Involutive Algebras Research Articles

On a comparison of real with complex involutive complete algebras

On a comparison of real with complex involutive complete algebras

QUATERNION HOMOLOGY OF BANACH SPACE

In this article we are concerned with the quaternion Banach spaces and their homology. We obtain the relation between the quaternion and the dihedral homology for a unital involutive Banach algebra.

The relative dihedral homology of involutive algebras

Let f : A → B be a homomorphism of involutive algebras A, B. The purpose of this paper is to define a free involutive algebra resolution of algebra B over f and use it to define and study the relative dihedral homology.

On the picard group of involutive algebras and coalgebras

We study how Hilbert bimodules correspond in the algebraic case to hermitian Morita equivalences and consequently we obtain a description of the hermitian Picard group of a commutative involutive algebra A as the semidirect product of the classical hermitian Picard group of A and the automorphisms of A commuting with the involution. We also obtain similar decomposition results on hermitian Picard groups of involutive coalgebras (Cωc), which show, at least in the cocommutative case, that this hermitian Picard group differs considerably from the non hermitian one.

Numerical ranges of elements of involutive Banach algebras and commutativity

Let $ {\cal A} $ be a unital involutive Banach algebra. Bonsall and Duncan defined the numerical range for each element x in $ {\cal A} $ by $ V(x) = \{f(x):f \in {\cal A'}, f(e) = 1 =|| f || \} $ , where e is the unit. We introduce another numerical range by $ W (x) = \{ f(x) : f \in {\cal A'}, f \geq 0, f(e) = 1 \} $ , and we call $ w(x) = {\rm sup} \{|z|: z \in W (x) \} $ the numerical radius of x. We give a few conditions for $ {\cal A} $ to be a C *-algebra, and we see that some mapping theorems for numerical ranges of elements of $ {\cal A} $ hold. We show that if w (x) ≤ 1 implies $ w (\varphi (x) ) \leq 1 $ for every Mobius transform $ \varphi $ of the unit disk, then $ {\cal A} $ is commutative.

Involutive Lie algebras graded by finite root systems and compact forms of IM algebras

Involutive Lie algebras graded by finite root systems and compact forms of IM algebras

Noncommutative structure of singularities in general relativity

Initial and final singularities in the closed Friedman world model are typical examples of malicious singularities. They form the single point of Schmidt’s b-boundary of this model and are not Hausdorff separated from the rest of space–time. The method of noncommutative geometry, developed by A. Connes and his co-workers, is applied to this case. We rephrase Schmidt’s construction in terms of the groupoid Ḡ of orthonormal frames over space–time and carry out the ‘‘desingularization’’ process. We define the line bundle τ:Ω1/2→Ḡ over Ḡ and change the space of its cross sections into an involutive algebra. This algebra is represented in the space of operators on a Hilbert space and, with the norm inherited from these operators, it becomes a C*-algebra. The initial and final singularities of the closed Friedman model are given by two distinct representations of this C*-algebra in the space of operators acting on the Hilbert space L2(O(3,1)).

On a certain construction of graded Lie algebras with derivation

Using a unitial associative ∗-algebra U over C a certain class of Hermitian finite projective modules together with a graded involutive differential algebra, both associated with U , we develop a procedure for constructing graded Lie algebras with derivation. Taking, in particular, the canonical differential algebra of Connes' theory, related to the simplest two-point K-cycle, we obtain a class of graded Lie algebras with derivation, which as one special case contains the graded Lie algebra used in the Mainz-Marseille approach to model building. Finally, we outline a new derivation of the standard model.

Dihedral homology of commutative algebras

Let A be an associative k-algebra with involution, where k is a commutative ring of characteristic not equal to two. Then the dihedral groups act on the Hochschild complex and, following Loday, a new homological theory, called dihedral homology, can be defined generalizing the notion of cyclic homology defined by Connes. Here we give a model to compute dihedral homology of a commutative algebra over a characteristic zero field. As, for an involutive algebra, we have a decomposition of Hochschild homology into a direct sum of two k-modules: Z 2- equivariant and skew Z 2- equivariant Hochschild homologies, we give smoothness criteria in terms of vanishing of some Z 2- equivariant Hochschild homology groups.

Morita equivalence for positive hochschild homology and dihedral homology

In this paper we prove the invariance of positive Hochschild homology and dihedral homology with respect to hermitian Morita equivalence between involutive algebras. We also define the notion of hermitian k‐congruence and prove some results on Morita invariance of HH∗ + and HD∗, in this context

Applications du calcul fonctionnel harmonique dans une algèbre de Banach involutive quotient

We dene the spectral radius and Ptak’s function on a quotient involutive Banach algebra and we extend certains results of V.Ptak to thes quotients. As an application of functional harmonic calculus we obtain theorem similar to those of K.Fan and Von Neumann.

Invariants of piecewise-linear 3-manifolds

This paper presents an algebraic framework for constructing invariants of closed oriented 3-manifolds by taking a state sum model on a triangulation. This algebraic framework consists of a tensor category with a condition on the duals which we have called a spherical category. A significant feature is that the tensor category is not required to be braided. The main examples are constructed from the categories of representations of involutive Hopf algebras and of quantised enveloping algebras at a root of unity.

Bordism groups of Poincare -coalgebras and symmetric -groups

A Poincare -coalgebra construction over involutive algebras is introduced in this paper. Various types of bordism between Poincare -coalgebras are defined and the relations between the corresponding bordism groups are studied. It is shown in particular that the Thom bordism groups of closed non-oriented smooth manifolds and the rational Wall groups of a unitary group have a common algebraic origin, that is, they are obtained by the same construction considered over the fields and , respectively.

Regular Fréchet–Lie Groups of Invertible Elements in Some Inverse Limits of Unital Involutive Banach Algebras

Abstract We consider a wide class of unital involutive topological algebras provided with a C*-norm and which are inverse limits of sequences of unital involutive Banach algebras; these algebras are taking a prominent position in noncommutative differential geometry, where they are often called unital smooth algebras. In this paper we prove that the group of invertible elements of such a unital solution smooth algebra and the subgroup of its unitary elements are regular analytic Fréchet–Lie groups of Campbell–Baker–Hausdorff type and fulfill a nice infinite-dimensional version of Lie's second fundamental theorem.

Regular Fréchet-Lie groups of invertible elements in some inverse limits of unital involutive Banach algebras

We consider a wide class of unital involutive topological algebras provided with aC*-norm and which are inverse limits of sequences of unital involutive Banach algebras; these algebra sare taking a prominent position in noncommutative differential geometry, where they are often called unital smooth algebras. In this paper we prove that the group of invertible elements of such a unital solution smooth algebra and the subgroup of its unitary elements are regular analytic Frechet-Lie groups of Campbell-Baker-Hausdorff type and fulfill a nice infinite-dimensional version of Lie's second fundamental theorem.

The logic of quantum systems with diagonal singularities

The work of the Brussels-Austin groups on irreversibility over the last years has shown that Quantum Large Poincare systems with diagonal singularity lead to an extension of the conventional formulation of dynamics at the level of mixtures which is manifestly time asymmetric. States with diagonal singularity acquire meaning as linear fractionals over the involutive Banach algebra of operators with diagonal singularity. We show in this paper that the logic of quantum systems with diagonal singularity is not the conventional logic of Hilbert space, because only finite combinations of prepositions are allowed.

ENERGY REPRESENTATIONS OF INFINITE DIMENSIONAL GAUGE GROUPS IN NONCOMMUTATIVE GEOMETRY

Let M be a compact smooth manifold, let [Formula: see text] be a unital involutive subalgebra of the von Neumann algebra £ (H) of bounded linear operators of some Hilbert space H, let [Formula: see text] be the unital involutive algebra [Formula: see text], let [Formula: see text] be an hermitian projective right [Formula: see text]-module of finite type, and let [Formula: see text] be the gauge group of unitary elements of the unital involutive algebra [Formula: see text] of right [Formula: see text]-linear endomorphisms of [Formula: see text]. We first prove that noncommutative geometry provides the suitable setting upon which a consistent theory of energy representations [Formula: see text] can be built. Three series of energy representations are constructed. The first consists of energy representations of the gauge group [Formula: see text], [Formula: see text] being the group of unitary elements of [Formula: see text], associated with integrable Riemannian structures of M, and the second series consists of energy representations associated with (d, ∞)-summable K-cycles over [Formula: see text]. In the case where [Formula: see text] is a von Neumann algebra of type II 1 a third series is given: we introduce the notion of regular quasi K-cycle, we prove that regular quasi K-cycles over [Formula: see text] always exist, and that each of them induces an energy representation.

The Structure and Represenation of a $\mathrm C^*$-Algebra Associated to the Super-Poincaré Group

The representation theory of a class of algebras associated to certain graded Lie groups is investigated. To a group whose even part is central is associated a natural involutive algebra all of whose {}^* -representations factor through a quotient algebra of continuous Clifford algebra-valued fields. The irreducible representations of crossed products of the algebra by a Lie algebra, such as the super-Poincaré group, are then constructed by Takesaki's method. It is then shown that they may also be constructed by Rieffel's \mathrm C^* -algebraic induction. Tensor product decompositions are briefly discussed.

Extreme positive operators on involution algebras

Extreme positive operators on involution algebras

Determining subspaces for discrete-type linear forms on commutative Banach algebras

In this paper we deal with some problems of the Korovkin-type Approximation theory concerning the convergence of nets of linear forms toward discrete-type linear forms on commutative Banach algebras. We also study the case of involutive symmetric commutative Banach algebras with identity or with bounded approximate identity. Examples and applications are presented in the context of the algebrasC0(X, C),C(X, C) andL1 (IR).