Arens Product Research Articles

THE SECOND TRANSPOSE OF A DERIVATION

Let A be a Banach algebra, and let D: A → A* be a continuous derivation, where A* is the topological dual space of A. The paper discusses the situation when the second transpose D**: A** → (A**)* is also a derivation in the case where A** has the first Arens product.

Synthesis and stability of 2-(1,1-difluoroalkyl) thiophenes and related 1,1-difluoroalkyl benzenes: fluorinated building blocks for liquid crystal synthesis

The synthesis of a series of 2-(1,1-difluoroalkyl) thiophenes, including some biphenylthienyl liquid crystalline materials, was examined using a variety of fluorination approaches. For comparison purposes, a series of 1,1-difluoroalkyl benzene analogs were also prepared. The direct fluorodeoxygenation of alkyl thienyl ketones and alkyl phenyl ketones using various aminofluorosulfuranes proceeded in only moderate to poor yields. In contrast, fluorodesulfurization of the corresponding 1,3-dithiolanes using NOBF 4/PPHF cleanly afforded the desired 2-(1,1-difluoroalkyl) thiophenes (and analogous 1,1-difluoroalkyl benzenes) in high yields. Fluorodesulfurization of 2-alkyl-2-thienyl-1,3-dithiolanes using NBS (or DBH)/PPHF was complicated by competing ring and/or side chain bromination pathways. These problems were avoided when using NIS/PPHF. Although the various 1,1-difluoroalkyl arene products were sensitive to hydrolytic decomposition on prolonged exposure to silica, the purified products proved quite stable and were well suited for use as building blocks for liquid crystal synthesis.

Aqueous DMF-potassium carbonate as a substitute for thallium and silver additives in the palladium-catalyzed conversion of aryl bromides to acetyl arenes.

Highly selective palladium-catalyzed internal alpha-arylations of alkyl vinyl ethers with aryl and heteroaryl bromides were conveniently conducted in aqueous DMF with potassium carbonate as base and with DPPP as bidentate ligand. The corresponding acetyl arene products were, after hydrolysis, isolated in good to excellent yields. This Heck reaction procedure does not require toxic thallium or expensive silver salt additives, is promoted by water, and is suggested to proceed via charged organopalladium intermediates. Single-mode microwave irradiation was utilized in one example to shorten the reaction time.

C−H Activation and Functionalization of Unsaturated Hydrocarbons by Transition-Metal Boryl Complexes

Transition-metal boryl complexes of the form Cp‘Fe(CO)LBcat and (CO)5MBcat, where Cp‘ = C5H5, C5Me5, M = Mn, Re, L = CO, PMe3, and cat = 1,2-O2C6H4, were synthesized by reaction of ClBcat with [Cp‘Fe(CO)L]- or [M(CO)5]-. X-ray crystal structures of CpFe(CO)2Bcat, Cp*Fe(CO)2Bcat, and (CO)5MnBcat were obtained. Upon irradiation, these metal boryl complexes reacted with arenes and alkenes to form aryl- and vinylboronate ester products in moderate to high yields. Monosubstituted arenes with methyl, chloro, trifluoromethyl, methoxy, and dimethylamino substituents were used as substrates, and the resulting ratios of ortho- to meta- to para-substituted arene products were measured. No significant electronic effects were observed, indicating that the chemistry is not occurring through a typical electrophilic aromatic substitution pathway. Competition experiments between toluene and other substituted arenes were conducted. Reactivity differences were small, but anisole was found to have the fastest rate of reactio...

Commutativity of the Arens product in lattice ordered algebras

Let \({\text{A}}\) be an Abelian Archimedean lattice ordered algebra. The order bidual \({A''}\) furnished with the Arens product is again a lattice ordered algebra. We show that the order continuous order bidual \((A')'_n\) is Abelian. This solves an open problem and improves a result of Scheffold, who proved it for the case of normed lattice ordered algebras. The proof is based on the ‘up-down-up’ approximation of positive elements in the order continuous order bidual \((A')'_n\) by elements in the canonical image \({\widehat A}\) of \(A\) in \((A')'_n\) Components of positive elements in \({\widehat A}\) are characterized and the result is applied to the Arens product of \(f\)-and almost \(f\)-algebras.

On asymmetry of topological centers of the second duals of Banach algebras

Let A be a Banach algebra with a bounded approximate identity and let Z1(2**) and Z2(2**) be the left and right topological centers of W**. It is shown that i) W*2 = AZ* is not sufficient for ZI(f**) = Z2(f**); ii) the inclusion 2fZj(2**) C Af is not sufficient for Z2(2**)2f C Af; iii) Zl(2**) = Z2(2**) = 2f is not sufficient for A to be weakly sequentially complete. These results answer three questions of Anthony To-Ming Lau and Ali Ulger. Suppose that St is a Banach algebra and let Wf* be the dual space of QI. Then Wf* can be made into a Banach Q-bimodule as follows: for f E W,a E Q, fa and af are defined by (fa, b) = (f, ab), (af, b) = (f, ba) (b E A) where (,) is used for the dual pairing between elements of W* and A. Let Q*%2 = {fa: f E t*,a E A} and %2%2* = {af :-a E Q(,f E :%*}. The dual space W(* is said to factor on the left (resp. right) if Wf* = Qf*Qf (resp. Wf* = At*). In a recent paper [5] Anthony To-Ming Lau and Ali Ulger have obtained various necessary and sufficient conditions for factoring of W.** Here we answer three questions left open in [5]. The second dual space (** of a Banach algebra St admits two Banach algebra products known as first (left) and second (right) Arens products. Each of these products extends the product of St as canonically embedded in (** (see [1] or [3]). We briefly recall the definition of these products. For m, n E Q**, their first (left) Arens product indicated by m O n is given by (m l n,f) =(m,nf) (fEQ*) where nf E W* is defined by (nff,a) = (nr,fa) ;(a E Q, where fa is as defined earlier. Received by the editors December 5, 1996. 1991 Mathematics Subject Classification. Primary 46H99.

On some special measures on βG

LetG be a discrete abelian group,Ĝ the character group ofG, andl ∞(G)* the conjugate ofl ∞(G) equipped with an Arens product. In many cases, we can find unitary functionsf such that χf is almost convergent to zero for all χ∈Ĝ. Some of these functions are then used to produce elements μ∈l ∞(G)* such that γμ=0 whenever γ is an annihilator ofC 0(G). Regarded as Borel measures on βG, these elements satisfyxμ=0 for allx∈βG/G. They belong to the radical ofl ∞(G)*, and each of them generates a left ideal ofl ∞(G)* that contains no minimal left ideal.

The second dual of C0 (S, A)

AbstractThe second dual of the vector-valued function space C0(S, A) is characterized in terms of generalized functions in the case where A* and A** have the Radon-Nikodým property. As an application we present a simple proof that C0 (S, A) is Arens regular if and only if A is Arens regular in this case. A representation theorem of the measure μh is given, where , h ∈ L∞ (|μ;|, A**) and μh is defined by the Arens product.

Module Homomorphisms of the Dual Modules of Convolution Banach Algebras

AbstractSuppose that A is either the group algebra L1 (G) of a locally compact group G, or the Volterra algebra or a weighted convolution algebra with a regulated weight. We characterize: a) Module homomorphisms of A*, when A* is regarded an A** left Banach module with the Arens product, b) all the weak*-weak* continuous left multipliers of A**.

Isomorphisms and multipliers on second dual algebras of Banach algebras

Suppose that A is a Banach algebra and let A be the second dual algebra of A equipped with the first Arens product 3. In this paper we characterize compact and weakly compact multipliers of A, when A possesses a bounded approximate identity and is a two sided ideal in A. We use this to study the isomorphisms between second duals of various classes of Banach algebras satisfying the above properties.

A simple example of a non-commutative Arens product

A simple example of a non-commutative Arens product

On the Arens products and reflexive Banach algebras

We give a characterization of reflexive Banach algebras involving the Arens product.

A Canonical Extension for Analytic Functions on Banach Spaces

Given Banach spaces $E$ and $F$, a Banach space ${G_{EF}}$ is presented in which $E$ is embedded and which seems a natural space to which extend $F$-valued analytic functions. Any $F$-valued analytic function defined on a subset $U$ of $E$ may be extended to an open neighborhood of $U$ in ${G_{EF}}$. This extension generalizes that of Aron and Berner. It is also related to the Arens product in Banach algebras, to the functional calculus for bounded linear operators, and to an old problem of duality in spaces of analytic functions. A characterization of the Aron-Berner extension is given in terms of continuity properties of first-order differentials.

A canonical extension for analytic functions on Banach spaces

Given Banach spaces E E and F F , a Banach space G E F {G_{EF}} is presented in which E E is embedded and which seems a natural space to which extend F F -valued analytic functions. Any F F -valued analytic function defined on a subset U U of E E may be extended to an open neighborhood of U U in G E F {G_{EF}} . This extension generalizes that of Aron and Berner. It is also related to the Arens product in Banach algebras, to the functional calculus for bounded linear operators, and to an old problem of duality in spaces of analytic functions. A characterization of the Aron-Berner extension is given in terms of continuity properties of first-order differentials.

Double multipliers andA*-algebras of the first kind

LetAbe anA*-algebra and letdenote its auxiliary norm closure. The multiplier algebras of dualA*-algebras of the first kind have been studed by Tomiuk [12], [13] and Wong[15]. In this paper we study the double multiplier algebra ofA*-algebras of the first kind. In particular, we prove that, ifAis anA*-algebra of the first kind, then the double multiplier algebraM(A) ofAis *-isomorphic and (auxiliary norm) isometric to a subalgebra ofM(), extending in the process some results established by Tomiuk[12]. We also consider the embedding of the double multiplier algebra ofAin**, when the latter is regarded as an algebra with the Arens product, and, in particular, we show that, for an annihilator A*-algebra,M(A) is *-isomorphic and (auxiliary norm) isometric to**.

Arens Product and the Algebra of Double Multipliers. II

Arens Product and the Algebra of Double Multipliers. II

Arens product and the algebra of double multipliers. II

We show that the algebra of double multipliers of a certain Banach algebra A A can be embedded in the second conjugate space A ∗ ∗ {A^{**}} of A A . This generalizes the previous work by the author for B ∗ {B^*} -algebras.

Numerical ranges in locally M-convex algebras. I

AbstractThe bidual of a unital infrabarrelled l.m.c. C* algebra E, equipped with the bidual topology and the regualr Arens product, is always an l.m.c. C*-algebra. On the other hand, a unital l.m.c. *-algebra E has the C*-property if and only if every self-adjoint element x of E is strongly hermitian (x has real numerical range), or the sets of normalized states and normalized continuous positive linear forms of E coincide. Finally, every unital cpmplete l.m.c. C* algebra satisfying, locally, the property ‘the extreme points are dense in that set of continuous positive linear forms” (antiliminal algebra) has the complexes as its only normal elements.

Imitative interactions and communicative development

Let G be a non-compact locally compact group, L1(G) be its group algebra, LUC(G) be the space of bounded functions on G which are uniformly continuous with respect to the right uniformity on G. Let L1(G)** be the second conjugate of L1(G) with the first Arens product and LUC(G)* be the conjugate of the space LUC(G) with the first Arens-type product. We see how the maximal ideals of LUC(G)* are related to those of L1(G), and give examples of weak* – dense maximal ideals in LUC(G)*. For a large class of locally compact groups, we compute the dimension of every right ideal in LUC(G)* and the dimension of every right ideal in L1(G)** which is not generated by a right annihilator of L1(G)**. In particuar, we see that they are infinite dimensional; a result which was known earlier only for the radicals of these algebras. Finally, we construct new elements in the radicals of LUC(G)* and L1(G)**.

Continuity of Arens multiplication on the dual space of bounded uniformly continuous functions on locally compact groups and topological semigroups

Let G be a topological semigroup, i.e. G is a semigroup with a Hausdorff topology such that the map (a, b) → a.b from G × G into G is continuous when G × G has the product topology. Let C(G) denote the space of complex-valued bounded continuous functions on G with the supremum norm. Let LUC (G) denote the space of bounded left uniformly continuous complex-valued functions on G i.e. all f ε C(G) such that the map a → laf of G into C(G) is continuous when C(G) has a norm topology, where (laf )(x) = f (ax) (a, x ε G). Then LUC (G) is a closed subalgebra of C(G) invariant under translations. Furthermore, if m ε LUC (G)*, f ε LUC (G), then the functionis also in LUC (G). Hence we may define a productfor n, m ε LUC(G)*. LUC (G)* with this product is a Banach algebra. Furthermore, ʘ is precisely the restriction of the Arens product defined on the second conjugate algebra l∞(G)* = l1(G)** to LUC (G)*. We refer the reader to [1] and [10] for more details.