Invertible Elements Research Articles

On a Banach algebra of entire functions with a weighted Hadamard multiplication

New algebraic-analytic properties of a previously studied Banach algebra A(p) of entire functions are established. For a given fixed sequence (p(n))n≥0 of positive real numbers, such that limn→∞⁡p(n)1n=∞, the Banach algebra A(p) is the set of all entire functions f such that f(z)=∑n=0∞fˆ(n)zn (z∈C), where the sequence (fˆ(n))n≥0 of Taylor coefficients of f satisfies fˆ(n)=O(p(n)−1) for n→∞, with pointwise addition and scalar multiplication, a weighted Hadamard multiplication ⁎ with weight given by p (i.e., (f⁎g)(z)=∑n=0∞p(n)fˆ(n)gˆ(n)zn for all z∈C), and the norm ‖f‖=supn≥0⁡p(n)|fˆ(n)|. The following results are shown:•The Topological stable rank of A(p) is 1.•The Bass stable rank of A(p) is 1.•A(p) is a Hermite ring.•A(p) is not a projective-free ring.•Idempotents in A(p) are described.•Exponentials in A(p) are described, and it is shown that every invertible element of A(p) has a logarithm, so that the first Čech cohomology group H1(M(A(p)),Z) with integer coefficients of the maximal ideal space M(A(p)) is trivial.•A generalised necessary and sufficient ‘corona-type condition’ on the matricial data (A,b) with entries from A(p) is given for the solvability of Ax=b with x also having entries from A(p).•The Krull dimension of A(p) is infinite.•A(p) is neither Artinian nor Noetherian.•A(p) is coherent.•The special linear group over A(p) is generated by elementary matrices.

Functional identities on Banach algebras and matrix rings

Let A be a unital Banach algebra and M be a unital A -bimodule. We show that additive mappings δ , τ : A → M satisfy the identity A − 1 δ ( A ) + δ ( A ) A − 1 + A τ ( A − 1 ) + τ ( A − 1 ) A = 0 for all invertible element A in A if and only if there exists a Jordan derivation D such that δ ( A ) = D ( A ) + A δ ( I ) and τ ( A ) = D ( A ) − δ ( I ) A for all A ∈ A . We also characterize additive mappings δ , τ : A → M satisfying the identity δ ( A ) B + A τ ( B ) = M for all A , B ∈ A with AB = N, where N is an invertible element in A and M is a fixed element in M . Similar conclusions are obtained for additive mappings from a matrix ring M n ( D ) into its unital bimodule satisfying any of the above identities, where D is a division ring and char D ≠ 2 .

A New Class of Leonardo Hybrid Numbers and Some Remarks on Leonardo Quaternions over Finite Fields

In this paper, we present a new class of Leonardo hybrid numbers that incorporate quantum integers into their components. This advancement presents a broader generalization of the q-Leonardo hybrid numbers. We explore some fundamental properties associated with these numbers. Moreover, we study special Leonardo quaternions over finite fields. In particular, we determine the Leonardo quaternions that are zero divisors or invertible elements in the quaternion algebra over the finite field Zp for special values of prime integer p.

Some decompositions of matrices over local rings

Let [Formula: see text] be a local ring with maximal ideal [Formula: see text], let [Formula: see text] be a natural number greater than [Formula: see text] and let [Formula: see text] be a matrix in the general linear group [Formula: see text] of degree [Formula: see text] over [Formula: see text]. We firstly show that if the matrix [Formula: see text] is nonscalar in [Formula: see text] and [Formula: see text] are invertible elements in [Formula: see text], then there exists an invertible element [Formula: see text] such that [Formula: see text] is similar to the product [Formula: see text] in which [Formula: see text] is a lower uni-triangular matrix and [Formula: see text] is an upper triangular matrix whose diagonal entries are [Formula: see text]. We then present some applications of this factorization to find decompositions of matrices in [Formula: see text] into product of commutators and involutions.

Universal constructions for Poisson algebras. Applications

We introduce the universal algebra of two Poisson algebras P and Q as a commutative algebra A:=P(P,Q) satisfying a certain universal property. The universal algebra is shown to exist for any finite dimensional Poisson algebra P and several of its applications are highlighted. For any Poisson P-module U, we construct a functor U⊗−:AM→QPM from the category of A-modules to the category of Poisson Q-modules which has a left adjoint whenever U is finite dimensional. Similarly, if V is an A-module, then there exists another functor −⊗V:PPM→QPM connecting the categories of Poisson representations of P and Q and the latter functor also admits a left adjoint if V is finite dimensional. If P is n-dimensional, then P(P):=P(P,P) is the initial object in the category of all commutative bialgebras coacting on P. As an algebra, P(P) can be described as the quotient of the polynomial algebra k[Xij|i,j=1,⋯,n] through an ideal generated by 2n3 non-homogeneous polynomials of degree ≤2. Two applications are provided. The first one describes the automorphisms group AutPoiss(P) as the group of all invertible group-like elements of the finite dual P(P)o. Secondly, we show that for an abelian group G, all G-gradings on P can be explicitly described and classified in terms of the universal coacting bialgebra P(P).

Lyapunov irregular set of Banach cocycles

Let be a topologically mixing subshift of finite type and let be a Hölder continuous map where is the set of invertible elements of a Banach ring B. We prove that either all ergodic measures have the same maximal Lyapunov exponent, or the irregular set is residual in and has full upper capacity topological entropy of . We also prove for generic fiber-bunched cocycles, similar statements can be strengthened by considering the Bowen topological entropy.

Group-like small cancellation theory for rings

In this paper, we develop a small cancellation theory for associative algebras with a basis of invertible elements. Namely, we study quotients of a group algebra of a free group and introduce three axioms for the corresponding defining relations. We show that the obtained ring is non-trivial. Moreover, we show that this ring enjoys a global filtration that agrees with relations, find a basis of the ring as a linear space and establish the corresponding structure theorems. We also provide a revision of a concept of Gröbner basis for our rings and establish a greedy algorithm for the Ideal Membership Problem.

Rings in which not invertible elements are uniquely clean

We call a ring R a generalized uniquely clean (or GUC for short) if every not invertible element in R is uniquely clean. Let R be a ring. It is shown that R is GUC if and only if it is a local ring or a uniquely clean ring. Thus the GUC ring is a generalization of the local ring. Some basic properties of GUC rings are proved.

Multiplicative closure of generalized invertible elements in a ring

Multiplicative closure of generalized invertible elements in a ring

Subset sums over Galois rings II

Suppose that p is an odd prime, and R=GR(p2,p2r) is a Galois ring of characteristic p2, whose cardinality is p2r. For any subset D⊆R, let N(D,n,b) and Nf(D,n,b), respectively, be the number of n-subsets, denoted by S, in D such that ∑x∈Sx=b and ∑x∈Sf(x)=b, where f(x)∈R[x] is a polynomial with degree d. In this paper, we give the asymptotic formulae of N(D,n,b) and Nf(D,n,b) under some certain constraints of D and f(x). The special cases D={ak:a∈R⁎} and f(x)=xk were studied by the authors in a previous paper, where R⁎ is the subset of all invertible elements of R and k is a given positive integer, which extends the former one in a more general setting. These extensions benefit from a refined form of the Li–Wan new sieve as well as a variant of the Weil bound on Galois rings.

Some new characterizations of EP elements, partial isometries and strongly EP elements in rings with involution

Abstract This paper mainly gives some sufficient and necessary conditions for an element in a ring to be EP {\mathrm{EP}} , partial isometry and strongly EP {\mathrm{EP}} by some equalities, using the solutions of certain equations and constructing the invertible elements in a ring.

Multi-recipient and threshold encryption based on hidden multipliers

Let $S$ be a pool of $s$ parties and Alice be the dealer. In this paper, we propose a scheme that allows the dealer to encrypt messages in such a way that only one authorized coalition of parties (which the dealer chooses depending on the message) can decrypt. At the setup stage, each of the parties involved in the process receives an individual key from the dealer. To decrypt information, an authorized coalition of parties must work together to use their keys. Based on this scheme, we propose a threshold encryption scheme. For a given message $f$ the dealer can choose any threshold $m = m(f).$ More precisely, any set of parties of size at least $m$ can evaluate $f$; any set of size less than $m$ cannot do this. Similarly, the distribution of keys among the included parties can be done in such a way that authorized coalitions of parties will be given the opportunity to put a collective digital signature on any documents. This primitive can be generalized to the dynamic setting, where any user can dynamically join the pool $S$. In this case the new user receives a key from the dealer. Also any user can leave the pool $S$. In both cases, already distributed keys of other users do not change. The main feature of the proposed schemes is that for a given $s$ the keys are distributed once and can be used multiple times. The proposed scheme is based on the idea of hidden multipliers in encryption. As a platform, one can use both multiplicative groups of finite fields and groups of invertible elements of commutative rings, in particular, multiplicative groups of residue rings. We propose two versions of this scheme.

Generalized Drazin invertible elements relative to a regularity

This paper is an attempt to give an axiomatic approach to the investigation of various kinds of generalizations of Drazin invertibility in Banach algebras. We shall say that an element a of a Banach algebra is generalized Drazin invertible relative to a regularity if there is such that and . The concept of Koliha-Drazin invertible elements, as well as some generalizations of this concept are described via the concept of generalized Drazin invertible elements relative to a regularity which satisfies two properties: (D1) if , p is an idempotent commuting with a and b, then ; (D2) if , then a is almost invertible. If a regularity satisfies the properties (D1) and (D2), we prove that is generalized Drazin invertible relative to if and only if 0 is not an accumulation point of . In particular we define and characterize generalized Drazin-T-Riesz invertible elements relative to an arbitrary (not necessarily bounded) Banach algebra homomorphism T and so extend the concept of generalized Drazin-Riesz invertible operators introduced in [Živković-Zlatanović SČ, Cvetković MD. Generalized Kato-Riesz decomposition and generalized Drazin-Riesz invertible operators. Linear Multilinear A. 2017;65(6):1171–1193]. Also we consider generalized Drazin invertibles relative to in the case when is the set of Drazin invertibles, as well as when is the set of Koliha-Drazin invertibles.

Maps preserving products of commuting elements in von Neumann algebras

A bijective map between specific structures in operator algebras is called a piecewise isomorphism if it preserves products of commuting elements in both directions. We shall show that any bicontinuous piecewise isomorphism between positive cones of invertible elements in von Neumann algebras or between unitary groups of von Neumann algebras can be described in terms of the following parameters: (i) Jordan *-isomorphism between given algebras (ii) one fixed central element in domain (or range) algebra (iii) a hermitian linear map from one algebra into the center of the other one. Especially, in case of factors any piecewise isomorphism between unitary groups is a Jordan *-isomorphism or Jordan *-isomorphism composed with inversion. This extends hitherto known results from von Neumann factors to general von Neumann algebras and brings new Jordan invariants of operator structures.

The Functor $K_{0}^{\operatorname {gr}}$ is Full and only Weakly Faithful

The Graded Classification Conjecture states that the pointed $K_{0}^{\operatorname {gr}}$ -group is a complete invariant of the Leavitt path algebras of finite graphs when these algebras are considered with their natural grading by $\mathbb {Z}$ . The strong version of this conjecture states that the functor $K_{0}^{\operatorname {gr}}$ is full and faithful when considered on the category of Leavitt path algebras of finite graphs and their graded homomorphisms modulo conjugations by invertible elements of the zero components. We show that the functor $K_{0}^{\operatorname {gr}}$ is full for the unital Leavitt path algebras of countable graphs and that it is faithful (modulo specified conjugations) only in a certain weaker sense.

Compact perturbations of Moore-Penrose invertible operators

Firstly, we study the stability of the Moore-Penrose invertibility of Hilbert space operators, establishing necessary and sufficient conditions for the invariance of the Moore-Penrose invertibility under small perturbations, compact perturbations and small compact perturbations. Secondly, we study the lifting problem of Moore-Penrose invertible elements of the Calkin algebra, showing that essentially Moore-Penrose invertible operators are small compact perturbations of Moore-Penrose invertible operators. Finally, we discuss the compactness of Moore-Penrose spectra, showing that each Hilbert space operator can be perturbed into an operator with nonempty compact Moore-Penrose spectrum by an arbitrarily small compact perturbation.

Approximately invertible elements in non-unital normed algebras

We introduce the concept of approximately invertible elements in non-unital normed algebras which is, on one side, a natural generalization of invertibility when having approximate identities at hand, and, on the other side, it is a direct extension of topological invertibility to non-unital algebras. Basic observations relate approximate invertibility with concepts of topological divisors of zero and density of (modular) ideals. We exemplify approximate invertibility in the group algebra, Wiener algebras, and operator ideals. For Wiener algebras with approximate identities (in particular, for the Fourier image of the convolution algebra), the approximate invertibility of an algebra element is equivalent to the property that it does not vanish. We also study approximate invertibility and its deeper connection with the Gelfand and representation theory in non-unital abelian Banach algebras as well as abelian and non-abelian C*-algebras.

On the set of (b,c)-invertible elements

Let b and c be two elements in a semigroup S. This paper is devoted to studying the structures of S||(b,c) and H(b,c) in a semigroup S, where S||(b,c) stands for the set of all (b, c)-invertible elements and H(b,c) = {y ? S | bS1 = yS1, S1y = S1c}. Denote the (b, c)-inverse of a ? S||(b,c) by a||(b,c). If S||(b,c) , ?, then H(b,c) = {a||(b,c)| a ? S||(b,c)}. We first find some new equivalent conditions for H(b,c) to be a group and analyze its structure from the viewpoint of generalized inverses. Then a necessary and sufficient condition under which S||(b,c) is a subsemigroup of S with the reverse order law holding for (b, c)-inverses is presented. At last, given a, b, c, d, x, y, z ? S and y ? S||(b,c), we prove that any two of the conditions x ? S||(a,c), z ? S||(b,d) and zy||(b,c)x ? S||(a,d) imply the rest one.

EP elements of Z[x]/(x2+x)

In this paper, we first show that the quotient ring Z[x]/(x2 + x) is an involution-ring with the involution ? given by (a1 +a2x)* = a1 ?a2 ?a2x, where a1, a2 ? Z. Then, we determine explicitly all invertible elements, regular elements, MP-inverses, group invertible elements, EP elements and SEP elements of Z[x]/(x2 + x). Furthermore, we give a new characterization for Abel rings.

Right e-core inverse and the related generalized inverses in rings

In this paper, some characterizations and properties of right e-core inverses by using right invertible element and {1, 3e}-inverse are investigated. Meanwhile, some characterizations for a new generalized right e-core inverse which is called right pseudo e-core inverse are also studied. The relationship between right pseudo e-core inverses and right e-core inverses are presented.