Topological Centers and Factorization of Certain Module Actions

In this paper, we extend some propositions of Banachalgebras into module actions and establish the relationships betweentopological centers of module actions. We introduce some newconcepts as -property and -property for Banach modules and obtain some conclusions inthe topological center of module actions and Arens regularity of Banachalgebras.

The paper begins with an introduction to the cohomology theory for Banach algebras needed in the remainder of the paper. A perturbation of the multiplication π of a Banach algebra A is a continuous associative multiplication ρ which is near ρ in the usual norm for bilinear maps. We consider the question of whether such a multiplication is necessarily of the form ρ(a, b) = W−1(WaWb) where W is a bounded linear map from A into A near the identity. A cohomological condition is given for this to hold. A linear map T from one Banach algebra A into another B is “almost” a homomorphism if the bilinear map \(\mathop{T}\limits^{ \circ } (a,b) = T(a,b) - T(a)T(b)\) is small. This occurs if T is near a genuine homomorphism and we consider whether every map which is almost a homomorphism is near a homomorphism. The main result is that if A is an amenable algebra and B is the dual of a suitable Banach space then the pair (A, B) has this property.

Let ⟨ A, B⟩ be a pairing of two regular multiplier Hopf algebras A and B. One method of constructing the Drinfel'd double 𝒟 = A⋈ B cop is by the use of an invertible twist map R: B⊗ A→ A⊗ B defining an associative product on A⊗ B. In Delvaux (2003) and Drabant and Van Daele (2001), the authors construct R by , where R 2 and is only related to the module actions between A and B. Another way is given in Delvaux and Van Daele (2004a) in which the authors also just consider the module actions and then construct the Drinfel'd double 𝒟 as an algebra of operators on the vector space B⊗ A. In this article we will give two different points of view of constructing the Drinfel'd double 𝒟 for multiplier Hopf algebras. The first is that the Drinfel'd double 𝒟 associated to the pairing ⟨ A, B⟩ is constructed by using not only the module actions but also the comodule coactions, i.e., the Drinfel'd double 𝒟 is given in the framework of a special twisted tensor product algebra structure A ⊡ A op ⊗ A B. The second is as follows: If P is a multiplier Hopf algebra and a reduced (A, B)-bicomodule algebra (an A-Long module algebra), then we present a twisting construction of the product of P via the coactions of A and B on P, and we show that the Drinfel'd double 𝒟 is isomorphic as a multiplier Hopf algebra to the opposite twisting of (A op, cop ⊗ B op, cop ). As an application of our theory, we consider the case of group-cograded multiplier Hopf algebras and the case of Hopf group-coalgebras.

Oestrogen receptors in the central nervous system and evidence for their role in the control of cardiovascular function

Relative Yoneda Cohomology for Operator Spaces

Let [Formula: see text] and [Formula: see text] be two Banach algebras such that [Formula: see text] is a Banach [Formula: see text]-bimodule with the left and right compatible actions of [Formula: see text] on [Formula: see text]. Let [Formula: see text] be a strongly splitting Banach algebra extension of [Formula: see text] by [Formula: see text]. In this paper, we investigate some homological aspects such as injectivity, projectivity and flatness of [Formula: see text] and give some necessary and sufficient conditions for injectivity, projectivity and flatness of [Formula: see text].

We provide a sufficient condition for strong (Arens) irregularity of certain bounded bilinear maps, which applies in particular to the adjoint of Banach module actions. We then apply our result to improve several known results concerning to the relation between Arens regularity of certain Banach module actions and reflexivity.

The convex-transitivity property can be seen as a convex generalization of the almost transitive (or quasi-isotropic) group action of the isometry group of a Banach space on its unit sphere. We will show that certain Banach algebras, including conformal invariant Douglas algebras, are weak-star convex-transitive. Geometrically speaking, this means that the investigated spaces are highly symmetric. Moreover, it turns out that the symmetry property is satisfied by using only 'inner' isometries, i.e. a subgroup consisting of isometries which are homomorphisms on the algebra. In fact, weighted composition operators arising from function theory on the unit disk will do. Some interesting examples are provided at the end.

Let \(\mathcal {A}\) be a unital Banach algebra and \(\mathcal {M}\) be a unital Banach \(\mathcal {A}\)-bimodule. The main results characterize a continuous linear map \(\varphi :\mathcal {A}\rightarrow \mathcal {M}\) that satisfies \(a\varphi (a^{-1})=\varphi (1)\) or \(a\varphi (a^{-1})+ \varphi (a^{-1}) a=2\varphi (1)\) for all \(a\) in principal component of invertible elements of \(\mathcal {A}\). The proof is based on the consideration of a continuous bilinear map satisfying a related condition.

The main theorem of this paper is a little involved (though the proof is straightforward using a well-known idea) but the immediate corollaries are interesting. For example, take a complex normed vector space A which is also a normed algebra with identity under each of two multiplications * and ∘. Then these multiplications coincide if and only if there exists α such that ‖a ∘ b ‖ ≦ α ‖ a * b ‖ for a, b in A. This is a condition for the two Arens multiplications on the second dual of a Banach algebra to be identical. By taking * to be the multiplication of a Banach algebra and ∘ to be its opposite, we obtain the condition for commutativity given in (3). Other applications are concerned with conditions under which a bilinear mapping between two algebras is a homomorphism, when an element lies in the centre of an algebra, and a one-dimensional subspace of an algebra is a right ideal. An example shows that the theorem is false for algebras over the real field, but Theorem 2 gives the parallel result in this case.

In this paper, we consider linear spaces and algebras with real scalars. It is well known that if X is a Banach space and is the set of all bounded linear operators which map X into itself, then is a Banach algebra. In this paper we shall show that can be partially ordered so that it becomes a partially ordered algebra in which norm convergence is equivalent to order convergence. This motivates a study of Banach algebras of operators in which one uses the order structure to obtain various results. In addition, it encourages a study of partially ordered algebras in general, since our result shows that among such algebras one finds all real Banach algebras of operators. Of course, there are many other real algebras which are naturally partially ordered and which have been studied from that point of view.

The canonical mapping on the product of a LCA group with its dual is shown to extend uniquely in a homomorphic and analytic way to the product of the respective complexifications.

We generalize the influential $C^*$ -algebraic results of Kawamura–Tomiyama and Archbold–Spielberg for crossed products of discrete groups actions to the realm of Banach algebras and twisted actions. We prove that topological freeness is equivalent to the intersection property for all reduced twisted Banach algebra crossed products coming from subgroups, and in the untwisted case to a generalized intersection property for a full $L^p$ -operator algebra crossed product for any $p\in [1,\,\infty ]$ . This gives efficient simplicity criteria for various Banach algebra crossed products. We also use it to identify the prime ideal space of some crossed products as the quasi-orbit space of the action. For amenable actions we prove that the full and reduced twisted $L^p$ -operator algebras coincide.

Bond‐extended stochastic and nonstochastic bilinear indices are introduced in this article as novel bond‐level molecular descriptors (MDs). These novel totals (whole‐molecule) MDs are based on bilinear maps (forms) similar to use defined in linear algebra. The proposed nonstochastic indices try to match molecular structure provided by the molecular topology by using the k th Edge(Bond)‐Adjacency Matrix ( E k , designed here as a nonstochastic E matrix). The stochastic parameters are computed by using the k th stochastic edge‐adjacency matrix, ES k , as matrix operators of bilinear transformations. This new edge (bond)‐adjacency relationship can be obtained directly from E k and can be considered like a new matrix‐transformation strategic to obtain new relations for a molecular graph. In both set of MDs, chemical information is codified by using different pair combinations of atomic weightings (in this case four atomic‐labels: atomic mass, polarizability, van der Waals volume, and electronegativity). In addition, a local‐fragment (bond‐type) formalism was also developed. The k th bond‐type bilinear indices are calculated by summing the k th bond bilinear indices of all bonds of the same bond type in the molecules. The new set of MDs can be easily and quickly calculated in our in house software TOMOCOMD‐CARDD (topological molecular computational design computer‐aided‐rational‐drug design). The reported application and utilization of these MDs for predictive capability correlations of structure with physicochemical and pharmacological properties are reviewed. Three benchmark datasets have been used to evaluate the QSPR/QSAR behavior of the new bond‐level TOMOCOMD‐CARDD MDs. We developed the QSPR models to describe several physicochemical properties of octane isomers (First Case) and, to analyze of the boiling point of 28 alkyl‐alcohols (Second Case) and to examine of the specific rate constant (log k ), the partition coefficient (log P ), as well as the antibacterial activity of 34 derivatives of 2‐furylethylenes (Third Case). For these three rounds, the quantitative models found are significant from a statistical point of view and permit a clear interpretation of the studied properties in terms of the structural features of molecules. A leave‐one‐out cross‐validation procedure revealed that the regression models had a good predictability. The comparison with other approaches reveals good performance of the method proposed. Therefore, it is clearly demonstrated that this suitability is higher than that shown by other 2D/3D well‐known sets of MDs. The approach described here appears to be a very promising structural invariant, useful for QSPR/QSAR studies and shown to provide an excellent alternative or guides for discovery and optimization of new lead compounds, reducing the time and cost of the traditional procedure. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

Vehicular cloud is the new paradigm of vehicular ad hoc networks (VANETs), which not only takes the recent advance of vehicular technology, but also keeps up with the trend of cloud computing for big data processing and complicated intelligent analysis over VANET environments. In this paper, we present a model for secure navigation systems based on vehicular cloud, which fully utilizes the advantages of cloud computing. In particular, we make up for the limitations of prior VANET-based secure navigation protocols and provide a stronger notion of security for secure navigation services in terms of insider threats and data leakage points of view. We propose a new secure and privacy-preserving navigation protocol based on vehicular cloud, which resolves the limitations of previous approaches. Moreover, the proposed protocol is highly efficient because it does not rely on bilinear pairing operations unlike prior secure navigation protocols in the literature. We demonstrate comprehensive analysis to confirm the fulfilment of the security objectives and the efficiency of the proposed protocol.