Involutive Algebras Research Articles

Planar Lyapunov Algebras

We study complex involutive algebras generated by a single nonselfadjoint idempotent and use them to construct a family of algebras, which we call planar Lyapunov algebras. As our main result, we prove that every 2-dimensional commutative real algebra whose homogeneous Riccati differential equation is stable at the origin must be isomorphic either to an algebra with zero multiplication or to some planar Lyapunov algebra.

On split involutive regular BiHom-Lie superalgebras

Abstract The goal of this paper is to examine the structure of split involutive regular BiHom-Lie superalgebras, which can be viewed as the natural generalization of split involutive regular Hom-Lie algebras and split regular BiHom-Lie superalgebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split involutive regular BiHom-Lie superalgebra L {\mathfrak{L}} is of the form L = U + ∑ α I α {\mathfrak{L}}=U+{\sum }_{\alpha }{I}_{\alpha } with U a subspace of a maximal abelian subalgebra H and any I α , a well-described ideal of L {\mathfrak{L}} , satisfying [I α , I β ] = 0 if [α] ≠ [β]. In the case of L {\mathfrak{L}} being of maximal length, the simplicity of L {\mathfrak{L}} is also characterized in terms of connections of roots.

Operators on Anti-dual pairs: Self-adjoint Extensions and the Strong Parrott Theorem

AbstractThe aim of this paper is to develop an approach to obtain self-adjoint extensions of symmetric operators acting on anti-dual pairs. The main advantage of such a result is that it can be applied for structures not carrying a Hilbert space structure or a normable topology. In fact, we will show how hermitian extensions of linear functionals of involutive algebras can be governed by means of their induced operators. As an operator theoretic application, we provide a direct generalization of Parrott’s theorem on contractive completion of 2 by 2 block operator-valued matrices. To exhibit the applicability in noncommutative integration, we characterize hermitian extendibility of symmetric functionals defined on a left ideal of a $C^{\ast }$-algebra.

Results on hyper equality algebras

Equality algebras as a generalization of EQ -algebras introduced by Jenei [ 8 ]. Hyper equal- ity algebras introduced and studied in [ 4 ], as a generalization of equality algebras. Now, in this paper, we investigate relations among hyper equality algebras and other hyper algebraic structures such as hyper K ( BE , MV )-algebras and hyper hoops. Specially, we prove that any linearly ordered hyper MV -algebra is a strongly commutative symmetric hyper equality algebra and under some conditions, any strongly commutative involutive hyper equality algebra is a hyper MV -algebra.

Kernels and cokernels in the category of augmented involutive stereotype algebras

We prove several properties of kernels and cokernels in the category of augmented involutive stereotype algebras: (1) this category has kernels and cokernels, (2) the cokernel is preserved under the passage to the group stereotype algebras, and (3) the notion of cokernel allows to prove that the continuous envelope [Formula: see text] of the group algebra of a compact buildup of an abelian locally compact group is an involutive Hopf algebra in the category of stereotype spaces [Formula: see text]. The last result plays an important role in the generalization of the Pontryagin duality for arbitrary Moore groups.

Involutive operator algebras

Examples of operator algebras with involution include the operator $*$-algebras occurring in noncommutative differential geometry studied recently by Mesland, Kaad, Lesch, and others, several classical function algebras, triangular matrix algebras, (complexifications) of real operator algebras, and an operator algebraic version of the {\em complex symmetric} operators studied by Garcia, Putinar, Wogen and others. We investigate the general theory of involutive operator algebras, and give many applications.

On the structure of split involutive Hom-Lie color algebras

In this paper we study the structure of arbitrary split involutive regular Hom-Lie color algebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split involutive regular Hom-Lie color algebra L is of the form L=U⊕∑[α∈Π/∼I[α], with U a subspace of the involutive abelian subalgebra H and any I[α], a well-described involutive ideal of L, satisfying [I[α], I[β]]=0 if [α]≠[β]. Under certain conditions, in the case of L being of maximal length, the simplicity of the algebra is characterized and it is shown that L is the direct sum of the family of its minimal involutive ideals, each one being a simple split involutive regular Hom-Lie color algebra. Finally, an example will be provided to characterise the inner structure of split involutive Hom-Lie color algebras.

Enveloping algebras of color hom-Lie algebras

In this paper, the universal enveloping algebra of color hom-Lie algebras is studied. A construction of the free involutive hom-associative color algebra on a hom-module is described and applied to ...

Ideals in bounded equality algebras

In this paper, the concept of ideal in bounded equality algebras is introduced. With respect to this concepts, some related results are given. In particular, we prove that there is an one-to-one corresponding between congruence relation on an involutive equality algebra and the set of ideals on it. Also, we prove the first isomorphism theorem on equality algebras. Moreover, the notions of prime and Boolean ideals in equality algebras are introduced. Finally, we prove that ideal I of involutive prelinear equality algebra E is a Boolean ideal if and only if E/I is a Boolean algebra.

Checking Admissibility Using Natural Dualities

This article presents a new method for obtaining small algebras to check the admissibility—equivalently, validity in free algebras—of quasi-identities in a finitely generated quasivariety. Unlike a previous algebraic approach of Metcalfe and Röthlisberger, which is feasible only when the relevant free algebra is not too large, this method exploits natural dualities for quasivarieties to work with structures of smaller cardinality and surjective rather than injective morphisms. A number of case studies are described here that could not be be solved using the algebraic approach, including (quasi)varieties of MS-algebras, double Stone algebras, and involutive Stone algebras.

Factorization results for left polynomials in some associative real algebras: State of the art, applications, and open questions

We discuss existence of factorizations with linear factors for (left) polynomials over certain associative real involutive algebras, most notably over Clifford algebras. Because of their relevance to kinematics and mechanism science, we put particular emphasis on factorization results for quaternion, dual quaternion and split quaternion polynomials. A general algorithm ensures existence of a factorization for generic polynomials over division rings but we also consider factorizations for non-division rings. We explain the current state of the art, present some new results and provide examples and counter examples.

Generalized state maps and states on pseudo equality algebras

Abstract In this paper, we attempt to cope with states in a universal algebraic setting, that is, introduce a notion of generalized state map from a pseudo equality algebra X to an arbitrary pseudo equality algebra Y. We give two types of special generalized state maps, namely, generalized states and generalized internal states. Also, we study two types of states, namely, Bosbach states and Riečan states. Finally, we discuss the relations among generalized state maps, states and internal states (or state operators) on pseudo equality algebras. We verify the results that generalized internal states are the generalization of internal states, and generalized states are the generalization of state-morphisms on pseudo equality algebras. Furthermore, we obtain that generalized states are the generalization of Bosbach states and Riečan states on linearly ordered and involutive pseudo equality algebras, respectively. Hence we can come to the conclusion that, in a sense, generalized state maps can be viewed as a possible united framework of the states and the internal states, the state-morphisms and the internal state-morphisms on pseudo equality algebras.

Involutive equality algebras

The present paper aims to study a special class of equality algebras, called involutive equality algebra. We obtain some properties of this structure and prove that every linearly ordered 0-compatible equality algebra includes a $$(\sim _0)$$ -involutive subalgebra. We prove that each $$(\sim _0)$$ -involutive equality algebra is a lattice, while it is distributive under a suitable condition. Then, we define $$(\sim _0)$$ -involutive deductive systems on bounded equality algebras and represent a condition under which the set of all dense elements of an equality algebra is a $$(\sim _0)$$ -involutive deductive system. Finally, we find the relations among 0-compatible equality algebras, residuated lattices and Boolean algebras.

GRADED CHAIN CONDITIONS AND LEAVITT PATH ALGEBRAS OF NO-EXIT GRAPHS

We obtain a complete structural characterization of Cohn–Leavitt algebras over no-exit objects as graded involutive algebras. Corollaries of this result include graph-theoretic conditions characterizing when a Leavitt path algebra is a directed union of (graded) matricial algebras over the underlying field and over the algebra of Laurent polynomials and when the monoid of isomorphism classes of finitely generated projective modules is atomic and cancelative. We introduce the nonunital generalizations of graded analogs of noetherian and artinian rings, graded locally noetherian and graded locally artinian rings, and characterize graded locally noetherian and graded locally artinian Leavitt path algebras without any restriction on the cardinality of the graph. As a consequence, we relax the assumptions of the Abrams–Aranda–Perera–Siles characterization of locally noetherian and locally artinian Leavitt path algebras.

Symmetric operators on modal pseudocomplemented De Morgan algebras

Modal pseudocomplemented De Morgan algebras are pseudocomplemented De Morgan algebras which satisfy the additional equation |$\mbox{(tm) }\, x\vee \sim x \leq x\vee x^\ast.$| There is a connection between these algebras and A. Monteiro’s tetravalent modal algebras. It is worth mentioning that these algebras have been studied previously using topological tools. In the present work, we enrich the language of modal pseudocomplemented De Morgan algebras by adding an automorphism of period |$2$|⁠, giving rise to a new algebraic structure which we call |$\cal S$|-algebras. From the logical point of view, we are enriching the original language with a new involutive unary connective. Among other things, it is proved that |$\cal S$| is a semisimple, locally finite variety. Besides, the generating algebras of this variety are determined. The structure of the free |$\cal S$|-algebra with a finite number of generators is studied, and a formula for computing its size is given. Finally, the lattice |$\Lambda({\cal S})$| is exhibited showing that the varieties of Boolean algebras, involutive Boolean algebras and three-valued Łukasiewicz algebras are proper subvarieties of |${\cal S}$|⁠.

About the convolution of distributions on groupoids

We review the properties of transversality of distributions with respect to submersions. This allows us to construct a convolution product for a large class of distributions on Lie groupoids. We get a unital involutive algebra \mathcal E_{r,s}'(G,\Omega^{1/2}) enlarging the convolution algebra C^\infty_c(G,\Omega^{1/2}) associated with any Lie groupoid G . We prove that G -operators are convolution operators by transversal distributions. We also investigate the microlocal aspects of the convolution product. We give sufficient conditions on wave front sets to compute the convolution product and we show that the wave front set of the convolution product of two distributions is essentially the product of their wave front sets in the symplectic groupoid T^*G of Coste–Dazord–Weinstein. This also leads to a subalgebra \mathcal E_{a}'(G,\Omega^{1/2}) of \mathcal E_{r,s}'(G,\Omega^{1/2}) which contains for instance the algebra of pseudodifferential G -operators and a class of Fourier integral G -operators which will be the central theme of a forthcoming paper.

ON THE HOCHSCHILD HOMOLOGY OF INVOLUTIVE ALGEBRAS

AbstractWe study the homological algebra of bimodules over involutive associative algebras. We show that Braun's definition of involutive Hochschild cohomology in terms of the complex of involution-preserving derivations is indeed computing a derived functor: the ℤ/2-invariants intersected with the centre. We then introduce the corresponding involutive Hochschild homology theory and describe it as the derived functor of the pushout of ℤ/2-coinvariants and abelianization.

On the structure of split involutive regular Hom-Lie algebras

On the structure of split involutive regular Hom-Lie algebras

On the structure of split involutive regular Hom-Lie algebras

On the structure of split involutive regular Hom-Lie algebras

Polynomial Representations of $$C^*$$ C ∗ -Algebras and their Applications

This is a sequel to our paper on nonlinear completely positive maps and dilation theory for real involutive algebras, where we have reduced all classification problems to the passage from a \(C^*\)-algebra \(\mathcal {A}\) to its symmetric powers \(S^n(\mathcal {A})\), resp., to holomorphic representations of the multiplicative \(*\)-semigroup \((\mathcal {A},\cdot )\). Here we study the correspondence between representations of \(\mathcal {A}\) and of \(S^n(\mathcal {A})\) in detail. As \(S^n(\mathcal {A})\) is the fixed point algebra for the natural action of the symmetric group \(S_n\) on \(\mathcal {A}^{\otimes n}\), this is done by relating representations of \(S^n(\mathcal {A})\) to those of the crossed product \(\mathcal {A}^{\otimes n} \rtimes S_n\) in which it is a hereditary subalgebra. For \(C^*\)-algebras of type I, we obtain a rather complete description of the equivalence classes of the irreducible representations of \(S^n(\mathcal {A})\) and we relate this to the Schur–Weyl theory for \(C^*\)-algebras. Finally we show that if \(\mathcal {A}\subseteq B(\mathcal {H})\) is a factor of type II or III, then its corresponding multiplicative representation on \(\mathcal {H}^{\otimes n}\) is a factor representation of the same type, unlike the classical case \(\mathcal {A}=B(\mathcal {H})\).