Invertible Elements Research Articles

A new invariant for finite dimensional Leibniz/Lie algebras

For an n-dimensional Leibniz/Lie algebra h over a field k we introduce a new invariant A(h), called the universal algebra of h, as a quotient of the polynomial algebra k[Xij|i,j=1,⋯,n] through an ideal generated by n3 polynomials. We prove that A(h) admits a unique bialgebra structure which makes it an initial object among all commutative bialgebras coacting on h. The new object A(h) is the key tool in answering two open problems in Lie algebra theory. First, we prove that the automorphism group AutLbz(h) of h is isomorphic to the group U(G(A(h)o)) of all invertible group-like elements of the finite dual A(h)o. Secondly, for an abelian group G, we show that there exists a bijection between the set of all G-gradings on h and the set of all bialgebra homomorphisms A(h)→k[G]. Based on this, all G-gradings on h are explicitly classified and parameterized. A(h) is also used to prove that there exists a universal commutative Hopf algebra associated to any finite dimensional Leibniz algebra h.

On sofic groups, Kaplansky's conjectures, and endomorphisms of pro-algebraic groups

Let G be a group. Let X be a connected algebraic group over an algebraically closed field K. Denote by A=X(K) the set of K-points of X. We study a class of endomorphisms of pro-algebraic groups, namely algebraic group cellular automata over (G,X,K). They are cellular automata τ:AG→AG whose local defining map is induced by a homomorphism of algebraic groups XM→X where M⊂G is a finite memory set of τ. Our first result is that when G is sofic, such an algebraic group cellular automaton τ is invertible whenever it is injective and char(K)=0. When G is amenable, we show that an algebraic group cellular automaton τ is surjective if and only if it satisfies a weak form of pre-injectivity called ▪-pre-injectivity. This yields an analogue of the classical Moore-Myhill Garden of Eden theorem. We also introduce the near ring R(K,G) which is K[Xg:g∈G] as an additive group but the multiplication is induced by the group law of G. The near ring R(K,G) contains naturally the group ring K[G] and we extend Kaplansky's conjectures to this new setting. Among other results, we prove that when G is an orderable group, then all one-sided invertible elements of R(K,G) are trivial, i.e., of the form aXg+b for some g∈G, a∈K⁎, and b∈K. This in turns allows us to show that when G is locally residually finite and orderable (e.g. Zd or a free group), all injective algebraic cellular automata τ:CG→CG are of the form τ(x)(h)=ax(g−1h)+b for all x∈CG,h∈G for some g∈G, a∈C⁎, and b∈C.

On invertible elements in reduced C⁎-algebras of acylindrically hyperbolic groups

Let G be an acylindrically hyperbolic group. We prove that if G has no non-trivial finite normal subgroups, then the set of invertible elements is dense in the reduced C⁎-algebra of G. The same result is obtained for finite direct products of acylindrically hyperbolic groups.

On, around, and beyond Frobenius' theorem on division algebras

ABSTRACT Frobenius' Theorem states that, besides the fields of real and complex numbers, the algebra of quaternions is the only finite-dimensional real division algebra. We first give a short elementary proof of this theorem, then characterize finite-dimensional real algebras that contain either a copy of , a copy of , or a pair of anticommuting invertible elements through the dimensions of their (left) ideals, and finally consider the problem of lifting algebraic elements modulo ideals.

One-sided central Drazin inverses

ABSTRACT We study the one-sided version of central Drazin inverses. An element a in a ring R is said to be left central Drazin invertible if there is such that , for some . The right central Drazin invertible elements can be defined similarly. It is shown that an element is central Drazin invertible if and only if a is both left and right central Drazin invertible. Some well-known results on Drazin inverses and left invertible elements including the famous Kaplansky theorem in [Jacobson N. Some remarks on one-sided inverses. Proc Amer Math Soc. 1950;1:352–355] are generalized. As applications, we give a new characterization of Dedekind-finite rings from the point of view of one-sided central Drazin invertible elements. The Cline's formula on Drazin invertible elements is also generalized.

Jacobson’s lemma and Cline’s formula for generalized inverses in a ring with involution

In a ring with involution, we prove that a Drazin invertible element is pseudo core invertible if and only if its spectral idempotent is {1, 4}-invertible. As its applications, we obtain necessary and sufficient conditions for 1 − ba (resp., ba) being pseudo core invertible while 1 − ab (resp., ab) has pseudo core inverse, and the pseudo core inverse of 1 − ba (resp., ba) is given in terms of 1 − ab (resp., ab). Inspired by the above idea, Jacobson’s lemma for Moore-Penrose inverse is considered.

On Homotopy Invariants of Tensor Products of Banach Algebras

We generalize results of Davie and Raeburn describing homotopy types of the group of invertible elements and of the set of idempotents of the projective tensor product of complex unital Banach algebras. We illustrate our results by specific examples.

On the classification of Schreier extensions of monoids with non-abelian kernel

Abstract We show that any regular (right) Schreier extension of a monoid M by a monoid A induces an abstract kernel Φ : M → End ⁡ ( A ) Inn ⁡ ( A ) {\Phi\colon M\to\frac{\operatorname{End}(A)}{\operatorname{Inn}(A)}} . If an abstract kernel factors through SEnd ⁡ ( A ) Inn ⁡ ( A ) {\frac{\operatorname{SEnd}(A)}{\operatorname{Inn}(A)}} , where SEnd ⁡ ( A ) {\operatorname{SEnd}(A)} is the monoid of surjective endomorphisms of A, then we associate to it an obstruction, which is an element of the third cohomology group of M with coefficients in the abelian group U ⁢ ( Z ⁢ ( A ) ) {U(Z(A))} of invertible elements of the center Z ⁢ ( A ) {Z(A)} of A, on which M acts via Φ. An abstract kernel Φ : M → SEnd ⁡ ( A ) Inn ⁡ ( A ) {\Phi\colon M\to\frac{\operatorname{SEnd}(A)}{\operatorname{Inn}(A)}} (resp. Φ : M → Aut ⁡ ( A ) Inn ⁡ ( A ) {\Phi\colon M\to\frac{\operatorname{Aut}(A)}{\operatorname{Inn}(A)}} ) is induced by a regular weakly homogeneous (resp. homogeneous) Schreier extension of M by A if and only if its obstruction is zero. We also show that the set of isomorphism classes of regular weakly homogeneous (resp. homogeneous) Schreier extensions inducing a given abstract kernel Φ : M → SEnd ⁡ ( A ) Inn ⁡ ( A ) {\Phi\colon M\to\frac{\operatorname{SEnd}(A)}{\operatorname{Inn}(A)}} (resp. Φ : M → Aut ⁡ ( A ) Inn ⁡ ( A ) {\Phi\colon M\to\frac{\operatorname{Aut}(A)}{\operatorname{Inn}(A)}} ), when it is not empty, is in bijection with the second cohomology group of M with coefficients in U ⁢ ( Z ⁢ ( A ) ) {U(Z(A))} .

Further results of special elements in rings with involution

In this paper, we consider Moore-Penrose invertible, group invertible, and core invertible elements in rings with involution to characterize EP, generalized normal, generalized Hermitian elements and generalized partial isometries. As a consequence, we obtain new characterizations for elements in rings with involution to be normal and Hermitian elements.

C∗-algebras defined by amalgamated duplication of C∗-algebras

Let [Formula: see text] and [Formula: see text] be two [Formula: see text]-algebras such that [Formula: see text] is a Banach [Formula: see text]-bimodule with the left and right compatible action of [Formula: see text] on [Formula: see text]. We define [Formula: see text] as a [Formula: see text]-algebra, where it is a strongly splitting [Formula: see text]-algebra extension of [Formula: see text] by [Formula: see text]. Normal, self-adjoint, unitary, invertible and projection elements of [Formula: see text] are characterized; sufficient and necessary conditions for existing unit and bounded approximate identity of [Formula: see text] as a Banach algebra and as a [Formula: see text]-algebra are given. We characterize ∗-automorphisms on [Formula: see text] and give some results related to ∗-homomorphisms, ∗-representations and completely bounded maps on this [Formula: see text]-algebra. Also, we have constructed a new Hilbert [Formula: see text]-module [Formula: see text] over [Formula: see text], where [Formula: see text] is a Hilbert [Formula: see text]-module over [Formula: see text] and [Formula: see text] is a Hilbert [Formula: see text]-module over [Formula: see text].

On the invertibility of solutions of first order linear homogeneous differential equations in Banach algebras

This work is devoted to the study of some properties of linear homogeneous differential equations of the first order in Banach algebras. It is found (for some types of Banach algebras), at what right-hand side of such an equation, from the invertibility of the initial condition it follows the invertibility of its solution at any given time. Associative Banach algebras over the field of real or complex numbers are considered. The right parts of the studied equations have the form [F(t)](x(t)), where {F(t)} is a family of bounded operators on the algebra, continuous with respect to t∈R. The problem is to find all continuous families of bounded operators on algebra, preserving the invertibility of elements from it, for a given Banach algebra. In the proposed article, this problem is solved for only three cases. In the first case, the algebra consists of all square matrices of a given order. For this algebra, it is shown that all continuous families of operators, preserving the invertibility of elements from the algebra at zero must be of the form [F(t)](y)=a(t)⋅y+y⋅b(t), where the families {a(t)} and {b(t)} are also continuous. In the second case, the algebra consists of all continuous functions on the segment. For this case, it is shown that all families of operators, preserving the invertibility of elements from the algebra at any time must be of the form [F(t)](y)=a(t)⋅y, where the family {a(t)} is also continuous. The third case concerns those Banach algebras in which all nonzero elements are invertible. For example, the algebra of complex numbers and the algebra of quaternions have this property. In this case, any continuous families of bounded operators preserves the invertibility of the elements from the algebra at any time. The proposed study is in contact with the research of the foundations of quantum mechanics. The dynamics of quantum observables is described by the Heisenberg equation. The obtained results are an indirect argument in favor of the fact, that the known form of the Heisenberg equation is the only correct one.

Groupoids on a skew lattice of objects

Motivated by some alternatives to the classical logical model of boolean algebra, this paper deals with algebraic structures which extend skew lattices by locally invertible elements. Following the meme of the Ehresmann-Schein-Nambooripad theorem, we consider a groupoid (small category of isomorphisms) in which the set of objects carries the structure of a skew lattice. The objects act on the morphisms by left and right restriction and extension mappings of the morphisms, imitating those of an inductive groupoid. Conditions are placed on the actions, from which pseudoproducts may be defined. This gives an algebra of signature (2, 2, 1), in which each binary operation has the structure of an orthodox semigroup. In the reverse direction, a groupoid of the kind described may be reconstructed from the algebra.

Sums and products of units and idempotents in duo rings

An element in a ring R is said to be clean (respectively unit-regular) if it is a sum (respectively product) of an invertible element and an idempotent. It is known that all elements in R are clean, if all of them are unit-regular. We study some classes of unit-regular and clean matrices over a duo ring. It is shown that a right adequate duo domain is almost 2-good ring. Cite as: A. I. Gatalevych, M. I. Kuchma, Non-singular solutions of one class of matrix equations over a polynomial ring, Prykl. Probl. Mekh. Mat. , Issue 17, 42–46 (2019) (in Ukrainian), https://doi.org/10.15407/apmm2019.17.42-46

Isometries on Positive Definite Operators with Unit Fuglede-Kadison Determinant

In this paper we explore the structure of certain surjective generalized isometries (which are transformations that leave any given member of a large class of generalized distance measures invariant) of the set of positive invertible elements in a finite von Neumann factor with unit Fuglede-Kadison determinant. We conclude that any such map originates from either an algebra $^*$-isomorphism or an algebra $^*$-antiisomorphism of the underlying operator algebra.

Groupoids and Coherent States

Schwinger’s algebra of selective measurements has a natural interpretation in terms of groupoids. This approach is pushed forward in this paper to show that the theory of coherent states has a natural setting in the framework of groupoids. Thus given a quantum mechanical system with associated Hilbert space determined by a representation of a groupoid, it is shown that any invariant subset of the group of invertible elements in the groupoid algebra determines a family of generalized coherent states provided that a completeness condition is satisfied. The standard coherent states for the harmonic oscillator as well as generalized coherent states for f-oscillators are exemplified in this picture.

The Kauffman bracket skein module of the handlebody of genus 2 via braids

In this paper we present two new bases, [Formula: see text] and [Formula: see text], for the Kauffman bracket skein module of the handlebody of genus 2 [Formula: see text], KBSM[Formula: see text]. We start from the well-known Przytycki-basis of KBSM[Formula: see text], [Formula: see text], and using the technique of parting we present elements in [Formula: see text] in open braid form. We define an ordering relation on an augmented set [Formula: see text] consisting of monomials of all different “loopings” in [Formula: see text], that contains the sets [Formula: see text], [Formula: see text] and [Formula: see text] as proper subsets. Using the Kauffman bracket skein relation we relate [Formula: see text] to the sets [Formula: see text] and [Formula: see text] via a lower triangular infinite matrix with invertible elements in the diagonal. The basis [Formula: see text] is an intermediate step in order to reach at elements in [Formula: see text] that have no crossings on the level of braids, and in that sense, [Formula: see text] is a more natural basis of KBSM[Formula: see text]. Moreover, this basis is appropriate in order to compute Kauffman bracket skein modules of closed–connected–oriented (c.c.o.) 3-manifolds [Formula: see text] that are obtained from [Formula: see text] by surgery, since isotopy moves in [Formula: see text] are naturally described by elements in [Formula: see text].

Existence and uniqueness of the Karcher mean on unital C⁎-algebras

The Karcher mean on the cone Ω of invertible positive elements of the C⁎-algebra B(E) of bounded operators on a Hilbert space E has recently been extended to a contractive barycentric map on the space of L1-probability measures on Ω. In this paper we first show that the barycenter satisfies the Karcher equation and then establish the uniqueness of the solution. Next we establish that the Karcher mean is real analytic in each of its coordinates, and use this fact to show that both the Karcher mean and Karcher barycenter map exist and are unique on any unital C⁎-algebra. The proof depends crucially on a recent result of the author giving a converse of the inverse function theorem.

Special clean elements in rings

A clean decomposition [Formula: see text] in a ring [Formula: see text] (with idempotent [Formula: see text] and unit [Formula: see text]) is said to be special if [Formula: see text]. We show that this is a left-right symmetric condition. Special clean elements (with such decompositions) exist in abundance, and are generally quite accessible to computations. Besides being both clean and unit-regular, they have many remarkable properties with respect to element-wise operations in rings. Several characterizations of special clean elements are obtained in terms of exchange equations, Bott–Duffin invertibility, and unit-regular factorizations. Such characterizations lead to some interesting constructions of families of special clean elements. Decompositions that are both special clean and strongly clean are precisely spectral decompositions of the group invertible elements. The paper also introduces a natural involution structure on the set of special clean decompositions, and describes the fixed point set of this involution.

Manifolds of classical probability distributions and quantum density operators in infinite dimensions

The manifold structure of subsets of classical probability distributions and quantum density operators in infinite dimensions is investigated in the context of C^{*}-algebras and actions of Banach-Lie groups. Specificaly, classical probability distributions and quantum density operators may be both described as states (in the functional analytic sense) on a given C^{*}-algebra mathscr {A} which is Abelian for Classical states, and non-Abelian for Quantum states. In this contribution, the space of states mathscr {S} of a possibly infinite-dimensional, unital C^{*}-algebra mathscr {A} is partitioned into the disjoint union of the orbits of an action of the group mathscr {G} of invertible elements of mathscr {A}. Then, we prove that the orbits through density operators on an infinite-dimensional, separable Hilbert space mathcal {H} are smooth, homogeneous Banach manifolds of mathscr {G}=mathcal {GL}(mathcal {H}), and, when mathscr {A} admits a faithful tracial state tau like it happens in the Classical case when we consider probability distributions with full support, we prove that the orbit through tau is a smooth, homogeneous Banach manifold for mathscr {G}.

WEIGHTED CORE–EP INVERSE AND WEIGHTED CORE–EP PRE-ORDERS IN A -ALGEBRA

We define extensions of the weighted core–EP inverse and weighted core–EP pre-orders of bounded linear operators on Hilbert spaces to elements of a $C^{\ast }$ -algebra. Some properties of the weighted core–EP inverse and weighted core–EP pre-orders are generalized and some new ones are proved. Using the weighted element, the weighted core–EP pre-order, the minus partial order and the star partial order of certain elements, new weighted pre-orders are presented on the set of all $wg$ -Drazin invertible elements of a $C^{\ast }$ -algebra. Applying these results, we introduce and characterize new partial orders which extend the core–EP pre-order to a partial order.