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Inspired by some recent development on the theory about projection valued dilations for operator valued measures or more generally bounded homomorphism dilations for bounded linear maps on Banach algebras, the authors explore a pure algebraic version of the dilation theory for linear systems acting on unital algebras and vector spaces. By introducing two natural dilation structures, namely the canonical and the universal dilation systems, they prove that every linearly minimal dilation is equivalent to a reduced homomorphism dilation of the universal dilation, and all the linearly minimal homomorphism dilations can be classified by the associated reduced subspaces contained in the kernel of synthesis operator for the universal dilation.

In this article, we investigate the privacy issues that arise from a new frame-based kernel analysis approach to reconstruct from frame coefficient erasures. We show that while an erasure recovery matrix is needed in addition to a decoding frame for a receiver to recover the erasures, the erasure recovery matrix can be designed in such a way that it protects the encoding frame. The set of such erasure recovery matrices is shown to be an open and dense subset of a certain matrix space. We present algorithms to construct concrete examples of encoding frame and erasure recovery matrix pairs for which the erasure reconstruction process is robust to additive channel noise. Using the Restricted Isometry Property, we also provide quantitative bounds on the amplification of sparse additive channel noise. Numerical experiments are presented on the amplification of additive normally distributed random channel noise. In both cases, the amplification factors are demonstrated to be quite small.

The duality principle for group representations developed in Dutkay et al. (J Funct Anal 257:1133–1143, 2009), Han and Larson (Bull Lond Math Soc 40:685–695, 2008) exhibits a fact that the well-known duality principle in Gabor analysis is not an isolated incident but a more general phenomenon residing in the context of group representation theory. There are two other well-known fundamental properties in Gabor analysis: the biorthogonality and the fundamental identity of Gabor analysis. The main purpose of this this paper is to show that these two fundamental properties remain to be true for general projective unitary group representations. Moreover, we also present a general duality theorem which shows that that muti-frame generators meet super-frame generators through a dual commutant pair of group representations. Applying it to the Gabor representations, we obtain that $$\{\pi _{\Lambda }(m, n)g_{1} \oplus \cdots \oplus \pi _{\Lambda }(m, n)g_{k}\}_{m, n \in {\mathbb {Z}}^{d}}$$ { π Λ ( m , n ) g 1 ⊕ ⋯ ⊕ π Λ ( m , n ) g k } m , n ∈ Z d is a frame for $$L^{2}({\mathbb {R}}\,^{d})\oplus \cdots \oplus L^{2}({\mathbb {R}}\,^{d})$$ L 2 ( R d ) ⊕ ⋯ ⊕ L 2 ( R d ) if and only if $$\cup _{i=1}^{k}\{\pi _{\Lambda ^{o}}(m, n)g_{i}\}_{m, n\in {\mathbb {Z}}^{d}}$$ ∪ i = 1 k { π Λ o ( m , n ) g i } m , n ∈ Z d is a Riesz sequence, and $$\cup _{i=1}^{k} \{\pi _{\Lambda }(m, n)g_{i}\}_{m, n\in {\mathbb {Z}}^{d}}$$ ∪ i = 1 k { π Λ ( m , n ) g i } m , n ∈ Z d is a frame for $$L^{2}({\mathbb {R}}\,^{d})$$ L 2 ( R d ) if and only if $$\{\pi _{\Lambda ^{o}}(m, n)g_{1} \oplus \cdots \oplus \pi _{\Lambda ^{o}}(m, n)g_{k}\}_{m, n \in {\mathbb {Z}}^{d}}$$ { π Λ o ( m , n ) g 1 ⊕ ⋯ ⊕ π Λ o ( m , n ) g k } m , n ∈ Z d is a Riesz sequence, where $$\pi _{\Lambda }$$ π Λ and $$\pi _{\Lambda ^{o}}$$ π Λ o is a pair of Gabor representations restricted to a time–frequency lattice $$\Lambda $$ Λ and its adjoint lattice $$\Lambda ^{o}$$ Λ o in $${\mathbb {R}}\,^{d}\times {\mathbb {R}}\,^{d}$$ R d × R d .

A scalar sequence ξ \xi is said to be admissible for a positive operator A A if A = ∑ ξ j P j A= \sum \xi _jP_j for some rank-one projections P j P_j , or, equivalently, if diag ξ \xi is the diagonal of V A V ∗ VAV^* for some partial isometry V V having as domain the closure of the range of A A . The main result of this paper is that if A A is the sum of infinitely many projections (converging in the strong operator topology) and ξ \xi is a nonsummable sequence in [ 0 , 1 ] [0,1] that satisfies the Kadison condition that requires that either ∑ { ξ i ∣ ξ i ≤ 1 2 } + ∑ { ( 1 − ξ i ) ∣ ξ i > 1 2 } = ∞ \sum \{\xi _i \mid \xi _i\le \frac {1}{2}\}+ \sum \{(1-\xi _i) \mid \xi _i> \frac {1}{2}\} = \infty or the difference ∑ { ξ i ∣ ξ i ≤ 1 2 } − ∑ { ( 1 − ξ i ) ∣ ξ i > 1 2 } \sum \{\xi _i \mid \xi _i\le \frac {1}{2}\}- \sum \{(1-\xi _i) \mid \xi _i> \frac {1}{2}\} is an integer, then ξ \xi is admissible for A A . This result extends Kadison’s carpenter’s theorem and provides an independent proof of it.

In a recent article, the authors discovered a new, efficient method for perfect reconstruction from frame erasures called nilpotent bridging. In order to perform the reconstruction, it is necessary to invert a matrix, called the bridge matrix, which consists of inner products of the frame vectors indexed by the erasure set, with vectors from the corresponding dual frame indexed by a set known as the bridge set, which is disjoint from the erasure set. If no search procedure is required to find a bridge set of the same size as the erasure set for which the bridge matrix is invertible, the dual frame pair is said to satisfy the full skew-spark property. Equivalently, a frame satisfies the full skew-spark property if every subset of the complement of the erasure set with the same cardinality as the erasure set yields an invertible bridge matrix. In this paper, we consider dual frame pairs which consist of a union of two bases, and the union of their corresponding dual bases. In finite dimensions, it is shown that if we restrict the class of bridge sets to block bridge sets, then a bridge set search is also unnecessary for most unions of two bases, and most unions of two orthonormal bases. By most, we mean the set of unions of two bases (resp. orthonormal bases) for which no bridge set search is necessary contains an open, dense set in the set of all unions of two bases (resp. orthonormal bases). So, while not every bridge set of the same cardinality as the erasure set yields an invertible bridge matrix, the bridge sets which have the special structure of block bridge sets are very likely to yield an invertible bridge matrix. These unions of two bases are said to satisfy the block skew-spark property. In infinite dimensions, we also show that the block skew-spark property holds for a dense (but not necessarily open) set of unions of two Riesz (resp. orthonormal) bases. Lastly, applications to Shannon–Whittaker sampling theory are discussed, and several open questions are posed pertaining to sampling theory.

Let $R(\mathbb{D})$ be the algebra generated in the Sobolev space $W^{22}(\mathbb{D})$ by the rational functions with poles outside the unit disk $\overline{\mathbb{D}}$ . This is called the Sobolev disk algebra. In this article, the commutant of the multiplication operator $M_{B(z)}$ on $R(\mathbb{D})$ is studied, where $B(z)$ is an n-Blaschke product. We prove that an operator $A\in\mathcal{L}(R(\mathbb{D}))$ is in $\mathcal{A}'(M_{B(z)})$ if and only if $A=\sum_{i=1}^{n}M_{h_{i}}M_{\Delta(z)}^{-1}T_{i}$ , where $\{h_{i}\}_{i=1}^{n}\subset R(\mathbb{D})$ , and $T_{i}\in\mathcal{L}(R(\mathbb{D}))$ is given by $(T_{i}g)(z)=\sum_{j=1}^{n}(-1)^{i+j}\Delta_{ij}(z)g(G_{j-1}(z))$ , $i=1,2,\ldots,n$ , $G_{0}(z)\equiv z$ .

Motivated by a general dilation theory for operator-valued measures, framings and bounded linear maps on operator algebras, we consider the dilation theory of the above objects with special structures. We show that every operator-valued system of imprimitivity has a dilation to a probability spectral system of imprimitivity acting on a Banach space. This completely generalizes a well-kown result which states that every frame representation of a countable group on a Hilbert space is unitarily equivalent to a subrepresentation of the left regular representation of the group. The dilated space in general can not be taken as a Hilbert space. However, it can be taken as a Hilbert space for positive operator valued systems of imprimitivity. We also prove that isometric group representation induced framings on a Banach space can be dilated to unconditional bases with the same structure for a larger Banach space This extends several known results on the dilations of frames induced by unitary group representations on Hilbert spaces., 21 pages

We investigate the geometric properties for Hilbert C*-modular frames. We show that any dual frame pair in a Hilbert C*-module is an orthogonal compression of a Riesz basis and its canonical dual for some larger Hilbert C*-module. This generalizes the Hilbert space dual frame pair dilation theory due to Casazza, Han and Larson to dual Hilbert C*-modular frame pairs. Additionally, for any Hilbert C*-modular dual frame pair induced by a group of unitary operators, we show that there is a dilated dual pair which inherits the same group structure.

We investigate the topological and metric structure of the set of idempotent operators and projections which have prescribed diagonal entries with respect to a fixed orthonormal basis of a Hilbert space. As an application, we settle some cases of conjectures of Larson, Dykema, and Strawn on the connectedness of the set of unit-norm tight frames., New title and introduction

We develop a natural generalization of vector-valued frame theory, we term operator-valued frame theory, using operator-algebraic methods. This extends work of the second author and D. Han which can be viewed as the mul- tiplicity one case and extends to higher multiplicity their dilation approach. We prove several results for operator-valued frames concerning duality, dis- jointeness, complementarity , and composition of operator valued frames and the relationship between the two types of similarity (left and right) of such frames. A key technical tool is the parametrization of Parseval operator val- ued frames in terms of a class of partial isometries in the Hilbert space of the analysis operator. We apply these notions to an analysis of multiframe gener- ators for the action of a discrete group G on a Hilbert space. One of the main results of the Han-Larson work was the parametrization of the Parseval frame generators in terms of the unitary operators in the von Neumann algebra gen- erated by the group representation, and the resulting norm path-connectedness of the set of frame generators due to the connectedness of the group of unitary operators of an arbitrary von Neumann algebra. In this paper we general- ize this multiplicity one result to operator-valued frames. However, both the parameterization and the proof of norm path-connectedness turn out to be necessarily more complicated, and this is at least in part the rationale for this paper. Our parameterization involves a class of partial isometries of a different von Neumann algebra. These partial isometries are not path-connected in the norm topology, but only in the strong operator topology. We prove that the set of operator frame generators is norm pathwise-connected precisely when the von Neumann algebra generated by the right representation of the group has no minimal projections. As in the multiplicity one theory there are analogous results for general (non-Parseval) frames.

In this article we show that there exist measurable sets W⊂ℝ2 with finite measure that tile ℝ2 in a measurable way under the action of a expansive matrix A, an affine Weyl group \(\widetilde{W}\) , and a full rank lattice \(\widetilde{\varGamma}\subset\mathbb{R}^{2}\) . This note is follow-up research to the earlier article “Coxeter groups and wavelet sets” by the first and second authors, and is also relevant to the earlier article “Coxeter groups, wavelets, multiresolution and sampling” by M. Dobrescu and the third author. After writing these two articles, the three authors participated in a workshop at the Banff Center on “Operator methods in fractal analysis, wavelets and dynamical systems,” December 2–7, 2006, organized by O. Bratteli, P. Jorgensen, D. Kribs, G. Olafsson, and S. Silvestrov, and discussed the interrelationships and differences between the articles, and worked on two open problems posed in the Larson-Massopust article. We solved part of Problem 2, including a surprising positive solution to a conjecture that was raised, and we present our results in this article.

We examine some recent results of Bownik on density and connectivity of the wavelet frames. We use orthogonality (strong disjointness) properties of frame and Bessel sequences, and also properties of Bessel multipliers (operators that map wavelet Bessel functions to wavelet Bessel functions). In addition we obtain an asymptotically tight approximation result for wavelet frames.

Let π be a projective unitary representation of a countable group G on a separable Hilbert space H. If the set B π of Bessel vectors for π is dense in H, then for any vector x ∈ H the analysis operator Θ x makes sense as a densely defined operator from B π to l 2 (G)-space. Two vectors x and y are called π-orthogonal if the range spaces of Θ x and Θ y are orthogonal, and they are π-weakly equivalent if the closures of the ranges of Θ x and Θ y are the same. These properties are characterized in terms of the commutant of the representation. It is proved that a natural geometric invariant (the orthogonality index) of the representation agrees with the cyclic multiplicity of the commutant of π(G). These results are then applied to Gabor systems. A sample result is an alternate proof of the known theorem that a Gabor sequence is complete in L 2 (ℝ d ) if and only if the corresponding adjoint Gabor sequence is l 2 -linearly independent. Some other applications are also discussed.

A single dyadic orthonormal wavelet on the plane [math] is a measurable square integrable function [math] whose images under translation along the coordinate axes followed by dilation by positive and negative integral powers of 2 generate an orthonormal basis for [math] . A planar dyadic wavelet set [math] is a measurable subset of [math] with the property that the inverse Fourier transform of the normalized characteristic function [math] of [math] is a single dyadic orthonormal wavelet. While constructive characterizations are known, no algorithm is known for constructing all of them. The purpose of this paper is to construct two new distinct uncountably infinite families of dyadic orthonormal wavelet sets in [math] . We call these the crossover and patch families. Concrete algorithms are given for both constructions.

The purpose of this chapter is to give a detailed tutorial-style exposition of a method we have recently discovered for complete recovery from frame and sampling erasures by inverting a matrix of dimension the cardinality of the erasure set. This is usually simpler and more efficient than inverting the frame operator of the remaining coefficients because the erasure set is usually much smaller than the dimension of the space. The method is not hard and can be easily implemented on a computer, even for infinite frames and sampling schemes as long as one is dealing with only a finite set of erasures. The set of erasures that can be handled in this way can be very large. We call the method nilpotent bridging. We introduced this method in a recent research article, along with some new classification results and methods of measuring redundancy that are based on the matricial infrastructure of a dual frame pair: the set of all submatrices of the cross-Gramian of the pair. The first author gave a talk on this material in a conference at Vanderbilt University in May 2014 entitled “Building bridges (reconstruction from frame and sampling omissions)”, and this chapter is based on the notes from that talk. In the process of developing the mathematics underlying the bridging technique, we discovered a second method for recovery from erasures in finitely many steps, and this chapter will also discuss this alternate method.

It is an open problem whether every one-dimensional extension of a triangular operator admits a separating vector. We prove that the answer is positive for many triangular Hilbert space operators, and in particular, for strictly triangular operators. This is revealing, because two-dimensional extensions of such operators can fail to have separating vectors.

We consider frames arising from the action of a unitary representation of a discrete countable abelian group. We show that the range of the analysis operator can be determined by computing which characters appear in the representation. This allows one to compare the ranges of two such frames, which is useful for determining similarity and also for multiplexing schemes. Our results then partially extend to Bessel sequences arising from the action of the group. We apply the results to sampling on bandlimited functions and to wavelet and Weyl-Heisenberg frames. This yields a sufficient condition for two sampling transforms to have orthogonal ranges, and two analysis operators for wavelet and Weyl-Heisenberg frames to have orthogonal ranges. The sufficient condition is easy to compute in terms of the periodization of the Fourier transform of the frame generators.

We offer a perturbation theory for finite ascent and descent properties of bounded operators.

We introduce the concept of the extension spectrum of a Hilbert space operator. This is a natural subset of the spectrum which plays an essential role in dealing with certain extension properties of operators. We prove that it has spectral-like properties and satisfies a holomorphic version of the Spectral Mapping Theorem. We establish structural theorems for algebraic extensions of triangular operators which use the extension spectrum in a natural way. The extension spectrum has some properties in common with the Kato spectrum, and in the final section we show how they are different and we examine their inclusion relationships.

Emerging intrastate transboundary issues focus on use of the Mahomet aquifer, which underlies about fifteen counties and many other political entities in east-central Illinois. This sand and gravel aquifer in the lower part of the buried Mahomet Bedrock Valley ranges between four and fourteen miles wide and from about 50 to 200 feet thick. Much of the region's rural population, several large communities, and many small towns obtain water from the Mahomet aquifer, as do industrial, agricultural, and commercial users. Increased development of the Mahomet aquifer to meet growing demands for water has caused conflicts over real or perceived adverse effects. One result has been the creation of fifteen resource protection zones and twelve water authorities. For groundwater supplies, resource protection zones help municipalities protect water-supply wells from potential adverse impacts. Many resource protection zones overlap one another, however, so this situation could lead to disputes over use of the ...

We give purely algebraic characterizations of the maps that are approximate compressions or skew-compressions of a unital representation of a C ∗ -algebra. The key techniques used also relate to closures of joint similarity orbits of matrices and elementary maps on B(H).

Deposition from at least three episodes of glaciation left a complex glacial-drift aquifer system in central Illinois. The deepest and largest of these aquifers, the Sankoty-Mahomet Aquifer, occupies the lower part of a buried bedrock valley and supplies water to communities throughout central Illinois. Thin, discontinuous aquifers are present within glacial drift overlying the Sankoty-Mahomet Aquifer. This study was commissioned by local governments to identify possible areas where a regional water supply could be obtained from the aquifer with minimal adverse impacts on existing users. Geologic information from more than 2200 existing water well logs was supplemented with new data from 28 test borings, water level measurements in 430 wells, and 35 km of surface geophysical profiles. A three-dimensional (3-D) hydrostratigraphic model was developed using a contouring software package, a geographic information system (GIS), and the 3-D geologic modeling package, Earth Vision®. The hydrostratigraphy of the glacial-drift sequence was depicted as seven uneven and discontinuous layers, which could be viewed from an infinite number of horizontal and vertical slices and as solid models of any layer. Several iterations were required before the 3-D model presented a reasonable depiction of the aquifer system. Layers from the resultant hydrostratigraphic model were imported into MODFLOW, where they were modified into continuous layers. This approach of developing a 3-D hydrostratigraphic model can be applied to other areas where complex aquifer systems are to be modeled and is also useful in helping lay audiences visualize aquifer systems.

We give some new methods for perfect reconstruction from frame and sampling erasures in a small number of steps. By bridging an erasure set we mean replacing the erased Fourier coefficients of a function with respect to a frame by appropriate linear combinations of the non-erased coefficients. We prove that if a minimal redundancy condition is satisfied bridging can always be done to make the reduced error operator nilpotent of index 2 using a bridge set of indices no larger than the cardinality of the erasure set. This results in perfect reconstruction of the erased coefficients. We also obtain a new formula for the inverse of an invertible partial reconstruction operator. This leads to a second method of perfect reconstruction from frame and sampling erasures in a small number of steps. This gives an alternative to the bridging method for many (but not all) cases. The methods we use employ matrix techniques only of the order of the cardinality of the erasure set, and are applicable to rather large finite erasure sets for infinite frames and sampling schemes as well as for finite frame theory. These methods are usually more efficient than inverting the frame operator for the remaining coefficients because the size of the erasure set is usually much smaller than the dimension of the underlying Hilbert space. Some new classification theorems for frames are obtained and some new methods of measuring redundancy are introduced based on our bridging theory.

Groundwater source heat pumps exploit the difference between the ground surface temperature and the nearly constant temperature of shallow groundwater. This project characterizes two areas for geothermal heating and cooling potential, Mason County in central Illinois and the American Bottoms area in southwestern Illinois. Both areas are underlain by thick sand and gravel aquifers and groundwater is readily available. Weather data, including monthly high and low temperatures and heating and cooling degree days, were compiled for both study areas. The heating and cooling requirements for a single-family house were estimated using two independent models that use weather data as input. The groundwater flow rates needed to meet these heating and cooling requirements were calculated using typical heat pump coefficient of performance values. The groundwater in both study areas has fairly high hardness and iron concentrations and is close to saturation with calcium and iron carbonates. Using the groundwater for cooling may induce the deposition of scale containing one or both of these minerals.Copyright © 2014 by ASME

A wandering vector multiplier is a unitary operator which maps the set of wandering vectors for a unitary system into itself. A special case of unitary system is a discrete unitary group. We prove that for many (and perhaps all) discrete unitary groups, the set of wandering vector multipliers is itself a group. We completely characterize the wandering vector multipliers for abelian and ICC unitary groups. Some characterizations of special wandering vector multipliers are obtained for other cases. In particular, there are simple characterizations for diagonal and permutation wandering vector multipliers. Similar results remain valid for irrational rotation unitary systems. We also obtain some results concerning the wandering vector multipliers for those unitary systems which are the ordered products of two unitary groups. There are applications to wavelet systems.

It is proved that associated with every wavelet set is a closely related “regularized” wavelet set which has very nice properties. Then it is shown that for many (and perhaps all) pairs E, F, of wavelet sets, the corresponding MSF wavelets can be connected by a continuous path in L2(ℝ) of MSF wavelets for which the Fourier transform has support contained in E ∪ F. Our technique applies, in particular, to the Shannon and Journe wavelet sets.

We consider systems of unitary operators on the complex Hilbert space L 2 ( R n ) of the form U ≔ U D A , T v 1 , …, T v n ≔{ D m T l 1 v 1 … T l n v n : m , l 1 , …, l n ∈ Z }, where D A is the unitary operator corresponding to dilation by an n × n real invertible matrix A and T v 1 , …, T v n are the unitary operators corresponding to translations by the vectors in a basis { v 1 , …, v n } for R n . Orthonormal wavelets ψ are vectors in L 2 ( R n ) which are complete wandering vectors for U in the sense that { U ψ : U ∈ U } is an orthonormal basis for L 2 ( R n ). It has recently been established that whenever A has the property that all of its eigenvalues have absolute values strictly greater than one (the expansive case) then U has orthonormal wavelets. The purpose of this paper is to determine when two ( n +1)-tuples of the form ( D A , T v 1 , …, T v n ) give rise to the “same wavelet theory.” In other words, when is there a unitary transformation of the underlying Hilbert space that transforms one of these unitary systems onto the other? We show, in particular, that two systems U D A , T e i , U D B , T e i , each corresponding to translation along the coordinate axes, are unitarily equivalent if and only if there is a matrix C with integer entries and determinant ±1 such that B = C −1 AC . This means that different expansive dilation factors nearly always yield unitarily inequivalent wavelet theories. Along the way we establish necessary and sufficient conditions for an invertible real n × n matrix A to have the property that the dilation unitary operator D A is a bilateral shift of infinite multiplicity.

A congruency theorem is proven for an ordered pair of groups of homeomorphisms of a metric space satisfying an abstract dilation-translation relationship. A corollary is the existence of wavelet sets, and hence of single-function wavelets, for arbitrary expansive matrix dilations on L 2 (ℝ n). Moreover, for any expansive matrix dilation, it is proven that there are sufficiently many wavelet sets to generate the Borel structure ofℝ n.

We examine the strongly extreme point structure of the unit balls of triangular UHF algebras. The semisimple triangular UHF algebras are characterized as those for which this structure is minimal in the sense that every strongly extreme point belongs to the diagonal. In contrast to this, for the class of full nest algebras we prove a Krein-Milman type theorem which asserts that every operator in the open unit ball of the algebra is a convex combination of strongly extreme points. Results concerning the geometry of the unit ball have a long history both in Banach space theory and in the theory of operator algebras. This geometry can be affiliated with structural and algebraic properties. Moreover, differences in geometric properties can prove useful in classification problems. Two fundamental results are the Russo-Dye Theorem and Kadison’s Theorem on isometries. More recently, there has been interest in the unit balls of nonselfadjoint operator algebras, especially nest algebras [1, 2, 3, 4, 15, 16]. This paper concerns the unit balls of triangular UHF algebras. These and the larger class of triangular AF algebras are nonselfadjoint analogues of the UHF and AF C*-algebras studied by Glimm and Bratteli. Their theory has grown rapidly, cf. [8, 9, 18, 19, 20]. We focus on the extreme point structure, and our results have a different flavor than those for nest algebras. Specifically, we study the strongly extreme points, those boundary points whose “stable character” with respect to approximations makes them behave well under direct limits. Triangular UHF algebras are direct limits of full upper triangular matrix algebras. Unit balls embed into unit balls in the direct limit scheme, and some types of embeddings respect the extreme point structure while others do not. This leads to structural differences in the limit algebras. The geometric structures of the unit balls of different triangular UHF algebras can be very dissimilar. The convex hull of the strongly extreme points, even without closure, always contains the unit ball of the diagonal. Theorem 7 shows that the two coincide if and only if the algebra is semisimple. This is a characterization of a purely geometric property in terms of a purely algebraic one. In contrast to Received by the editors January 11, 1996 and, in revised form, March 28, 1996. 1991 Mathematics Subject Classification. Primary 47D25, 46K50, 46B20.

We prove that every one-dimensional extension of a bitriangular operator has a cyclic commutant. We also prove that ifT is an extension of a bitriangular operator by an algebraic operator, then the weakly closed algebraW(T) generated byT has a separating vector.

We prove that every one dimensional extension of a separably acting normal operator has a cyclic commutant, and that every non-algebraic normal operator has a two-dimensional extension which fails to have a cyclic commutant. Contrasting this, we prove that ifT is an extension of a normal operator by an algebraic operator then the weakly closed algebraW(T) has a separating vector.

A Bessel generator multiplier for a unitary system is a bounded linear operator that sends Bessel generator vectors to Bessel generator vectors. We characterize some special (but useful) Bessel generator multipliers for the unitary systems that are ordered products of two unitary groups. These include the wavelet and the Gabor unitary systems. We also provide a detailed exposition of some of the history leading up to this work.

We focus on the irreducibility of wavelet representations. We present some connections between the following notions: covariant wavelet representations, ergodic shifts on solenoids, fixed points of transfer (Ruelle) operators and solutions of refinement equations. We investigate the irreducibility of the wavelet representations, in particular the representation associated to the Cantor set, introduced in \cite{DuJo06b}, and we present several equivalent formulations of the problem.

Arveson has shown in [ 1 ] that if A is a nest algebra of operators acting on a separable Hilbert space H with nest of order type the extended natural numbers and if K(H) is the ideal of compact operators on H, then the quasitriangular algebra A + K(H) is norm closed. In [6] Fall, Arveson, and Muhly extended Arveson’s result proving that in B(H), general quasitriangular algebras (i.e., compact perturbations of nest algebras) are always norm closed. A key step in their proof, and a result of independent interest, first obtained by Erdos in [S], is the a-weak density of A n K(H) in A. From this fact they deduced that quasitriangular algebras were norm closed by using an argument depending on the duality K(H)** = B(H). Every semifinite von Neumann algebra M has an ideal of compact operators that behaves like K(H), namely the norm-closed two-sided ideal K generated by the finite projections of the algebra. It is a natural question

The algebraic equivalence and similarity classes of idempotents within a nest algebra Alg β are completely characterized. We obtain necessary and sufficient conditions for two idempotents to be equivalent or similar. Our criterion yields examples illustrating pathology and also shows that to each equivalence class of idempotents there corresponds a “dimension function” from β × β into N ∪ { ∞ }. We complete the characterization of the algebraic equivalence classes by proving that any dimension function corresponds to an equivalence class of idempotents. Also, to each sequence of dimension functions, there corresponds a commuting sequence of idempotents. A criterion is obtained for when an idempotent is similar to a subidempotent of another. The mapping which sends an equivalence class (or idempotent) to its associated dimension function plays a role in the nest algebra theory analogous to the role played by the mapping sending a projection in a Type I W ∗ -algebra to its center valued trace. We prove that almost commuting, similar idempotents are homotopic; this contrasts with the situation in certain C ∗ -algebras. Using this, we show that similar, simultaneously diagonalizable idempotents are homotopic, and in the continuous nest case, every diagonal idempotent is homotopic to a core projection.

It is proved that if an interpolation map between two wavelet sets preserves the union of the sets, then the pair must be an interpolation pair. We also construct an example of a pair of wavelet sets for which the congruence domains of the associated interpolation map and its inverse are equal, and yet the pair is not an interpolation pair. The first result solves affirmatively a problem that the second author had posed several years ago, and the second result solves an intriguing problem of D. Han. The key to this counterexample is a special technical lemma on constructing wavelet sets. Several other applications of this result are also given. In addition, some problems are posed. We also take the opportunity to give some general exposition on wavelet sets and operator-theoretic interpolation of wavelets.

Introduction This resource contains a focused, 2.5 hour interactive session explicitly designed to orient new housestaff to the six ACGME General Competencies, and address key topics related to risk management and do-not-resuscitate (DNR) orders. Methods Trained facilitators (residency program directors, associate program directors, or faculty) guide discussions in small, six-person groups within their own classroom of housestaff. In the small groups, housestaff read a scenario about an intern who must respond in a difficult situation. They then answer questions about the scenario including what they should do if they were the intern, what resources are available, and how and where the ACGME General Competencies appear in this situation. After small groups discuss a scenario, facilitators led a discussion with the large group (whole classroom) about the answers to the scenario questions. Results Three data sets highlight the success of this program. Results of a multiple-choice pre- and posttest requiring residents to identify the ACGME competencies associated with 20 statements revealed that housestaff arrived with a general understanding of the competencies and that this session enhances that comprehension. Mean percentages for the housestaff scores changed from 76% to 81% (p < .001) following the session. Results from the housestaff evaluation of the session revealed that they enjoyed the case-based discussions, citing the opportunities to interact in small groups, discuss relevant and interesting scenarios, and learning the ACGME General Competencies. Faculty facilitators evaluated the sessions highly and volunteered to serve as facilitators the following year. Discussion The strength of this overall approach is that the length of the session may be reduced or lengthened dependent on the degree to which objectives have been achieved, and facilitators can change the content or characters in the scenarios to enable the discussion to continue in a related direction. Also, housestaff may want to contribute their own experiences to the discussions, especially new fellows. It was decided not to separate the groups by level of experience, as the fellows would likely add more to the discussions when discussing the scenarios with new residents. In some cases, fellows were able to say, “This actually happened to me and here's what I did …” providing valuable insight for the new residents in their group.