Arens Product Research Articles

On topological centre problems and SIN quantum groups

Let A be a Banach algebra with a faithful multiplication and 〈 A ∗ A 〉 ∗ be the quotient Banach algebra of A ∗ ∗ with the left Arens product. We introduce a natural Banach algebra, which is a closed subspace of 〈 A ∗ A 〉 ∗ but equipped with a distinct multiplication. With the help of this Banach algebra, new characterizations of the topological centre Z t ( 〈 A ∗ A 〉 ∗ ) of 〈 A ∗ A 〉 ∗ are obtained, and a characterization of Z t ( 〈 A ∗ A 〉 ∗ ) by Lau and Ülger for A having a bounded approximate identity is extended to all Banach algebras. The study of this Banach algebra motivates us to introduce the notion of SIN locally compact quantum groups and the concept of quotient strong Arens irregularity. We give characterizations of co-amenable SIN quantum groups, which are even new for locally compact groups. Our study shows that the SIN property is intrinsically related to topological centre problems. We also give characterizations of quotient strong Arens irregularity for all quantum group algebras. Within the class of Banach algebras introduced recently by the authors, we characterize the unital ones, generalizing the corresponding result of Lau and Ülger. We study the interrelationships between strong Arens irregularity and quotient strong Arens irregularity, revealing the complex nature of topological centre problems. Some open questions by Lau and Ülger on Z t ( 〈 A ∗ A 〉 ∗ ) are also answered.

ULTRAPOWERS OF BANACH ALGEBRAS AND MODULES

AbstractThe Arens products are the standard way of extending the product from a Banach algebrato its bidual″. Ultrapowers provide another method which is more symmetric, but one that in general will only give a bilinear map, which may not be associative. We show that ifis Arens regular, then there is at least one way to use an ultrapower to recover the Arens product, a result previously known for C*-algebras. Our main tool is a principle of local reflexivity result for modules and algebras.

Characterization of separating couples

Let A, B be two archimedean l-algebras and let U,V be two positive linear maps from A to B. We call that the couple (U,V) is separating with respect to A and B if |a||b| = 0 in A implies |U (a)||V (b)| = 0 in B. In this paper, we prove that if A is an f-algebra with unit elment e, if B is an l-algebra and if (U,V) is a separating couple with respect to A and B then (U∼∼,V∼∼), where U∼∼ (resp V∼∼) is the bi-adjoint of U (resp of V), is again a separating couple with respect to the order continuous order biduals (A′)′n and (B′)′n of A and B respectively furnished with their Arens products respectively. Moreover, in the case where B′ separates the points of B, we give a characterization of any separating couple with respect to A and B.

The third dual of a Banach algebra

Let A be a Banach algebra and ( A ″, □) be its second dual with first Arens product. The third dual of A can be regarded as dual of A ″ ( A ‴ = ( A ″)′) or as the second dual of A ′ ( A ‴ = ( A ′)″), so there are two ( A ″, □)-bimodule structures on A ‴ that are not always equal. This paper determines the conditions that make these structures equal. As a consequence, there are some relations between weak amenability of A and ( A ″, □).

Cobalt-Catalyzed Cyclotrimerization of Alkynes: The Answer to the Puzzle of Parallel Reaction Pathways

To understand some experimental data at odds with the computed mechanism of the CpCo(L2)-catalyzed [2 + 2 + 2] cyclotrimerization of ethyne, DFT computations were carried out following the fate of methyl- and hydroxycarbonyl-substituted alkynes to give the corresponding arenes. The key intermediate in all cases is a triplet cobaltacyclopentadiene obtained by oxidative coupling of the corresponding CpCo(bisalkyne) complex and subsequent spin change via a minimum energy crossing point (MECP). From that species, two different catalytic cycles lead to an arene product, depending on the nature of the alkyne and other ligands present: either alkyne ligation to furnish a cobaltacyclopentadiene(alkyne) intermediate or trapping by a sigma-donor ligand to generate a coordinatively saturated cobaltacyclopentadiene(PR3) complex. The former leads to the CpCo-complexed arene product via intramolecular cobalt-assisted [4 + 2] cycloaddition, whereas the latter may, in the case of a reactive dienophile (butynedioic acid), undergo direct intermolecular [4 + 2] cycloaddition to generate a cobaltanorbornene. The bridgehead cobalt atom is then reductively eliminated after another change in spin state from singlet to triplet. The necessary conditions for one or the other mechanistic pathway are elaborated.

The topological center of L^{1**}(X,\mu) with respect to first Arens product

The topological center of L^{1**}(X,\mu) with respect to first Arens product

Ideal amenability of second duals of Banach algebras

In this paper we study the ideal amenability of second duals of Banach algebras. We investigate relations between ideal amenability of the second dual of a Banach algebra with the first and the second Arens products. Mathematics Subject Classification: 46H25

Narrow‐Rim Functionalization of Calix[4]arenes via Sonogashira Coupling Reactions.

The narrow (or “lower”)-rim hydroxyl groups of calixarenes are known to be resistant to substitution/displacement. The Sonogashira coupling reaction with TMSA and phenylacetylene, however, has now been extended to the bistriflate of p-tert-butylcalix[4]arene, previously known to be resistant to Stille, Neigishi, or Suzuki−Miyaura reactions. Under some of the reaction conditions investigated, the previously unknown narrow-rim mono- and diiodo-p-tert-butylcalix[4]arene products were also produced, in addition to the narrow-rim mono- and dialkynyl products. Homocoupling of the narrow-rim monoethynyl-p-tert-butyl-calix[4]arene produced a new narrow-rim rigid butadiyne-linked bis-p-tert-butylcalix[4]arene.

A Transition-Metal-Catalyzed Enediyne Cycloaromatization

The pentamethylcyclopentadienyl iron cation, generated from [(eta5-C5Me5)Fe(NCMe)3]PF6, triggers the room temperature cycloaromatization of acyclic and alicyclic enediynes, in the presence of either 1,4-cyclohexadiene or terpinene as the hydrogen-atom donor, to give metal-arene products in good to excellent yields. Photolysis of the metal-arene complexes liberates the arene from the metal in excellent yield. The first demonstration of a transition-metal-catalyzed cycloaromatization of conjugated enediynes has been achieved under photochemical conditions utilizing either [(eta5-C5Me5)Fe(NCMe)3]PF6 or [(eta5-C5Me5)Fe(eta6-1,2-(Prn)2C6H4)]PF6 as the catalyst precursor. The use of a metal and light has led to a convenient method for cycloaromatization of a trans-enediyne.

Narrow-Rim Functionalization of Calix[4]arenes via Sonogashira Coupling Reactions

The narrow (or "lower")-rim hydroxyl groups of calixarenes are known to be resistant to substitution/displacement. The Sonogashira coupling reaction with TMSA and phenylacetylene, however, has now been extended to the bistriflate of p-tert-butylcalix[4]arene, previously known to be resistant to Stille, Neigishi, or Suzuki-Miyaura reactions. Under some of the reaction conditions investigated, the previously unknown narrow-rim mono- and diiodo-p-tert-butylcalix[4]arene products were also produced, in addition to the narrow-rim mono- and dialkynyl products. Homocoupling of the narrow-rim monoethynyl-p-tert-butyl-calix[4]arene produced a new narrow-rim rigid butadiyne-linked bis-p-tert-butylcalix[4]arene.

Effects of zeolite structure and aluminum content on thiophene adsorption, desorption, and surface reactions

The adsorption and desorption of thiophene and the reactions of thiophene-derived adsorbed species in He, H2, and O2 were examined on H-ZSM5, H-Beta, and H-Y with varying Si/Al ratios. Thiophene adsorption uptakes (per Al) were independent of Al content, but were above unity and influenced by zeolite structure (1.7, 2.2, and 2.9 on H-ZSM5, H-Beta, and H-Y). These data indicate that thiophene oligomers form during adsorption and that their size depends on spatial constraints within zeolite channels. Adsorption and oligomerization occur on Brønsted acid sites at 363K. Thiophene/toluene adsorption from their mixtures show significant thiophene selectivity ratios (10.3, 7.9, and 6.4, for H-ZSM5, H-Beta, and H-Y zeolites), which exceed those expected from van der Waals interactions and reflect specific interactions with Brønsted acid sites and formation of toluene–thiophene reaction products. Treatment of thiophene-derived adsorbed species above 363K in He or H2 led to depolymerization of thiophene oligomers and to the formation of unsaturated adsorbed species with a 1:1 thiophene/Al stoichiometry on all zeolites and at all Si/Al ratios. These unsaturated species desorb as stable molecules, such as H2S, hydrocarbons, and larger organosulfur compounds, formed via ring opening and hydrogen transfer from H2 or co-adsorbed species, and also form stranded unsaturated organic deposits. Smaller channels and higher Al contents preferentially formed H2S, benzotiophenes, and arene products during treatment in He or H2, as a result of diffusion-enhanced of secondary reactions of desorbed thiophene molecules with adsorbed thiophene-derived species. Only oxidative regeneration treatments led to full recovery of thiophene uptake capacities. A preceding treatment in H2, however, led to the partial recovery of thiophene-derived carbon atoms as useful hydrocarbons and decreased the amount of CO2 and SO2 formed during subsequent oxidative treatments required for regeneration.

On a conjecture by Ghahramani–Lau and related problems concerning topological centres

Let A be a Banach algebra, and consider A * * equipped with the first Arens product. We establish a general criterion which ensures that A is left strongly Arens irregular, i.e., the first topological centre of A * * is reduced to A itself. Using this criterion, we prove that for a very large class of locally compact groups, Ghahramani–Lau's conjecture (cf. [Lau 94] and [Gha-Lau 95]) stating the left strong Arens irregularity of the measure algebra M ( G ) , holds true. (Our methods obviously yield as well the right strong Arens irregularity in the situation considered.) Furthermore, the same condition used above implies that every linear left A * * -module homomorphism on A * is automatically bounded and w * -continuous. We finally show that our criterion also yields a partial answer to a question raised by Lau-Ülger (Trans. Amer. Math. Soc. 348 (3) (1996) 1191) on the topological centre of the algebra ( A * ⊙ A ) * , for A having a right approximate identity bounded by 1.

Semisimplicity of [formula omitted

We study the semi-simplicity of the second dual of the Banach algebra of operators on a Banach space, B(E)″ , endowed with either Arens product. It was previously shown that if E is a Hilbert space, then B(E) is Arens regular and B(E)″ is semi-simple. We show that for a large class of Banach spaces E, including subspaces of L p spaces not isomorphic to a Hilbert space, B(E)″ is not semi-simple. This is achieved by deriving a new representation of B(l p)′ , and then constructing a member of the radical of B(l p)″ , for p≠2.

Solution to a conjecture by Hofmeier–Wittstock

For a locally compact (non-compact) group G , consider L ∞( G) as a left module over the algebra L 1( G) ∗∗ , 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 L ∞( G) . 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) L ∞( G) ∗ -module homomorphisms on L ∞( G) , can be sharpened in a similar fashion. To this end, a general factorization theorem for bounded families in L ∞( G) is proved from which we shall derive both results.

Designed Enediyne Antitumor Agents

The enediynes remain among the most potent antitumoral agents to have been discovered in the past decade. Following prodrug activation, the enediynes undergo cycloaromatization reactions resulting in formation of highly reactive diradical intermediates. The diradical species engage in atom-transfer chemistry to produce neutral arene products, in the process inducing damage to key macromolecules. Several of the naturally occurring members of the enediyne family of antibiotics have entered clinical trials, and this has prompted the design of synthetic enediynes, where the enediyne lquo;warheadrquo; is conjugated to a targeted delivery vehicle. This review will describe ecent efforts using chemical synthesis to identify and improve the target specificity of designed enediynes, and to establish efficient methods to achieve prodrug activation. Finally, new horizons will be examined, including the use of post-cycloaromatized enediyne templates as recognition elements for unique DNA and RNA microenvironments.

Palladium(0)-catalyzed intramolecular [2+2+2] alkyne cyclotrimerizations with electron-deficient diynes and triynes.

In the presence of 2.5 mol % of [Pd(2)(dba)(3)] (dba=dibenzylideneacetone) and 5 mol % of PPh(3), nearly equimolar amounts of dimethyl nona-2,7-diyne-1,9-dioate derivatives (diyne diesters) and dialkyl acetylenedicarboxylates were allowed to react in toluene at 110 degrees C to afford [2+2+2] cycloadducts in moderate-to-good yields. Similarly, dimethyl trideca-2,7,12-triyne-1,13-dioate derivatives (triyne diesters) were catalytically transformed into phthalic acid ester analogues in excellent yields. To gain insight into the mechanism of these intramolecular alkyne cyclotrimerizations, stoichiometric reactions of [Pd(2)(dba)(3)] with a diyne diester and a triyne diester bearing ether tethers were conducted in acetone at room temperature to furnish an oligomeric bicyclopalladacyclopentadiene and a Pd(0) triyne complex, respectively. The structures of these novel complexes were unequivocally determined by Xray structure analysis. The isolated triyne complex was heated at 50 degrees C or treated with PPh(3) in acetone at room temperature to afford the arene product. Furthermore, the same complex catalyzed the triyne cyclization with or without PPh(3).

Designed enediyne antitumor agents.

The enediynes remain among the most potent antitumoral agents to have been discovered in the past decade. Following prodrug activation, the enediynes undergo cycloaromatization reactions resulting in formation of highly reactive diradical intermediates. The diradical species engage in atom-transfer chemistry to produce neutral arene products, in the process inducing damage to key macromolecules. Several of the naturally occurring members of the enediyne family of antibiotics have entered clinical trials, and this has prompted the design of synthetic enediynes, where the enediyne lquo;warheadrquo; is conjugated to a targeted delivery vehicle. This review will describe ecent efforts using chemical synthesis to identify and improve the target specificity of designed enediynes, and to establish efficient methods to achieve prodrug activation. Finally, new horizons will be examined, including the use of post-cycloaromatized enediyne templates as recognition elements for unique DNA and RNA microenvironments.

Kinetic method for determining antioxidant distributions in model food emulsions: distribution constants of t-butylhydroquinone in mixtures of octane, water, and a nonionic emulsifier.

The absence of reliable estimates of distributions of antioxidants in food emulsions hinders the development of a useful method for comparing the efficiencies of antioxidants. Here we describe the application of a pseudophase kinetic model, originally developed for homogeneous microemulsions, to the determination of distribution constants of tert-butylhydroquinone, TBHQ, in a fluid, opaque, model food emulsion composed of the nonionic emulsifier C(12)E(6), octane, and water. This kinetic method should be applicable to a wide variety of charged and uncharged antioxidants in emulsions composed of charged and uncharged emulsifiers. The distribution constants for partitioning of TBHQ between the oil and surfactant film regions, K(O)(I), and the aqueous and surfactant film regions, K(W)(I), were obtained by fitting changes in first-order rate constants, k(obs), with emulsifier volume fraction for the reaction of 4-hexadecyl-2,6-dimethylbenzenediazonium ion, 16-ArN(2)(+), with TBHQ. The rate of formation of the reduced arene product hexadecyl-2,6-dimethylbenzene, 16-ArH, was followed by HPLC. About 90% of the TBHQ is in the surfactant film at about 2% volume fraction of C(12)E(6), which suggests that this region may be the primary site of antioxidant activity for neutral phenolic antioxidants.

Amenability and topological centres of the second duals of Banach algebras

Let  be a Banach algebra and let ** be the second dual algebra of  endowed with the first or the second Arens product. We investigate relations between amenability of ** and Arens regularity of  and the rôle topological centres in amenability of **. We also find conditions under which weak amenability of ** implies weak amenability of .

On amenability of the Banach algebras l∞(S, [Ufr

Let [Ufr ] be an associative Banach algebra. Given a set S, we write l∞(S, [Ufr ]) for the Banach algebra of all bounded functions f: S→[Ufr ] with the usual norm ∥f∥∞ = sups∈S∥f(s)∥[Ufr ] and pointwise multiplication. When S is countable, we simply write l∈([Ufr ]).In this short note, we exhibit examples of amenable (resp. weakly amenable) Banach algebras [Ufr ] for which l∈(S, [Ufr ]) fails to be amenable (resp. weakly amenable), thus solving a problem raised by Gourdeau in [7] and [8]. We refer the reader to [4, 9, 10] for background on amenability and weak amenability. For basic information about the Arens product in the second dual of a Banach algebra the reader can consult [5, 6].