Inner Derivations and Primal Ideals of C*-Algebras

Let A be a C*-algebra. For a A let D(a, A) denote the innerderivation induced by a, regarded as a bounded operator on A,and let d(a, Z(A)) denote the distance of a from Z(A), the centreof A. Let K(A) be the smallest number in [0, ] such that d(a,Z(A)) K(A)||D(a, A)|| for all a A. It is shown that if A isnon-commutative and has an identity then either K(A) = , or K(A) = 1 / 3, or K(A) 1. Necessaryand sufficient conditions for these three possibilities aregiven in terms of the primitive and primal ideals of A. If Ais a quotient of an AW*-algebra then K(A) . Helly's Theorem is used to show that if A is aweakly central C*-algebra then K(A) 1.     

Separation properties in the primitive ideal space of a multiplier algebra

We use recent work on spectral synthesis in multiplier algebras to give an intrinsic characterization of the separable C*-algebras A for which Orc(M(A)) = 1, i.e., for which the relation of inseparability on the topological space of primitive ideals of the multiplier algebra M(A) is an equivalence relation. This characterization has applications to the calculation of norms of inner derivations and other elementary operators on A and M(A). For example, we give necessary and sufficient conditions on the ideal structure of a separable C*-algebra A for the norm of every inner derivation to be twice the distance of the implementing element to the centre of M(A).     

Electrochemical and Spectroscopic Properties of Fly Ash–Polyaniline Matrix Nanorod Composites

Polyaniline (PANI) nanocomposites were prepared with fly ash (FA) either by aging the starting materials (aniline and FA) before oxidative polymerisation or by including poly(styrene sulphonic acid) (PSSA) eliminating the aging step. The aging procedure formed polymer nanotubes that have cross-sectional diameters of 50–110 nm. The procedure involving PSSA produced nanorods and nanofibres composites that have diameters of 100–500 nm and length of up to 10 μm attributed to the presence of metal oxides and silica in FA. The electrochemical analysis of the PANI–PSSA–FA nanorod composites shows three redox couples with formal potentials, , values of 105 mV, 455 mV and 670 mV, and conductance, C, value of 1.21 × 10⁻² S. The UV-Vis spectroscopy of the polymeric nanorod shows absorption maxima at 340 and 370 nm (due to π–π^* transition of the benzoid rings), and 600–650 nm (due to charge transfer excitons of the quinoid structure), which are characteristic of emeraldine base.     

Spectroelectrochemical Reactivities of Novel Polyaniline Nanotube Pesticide Biosensors

Summary: The preparation and characterisation of electrosynthetic polyaniline nanomaterials doped with phenanthrene sulphonic acid (PSA) is being presented. The polymeric nanomaterials prepared include processable poly(o-methoxy aniline) (POMA) and poly(2,5-dimethoxy aniline) (PDMA). Spectroelectrochemical reactivities of the electroactive polymeric nanotube systems as well as the nanobiosensor systems were studied by SEM, FTIR, UV-Vis and Subtractively Normalised Fourier Transformed Infrared Spectroscopy (SNIFTIRS) techniques. Furthermore, cyclic and differential pulse voltammetric studies of the nanomaterials were also performed using platinum or thiol-modulated gold electrodes. The SEM studies confirmed the nanorod morphology of the polyanilines. The heterogeneous rate constant, k_o, for the nanopolymeric material and the diffusion coefficient of electrons, D_e, was calculated and found to be in agreement with values expected for electron hopping along conducting polymer chains. Organophosphate pesticide nanobiosensor devices were prepared by encapsulating acetylcholinesterase (AChE) in the nanopolymeric composite. The biosensor amperometric response to the organophosphate pesticide called diazinon and the carbamate pesticide called carbofuran were studied. The sensor responses to pesticides followed typical electrochemical Michaelis-Menten kinetics.     

The use of X-ray fluorescence (XRF) analysis in predicting the alkaline hydrothermal conversion of fly ash precipitates into zeolites

The use and application of synthetic zeolites for ion exchange, adsorption and catalysis has shown enormous potential in industry. In this study, X-ray fluorescence (XRF) analysis was used to determine Si and Al in fly ash (FA) precipitates. The Si and Al contents of the fly ash precipitates were used as indices for the alkaline hydrothermal conversion of the fly ash compounds into zeolites. Precipitates were collected by using a co-disposal reaction wherein fly ash is reacted with acid mine drainage (AMD). These co-disposal precipitates were then analysed by XRF spectrometry for quantitative determination of SiO₂ and Al₂O₃. The [SiO₂]/[Al₂O₃] ratio obtained in the precipitates range from 1.4 to 2.5. The [SiO₂]/[Al₂O₃] ratio was used to predict whether the fly ash precipitates could successfully be converted to faujasite zeolitic material by the synthetic method of [J. Haz. Mat. B 77 (2000) 123]. If the [SiO₂]/[Al₂O₃] ratio is higher than 1.5 in the fly ash precipitates, it favours the formation of faujasite. The zeolite synthesis included an alkaline hydrothermal conversion of the co-disposal precipitates, followed by aging for 8 h and crystallization at 100 °C. Different factors were investigated during the synthesis of zeolite to ascertain their influence on the end product. The factors included the amount of water in the starting material, composition of fly ash related starting material and the FA:NaOH ratio used for fusing the starting material. The mineralogical and physical analysis of the zeolitic material produced was performed by X-ray diffraction (XRD) and nitrogen Brunauer-Emmett-Teller (N₂ BET) surface analysis. Scanning electron microscopy (SEM) was used to determine the morphology of the zeolites, while inductively coupled mass spectrometry (ICP-MS), Fourier transformed infrared spectrometry (FT-IR) and Cation exchange capacity (CEC) [Report to Water Research Commission, RSA (2003) 15] techniques were used for chemical characterisation. The heavy and trace metal concentrations of the zeolite products were compared to that of the post-synthesis filtrate and of the precipitate materials used as Si and Al feed stock for zeolite formation, in order to determine the trends (increase or decrease) and ultimate fate of any toxic metals incorporated in the co-disposed precipitated residues.     

A Dauns-Hofmann theorem for TAF-algebras

Let be a TAF-algebra, the centre of the ideal lattice of , and the space of meet-irreducible elements of , equipped with the hull-kernel topology. It is shown that is a compact, locally compact, second countable, -space, that is an algebraic lattice isomorphic to the lattice of open subsets of , and that is isomorphic to the algebra of continuous, complex functions on . If is semisimple, then is isomorphic to the algebra of continuous, complex functions on , the primitive ideal space of . If is strongly maximal, then the sum of two closed ideals of is closed.

     

Spectral Synthesis and Topologies on Ideal Spaces for Banach*-Algebras

This paper continues the study of spectral synthesis and the topologies τ_∞ and τ_r on the ideal space of a Banach algebra, concentrating on the class of Banach ^*-algebras, and in particular on L¹-group algebras. It is shown that if a group G is a finite extension of an abelian group then τ_r is Hausdorff on the ideal space of L¹(G) if and only if L¹(G) has spectral synthesis, which in turn is equivalent to G being compact. The result is applied to nilpotent groups, [FD]⁻-groups, and Moore groups. An example is given of a non-compact, non-abelian group G for which L¹(G) has spectral synthesis. It is also shown that if G is a non-discrete group then τ_r is not Hausdorff on the ideal lattice of the Fourier algebra A(G).     

Spectral synthesis for Banach algebras, II

This paper continues the study of spectral synthesis and the topologies and 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 . Under a weak extra condition, spectral synthesis is shown to be equivalent to the Hausdorffness of . Received: 6 October 1999; in final form: 15 May 2000 / Published online: 25 June 2001     

Elementary operators on antiliminal C^*-algebras

We establish the equivalence of the following three properties of a -algebra A. (a) Every positive elementary operator on A is completely positive. (b) The norm and the cb-norm coincide for every elementary operator on A. (c) A is an extension of an antiliminal -algebra by an abelian one. Received: 15 July 1998 / in revised form: 22 September 1998