High-dimensional Z-stable AH algebras

Peer reviewed

Multidimensional operator multipliers

We introduce multidimensional Schur multipliers and characterise them, generalising well-known results by Grothendieck and Peller. We define a multidimensional version of the two-dimensional operator multipliers studied recently by Kissin and Shulman. The multidimensional operator multipliers are defined as elements of the minimal tensor product of several C *-algebras satisfying certain boundedness conditions. In the case of commutative C*-algebras, the multidimensional operator multipliersreduce to continuousmul-tidimensional Schur multipliers. We show that the multiplierswith respect to some given representations of the corresponding C*-algebrasdo not change if the representations are replaced by approximately equivalent ones. We establish a non-commutative and multidimensional version of the characterisations by Grothendieck and Peller which shows that universal operator multipliers can be obtained ascertain weak limits of elements of the algebraic tensor product of the corresponding C *-algebras.

The C*-algebra of ax+b-like groups

We describe the C*-algebras of " ax+b" -like groups in terms of algebras of operator fields defined over their dual spaces.

The solvable Lie group N_{6, 28}: an example of an almost C_0(\mathcal{K})-C*-algebra

Motivated by the description of the C*-algebra of the affine automorphism group N6,28 of the Siegel upper half-plane of degree 2 as an algebra of operator fields defined over the unitary dual View the MathML source of the group, we introduce a family of C*-algebras, which we call almost C0(K), and we show that the C*-algebra of the group N6,28 belongs to this class.

INTEGRAL FUNCTIONALS ON C*-ALGEBRA OF VECTOR-VALUED REGULATED FUNCTIONS

In this paper we deal with the notion of regulated functions with values in a C*-algebra A and present examples using a special bi-dimensional C*-algebra of triangular matrices. We consider the Dushnik integral for these functions and shows that a convenient choice of the integrator function produces an integral homomorphism on the C*-algebra of all regulated functions ([a, b], A). Finally we construct a family of linear integral functionals on this C*-algebra.

Toeplitz operators on finite and infinite dimensional spaces with associated psi *-Fréchet algebras

The present thesis is a contribution to the multi-variable theory of Bergman and Hardy Toeplitz operators on spaces of holomorphic functions over finite and infinite dimensional domains. In particular, we focus on certain spectral invariant Frechet operator algebras F closely related to the local symbol behavior of Toeplitz operators in F. We summarize results due to B. Gramsch et.al. on the construction of Psi_0- and Psi^*-algebras in operator algebras and corresponding scales of generalized Sobolev spaces using commutator methods, generalized Laplacians and strongly continuous group actions. In the case of the Segal-Bargmann space H^2(C^n,m) of Gaussian square integrable entire functions on C^n we determine a class of vector-fields Y(C^n) supported in complex cones K. Further, we require that for any finite subset V of Y(C^n) the Toeplitz projection P is a smooth element in the Psi_0-algebra constructed by commutator methods with respect to V. As a result we obtain Psi_0- and Psi^*-operator algebras F localized in cones K. It is an immediate consequence that F contains all Toeplitz operators T_f with a symbol f of certain regularity in an open neighborhood of K. There is a natural unitary group action on H^2(C^n,m) which is induced by weighted shifts and unitary groups on C^n. We examine the corresponding Psi^*-algebra A of smooth elements in Toeplitz-C^*-algebras. Among other results sufficient conditions on the symbol f for T_f to belong to A are given in terms of estimates on its Berezin-transform. Local aspects of the Szegö projection P_s on the Heisenbeg group and the corresponding Toeplitz operators T_f with symbol f are studied. In this connection we apply a result due to Nagel and Stein which states that for any strictly pseudo-convex domain U the projection P_s is a pseudodifferential operator of exotic type (1/2, 1/2). The second part of this thesis is devoted to the infinite dimensional theory of Bergman and Hardy spaces and the corresponding Toeplitz operators. We give a new proof of a result observed by Boland and Waelbroeck. Namely, that the space of all holomorphic functions H(U) on an open subset U of a DFN-space (dual Frechet nuclear space) is a FN-space (Frechet nuclear space) equipped with the compact open topology. Using the nuclearity of H(U) we obtain Cauchy-Weil-type integral formulas for closed subalgebras A in H_b(U), the space of all bounded holomorphic functions on U, where A separates points. Further, we prove the existence of Hardy spaces of holomorphic functions on U corresponding to the abstract Shilov boundary S_A of A and with respect to a suitable boundary measure on S_A. Finally, for a domain U in a DFN-space or a polish spaces we consider the symmetrizations m_s of measures m on U by suitable representations of a group G in the group of homeomorphisms on U. In particular,in the case where m leads to Bergman spaces of holomorphic functions on U, the group G is compact and the representation is continuous we show that m_s defines a Bergman space of holomorphic functions on U as well. This leads to unitary group representations of G on L^p- and Bergman spaces inducing operator algebras of smooth elements related to the symmetries of U.

Pseudodifferential analysis in Psi*-algebras on transmission spaces, infinite solving ideal chains and K-theory for conformally compact spaces

The present thesis is a contribution to the theory of algebras of pseudodifferential operators on singular settings. In particular, we focus on the $b$-calculus and the calculus on conformally compact spaces in the sense of Mazzeo and Melrose in connection with the notion of spectral invariant transmission operator algebras. We summarize results given by Gramsch et. al. on the construction of $Psi_0$-and $Psi*$-algebras and the corresponding scales of generalized Sobolev spaces using commutators of certain closed operators and derivations. In the case of a manifold with corners $Z$ we construct a $Psi*$-completion $A_b(Z,{}^bOmega^{1/2})$ of the algebra of zero order $b$-pseudodifferential operators $Psi_{b,cl}(Z, {}^bOmega^{1/2})$ in the corresponding $C*$-closure $B(Z,{}^bOmega^{12})hookrightarrow L(L^2(Z,{}^bOmega^{1/2}))$. The construction will also provide that localised to the (smooth) interior of Z the operators in the $A_b(Z, {}^bOmega^{1/2})$ can be represented as ordinary pseudodifferential operators. In connection with the notion of solvable $C*$-algebras - introduced by Dynin - we calculate the length of the $C*$-closure of $Psi_{b,cl}^0(F,{}^bOmega^{1/2},R^{E(F)})$ in $B(F,{}^bOmega^{1/2}),R^{E(F)})$ by localizing $B(Z, {}^bOmega^{1/2})$ along the boundary face $F$ using the (extended) indical familiy $I^B_{FZ}$. Moreover, we discuss how one can localise a certain solving ideal chain of $B(Z, {}^bOmega^{1/2})$ in neighbourhoods $U_p$ of arbitrary points $pin Z$. This localisation process will recover the singular structure of $U_p$; further, the induced length function $l_p$ is shown to be upper semi-continuous. We give construction methods for $Psi*$- and $C*$-algebras admitting only infinite long solving ideal chains. These algebras will first be realized as unconnected direct sums of (solvable) $C*$-algebras and then refined such that the resulting algebras have arcwise connected spaces of one dimensional representations. In addition, we recall the notion of transmission algebras on manifolds with corners $(Z_i)_{iin N}$ following an idea of Ali Mehmeti, Gramsch et. al. Thereby, we connect the underlying $C^infty$-function spaces using point evaluations in the smooth parts of the $Z_i$ and use generalized Laplacians to generate an appropriate scale of Sobolev spaces. Moreover, it is possible to associate generalized (solving) ideal chains to these algebras, such that to every $ninN$ there exists an ideal chain of length $n$ within the algebra. Finally, we discuss the $K$-theory for algebras of pseudodifferential operators on conformally compact manifolds $X$ and give an index theorem for these operators. In addition, we prove that the Dirac-operator associated to the metric of a conformally compact manifold $X$ is not a Fredholm operator.

Orientation in operator algebras

A concept of orientation is relevant for the passage from Jordan structure to associative structure in operator algebras. The research reported in this paper bridges the approach of Connes for von Neumann algebras and ourselves for C*-algebras in a general theory of orientation that is of geometric nature and is related to dynamics.

Spectral synthesis for Banach Algebras II

This paper continues the study of spectral synthesis and the topologies tau-infinity and tau-r on the ideal space of a Banach algebra, concentrating particularly on the class of Haagerup tensor products of C*-algebras. For this class, it is shown that spectral synthesis is equivalent to the Hausdorffness of tau_infinity. Under a weak extra condition, spectral synthesis is shown to be equivalent to the Hausdorffness of tau_r.

Reconstruction of graded groupoids from graded Steinberg algebras

We show how to reconstruct a graded ample Hausdorff groupoid with topologically principal neutrally-graded component from the ring structure of its graded Steinberg algebra over any commutative integral domain with 1, together with the embedding of the canonical abelian subring of functions supported on the unit space. We deduce that diagonal-preserving ring isomorphism of Leavitt path algebras implies $C^*$-isomorphism of $C^*$-algebras for graphs $E$ and $F$ in which every cycle has an exit. This is a joint work with Joan Bosa, Roozbeh Hazrat and Aidan Sims.

The influence of ambient temperature on birth outcomes in Brisbane, Australia

Lately, there has been increasing interest in the association between temperature and adverse birth outcomes including preterm birth (PTB) and stillbirth. PTB is a major predictor of many diseases later in life, and stillbirth is a devastating event for parents and families. The aim of this study was to assess the seasonal pattern of adverse birth outcomes, and to examine possible associations of maternal exposure to temperature with PTB and stillbirth. We also aimed to identify if there were any periods of the pregnancy where exposure to temperature was particularly harmful. A retrospective cohort study design was used and we retrieved individual birth records from the Queensland Health Perinatal Data Collection Unit for all singleton births (excluding twins and triplets) delivered in Brisbane between 1 July 2005 and 30 June 2009. We obtained weather data (including hourly relative humidity, minimum and maximum temperature) and air-pollution data (including PM10, SO2 and O3) from the Queensland Department of Environment and Resource Management. We used survival analyses with the time-dependent variables of temperature, humidity and air pollution, and the competing risks of stillbirth and live birth. To assess the monthly pattern of the birth outcomes, we fitted month of pregnancy as a time-dependent variable. We examined the seasonal pattern of the birth outcomes and the relationship between exposure to high or low temperatures and birth outcomes over the four lag weeks before birth. We further stratified by categorisation of PTB: extreme PTB (< 28 weeks of gestation), PTB (28–36 weeks of gestation), and term birth (≥ 37 weeks of gestation). Lastly, we examined the effect of temperature variation in each week of the pregnancy on birth outcomes. There was a bimodal seasonal pattern in gestation length. After adjusting for temperature, the seasonal pattern changed from bimodal, to only one peak in winter. The risk of stillbirth was statistically significant lower in March compared with January. After adjusting for temperature, the March trough was still statistically significant and there was a peak in risk (not statistically significant) in winter. There was an acute effect of temperature on gestational age and stillbirth with a shortened gestation for increasing temperature from 15 °C to 25 °C over the last four weeks before birth. For stillbirth, we found an increasing risk with increasing temperatures from 12 °C to approximately 20 °C, and no change in risk at temperatures above 20 °C. Certain periods of the pregnancy were more vulnerable to temperature variation. The risk of PTB (28–36 weeks of gestation) increased as temperatures increased above 21 °C. For stillbirth, the fetus was most vulnerable at less than 28 weeks of gestation, but there were also effects in 28–36 weeks of gestation. For fetuses of more than 37 weeks of gestation, increasing temperatures did not increase the risk of stillbirth. We did not find any adverse affects of cold temperature on birth outcomes in this cohort. My findings contribute to knowledge of the relationship between temperature and birth outcomes. In the context of climate change, this is particularly important. The results may have implications for public health policy and planning, as they indicate that pregnant women would decrease their risk of adverse birth outcomes by avoiding exposure to high temperatures and seeking cool environments during hot days.

Comparison of antigen detection and quantitative PCR in the detection of chlamydial infection in koalas (Phascolarctos cinereus)

The gold standard method for detecting chlamydial infection in domestic and wild animals is PCR, but the technique is not suited to testing animals in the field when a rapid diagnosis is frequently required. The objective of this study was to compare the results of a commercially available enzyme immunoassay test for Chlamydia against a quantitative Chlamydia pecorum-specific PCR performed on swabs collected from the conjunctival sac, nasal cavity and urogenital sinuses of naturally infected koalas (Phascolarctos cinereus). The level of agreement for positive results between the two assays was low (43.2%). The immunoassay detection cut-off was determined as approximately 400 C. pecorum copies, indicating that the test was sufficiently sensitive to be used for the rapid diagnosis of active chlamydial infections.

Properties of intramuscular connective tissue in pork and poultry with reference to weakening of structure

The most common connective tissue research in meat science has been conducted on the properties of intramuscular connective tissue (IMCT) in connection with eating quality of meat. From the chemical and physical properties of meat, researchers have concluded that meat from animals younger than physiological maturity is the most tender. In pork and poultry, different challenges have been raised: the structure of cooked meat has weakened. In extreme cases raw porcine M. semimembranosus (SM) and in most turkey M. pectoralis superficialis (PS) can be peeled off in strips along the perimysium which surrounds the muscle fibre bundles (destructured meat), and when cooked, the slices disintegrate. Raw chicken meat is generally very soft and when cooked, it can even be mushy. The overall aim of this thesis was to study the thermal properties of IMCT in porcine SM in order to see if these properties were in association with destructured meat in pork and to characterise IMCT in poultry PS. First a 'baseline' study to characterise the thermal stability of IMCT in light coloured (SM and M. longissimus dorsi in pigs and PS in poultry) and dark coloured (M. infraspinatus in pigs and a combination of M. quadriceps femoris and M. iliotibialis lateralis in poultry) muscles was necessary. Thereafter, it was investigated whether the properties of muscle fibres differed in destructured and normal porcine muscles. Collagen content and also solubility of dark coloured muscles were higher than in light coloured muscles in pork and poultry. Collagen solubility was especially high in chicken muscles, approx. 30 %, in comparison to porcine and turkey muscles. However, collagen content and solubility were similar in destructured and normal porcine SM muscles. Thermal shrinkage of IMCT occurred at approximately 65 °C in pork and poultry. It occurred at lower temperature in light coloured muscles than in dark coloured muscles, although the difference was not always significant. The onset and peak temperatures of thermal shrinkage of IMCT were lower in destructured than in normal SM muscles, when the IMCT from SM muscles exhibiting ten lowest and ten highest ultimate pH values were investigated (onset: 59.4 °C vs. 60.7 °C, peak: 64.9 °C vs. 65.7 °C). As the destructured meat was paler than normal meat, the PSE (pale, soft, exudative) phenomenon could not be ruled out. The muscle fibre cross sectional area (CSA), the number of capillaries per muscle fibre CSA and per fibre and sarcomere length were similar in destructured and normal SM muscles. Drip loss was clearly higher in destructured than in normal SM muscles. In conclusion, collagen content and solubility and thermal shrinkage temperature vary between porcine and poultry muscles. One feature in the IMCT could not be directly associated with weakening of the meat structure. Poultry breast meat is very homogenous within the species.

Running-in wear of a compression ignition engine: Factors influencing the conformance between cylinder liner and piston rings

The conformance between the liner and rings of an internal combustion engine depends mainly on their linear wear (dimensional loss) during running-in. Running-in wear studies, using the factorial design of experiments, on a compression ignition engine show that at certain dead centre locations of piston rings the linear wear of the cylinder liner increases with increase in the initial surface roughness of the liner. Rough surfaces wear rapidly without seizure during running-in to promote quick conformance, so an initial surface finish of the liner of 0.8 μm c.l.a. is recommended. The linear wear of the cast iron liner and rings decreases with increasing load but the mass wear increases with increasing load. This discrepancy is due to phase changes in the cast iron accompanied by dimensional growth at higher thermal loads. During running-in the growth of cast iron should be minimised by running the engine at an initial load for which the exhaust gas temperature is approximately 180 °C.

Coherent states on Hilbert modules

We generalize the concept of coherent states, traditionally defined as special families of vectors on Hilbert spaces, to Hilbert modules. We show that Hilbert modules over C*-algebras are the natural settings for a generalization of coherent states defined on Hilbert spaces. We consider those Hilbert C*-modules which have a natural left action from another C*-algebra, say A. The coherent states are well defined in this case and they behave well with respect to the left action by A. Certain classical objects like the Cuntz algebra are related to specific examples of coherent states. Finally we show that coherent states on modules give rise to a completely positive definite kernel between two C*-algebras, in complete analogy to the Hilbert space situation. Related to this, there is a dilation result for positive operator-valued measures, in the sense of Naimark. A number of examples are worked out to illustrate the theory. Some possible physical applications are also mentioned.