a continuous crossed product N?_θR which contains N?_θR₊. We show that there exists a proper σ-weakly closed subalgebra of N?_θR which contains N?_θR₊ and is not an analytic crossed product. Moreover we give an example that an analytic subalgebra is not a continuous analytic crossed product using the continuous decomposition of a factor of type III_λ(0?λ<1).     

Properties of cyclic subspace regression

Various properties of the regression vector produced by cyclic subspace regression with regard to the meancentered linear regression equation are put forth. In particular, the subspace associated with the creation of is shown to contain a basis that maximizes certain covariances with respect to , the orthogonal projection of onto a specific subspace of the range of X. This basis is constructed. Moreover, this paper shows how the maximum covariance values effect the . Several alternative representations of are also developed. These representations show that is a modified version of the l-factor principal components regression vector , with the modification occurring by a nonorthogonal projection. Additionally, these representations enable prediction properties associated with to be explicitly identified. Finally, methods for choosing factors are spelled out.     

On the product system of a completely positive semigroup

Given a W∗-continuous semigroup φ of unital, normal, completely positive maps of B(H), we introduce its continuous tensor product system . If α is a minimal dilation E₀-semigroup of φ with Arveson product system F, then is canonically isomorphic to F. We apply this construction to a class of semigroups of arising from a modified Weyl-Moyal quantization of convolution semigroups of Borel probability measures on . This class includes the heat flow on the CCR algebra studied recently by Arveson. We prove that the minimal dilations of all such semigroups are completely spatial, and additionally, we prove that the minimal dilation of the heat flow is cocyle conjugate to the CAR/CCR flow of index two.     

Delooping the K-theory of exact categories

We generalize, from additive categories to exact categories, the concept of “Karoubi filtration” and the associated homotopy fibration in algebraic K-theory. As an application, we construct for any idempotent complete exact category an exact category such that .     

Polynomial identities and noncommutative versal torsors

To any cleft Hopf Galois object, i.e., any algebra obtained from a Hopf algebra H by twisting its multiplication with a two-cocycle α, we attach two “universal algebras” and . The algebra is obtained by twisting the multiplication of H with the most general two-cocycle σ formally cohomologous to α. The cocycle σ takes values in the field of rational functions on H. By construction, is a cleft H-Galois extension of a “big” commutative algebra . Any “form” of can be obtained from by a specialization of and vice versa. If the algebra is simple, then is an Azumaya algebra with center . The algebra is constructed using a general theory of polynomial identities that we set up for arbitrary comodule algebras; it is the universal comodule algebra in which all comodule algebra identities of are satisfied. We construct an embedding of into ; this embedding maps the center of into when the algebra is simple. In this case, under an additional assumption, , thus turning into a central localization of . We completely work out these constructions in the case of the four-dimensional Sweedler algebra.     

Tensor products of holomorphic representations and bilinear differential operators

Let be the weighted Bergman space on a bounded symmetric domain D=G/K. It has analytic continuation in the weight ν and for ν in the so-called Wallach set still forms unitary irreducible (projective) representations of G. We give the irreducible decomposition of the tensor product of the representations for any two unitary weights ν and we find the highest weight vectors of the irreducible components. We find also certain bilinear differential intertwining operators realizing the decomposition, and they generalize the classical transvectants in invariant theory of . As applications, we find a generalization of the Bol's lemma and we characterize the multiplication operators by the coordinate functions on the quotient space of the tensor product modulo the subspace of functions vanishing of certain degree on the diagonal.     

Genericity of wild holomorphic functions and common hypercyclic vectors

Let T_α be the translation operator by α in the space of entire functions defined by . We prove that there is a residual set G of entire functions such that for every f∈G and every the sequence is dense in , that is, G is a residual set of common hypercyclic vectors ( functions) for the family . Also, we prove similar results for many families of operators as: multiples of differential operator, multiples of backward shift, weighted backward shifts.     

A formula for inserting point masses

Let be a probability measure on the unit circle and be the measure formed by adding a pure point to . We give a formula for the Verblunsky coefficients of , based on a result of Simon.     

The Bourgain ?-index of mixed Tsirelson space

Suppose that is a sequence of regular families of finite subsets of such that contains all singletons, and (θ_n)_n=1^∞ is a nonincreasing null sequence in (0,1). The mixed Tsirelson space is the completion of c₀₀ with respect to the implicitly defined norm

     

Cap pairings and spectral sequences

Let be a small category. For an -diagram X and -diagrams A and B of pointed spaces, each pairing X∧A→B satisfying the projection formula induces a pairing . In this note we show that there is an induced pairing of homotopy spectral sequences compatible with abutments in the sense that