Approximate identities on some homogeneous Banach spaces

We study convergence of approximate identities on some complete semi-normed or normed spaces of locally L ^p functions where translations are isometries, namely Marcinkiewicz spaces M^p{\mathcal{M}^{p}} and Stepanoff spaces S^p{\mathcal{S}^p}, 1 ≤ p < ∞, as well as others where translations are not isometric but bounded (the bounded p-mean spaces M ^p ) or even unbounded (M^p₀{M^{p}_{0}}). We construct a function f that belongs to these spaces and has the property that all approximate identities f_e * f{\phi_\varepsilon * f} converge to f pointwise but they never converge in norm.     

Approximation capabilities of immersed finite element spaces for elasticity Interface problems

We construct and analyze a group of immersed finite element (IFE) spaces formed by linear, bilinear, and rotated Q₁ polynomials for solving planar elasticity equation involving interface. The shape functions in these IFE spaces are constructed through a group of approximate jump conditions such that the unisolvence of the bilinear and rotated Q₁ IFE shape functions are always guaranteed regardless of the Lamé parameters and the interface location. The boundedness property and a group of identities of the proposed IFE shape functions are established. A multi‐point Taylor expansion is utilized to show the optimal approximation capabilities for the proposed IFE spaces through the Lagrange type interpolation operators.     

Functional equations for higher logarithms

Following earlier work by Abel and others, Kummer gave in 1840 functional equations for the polylogarithm function Li _m (z) up to m = 5, but no example for larger m was known until recently. We give the first genuine 2-variable functional equation for the 7-logarithm. We investigate and relate identities for the 3-logarithm given by Goncharov and Wojtkowiak and deduce a certain family of functional equations for the 4-logarithm.     

Identities of PI-Algebras Graded by a Finite Abelian Group

We consider associative PI-algebras over an algebraically closed field of zero characteristic graded by a finite abelian group G. It is proved that in this case the ideal of graded identities of a G-graded finitely generated PI-algebra coincides with the ideal of graded identities of some finite dimensional G-graded algebra. This implies that the ideal of G-graded identities of any (not necessary finitely generated) G-graded PI-algebra coincides with the ideal of G-graded identities of the Grassmann envelope of a finite dimensional (G × ?₂)-graded algebra, and is finitely generated as GT-ideal. Similar results take place for ideals of identities with automorphisms.     

Identities of Group Algebras

We consider polynomial identities of group algebras over a field F of characteristic zero. We prove that any PI group algebra satisfies the same identities as a matrix algebra M _n (F ), where n is the maximal degree of finite dimensional representations of the group over algebraic extensions of F.     

Involutions for upper triangular matrix algebras

In this paper we describe completely the involutions of the first kind of the algebra UT_n(F) of n×n upper triangular matrices. Every such involution can be extended uniquely to an involution on the full matrix algebra. We describe the equivalence classes of involutions on the upper triangular matrices. There are two distinct classes for UT_n(F) when n is even and a single class in the odd case.Furthermore we consider the algebra UT₂(F) of the 2×2 upper triangular matrices over an infinite field F of characteristic different from 2. For every involution *, we describe the *-polynomial identities for this algebra. We exhibit bases of the corresponding ideals of identities with involution, and compute the Hilbert (or Poincaré) series and the codimension sequences of the respective relatively free algebras.Then we consider the *-polynomial identities for the algebra UT₃(F) over a field of characteristic zero. We describe a finite generating set of the ideal of *-identities for this algebra. These generators are quite a few, and their degrees are relatively large. It seems to us that the problem of describing the *-identities for the algebra UT_n(F) of the n×n upper triangular matrices may be much more complicated than in the case of ordinary polynomial identities.     

Identities of the Semigroup of Upper Triangular Tropical Matrices

A new family of identities satisfied by the semigroups U _n (𝕋) of n × n upper triangular tropical matrices is constructed and an elementary proof is given.     

On two fifth order mock theta functions

We consider the fifth order mock theta functions χ ₀ and χ ₁, defined by Ramanujan, and find identities for these functions, which relate them to indefinite theta functions. Similar identities have been found by Andrews for the other fifth order mock theta functions and the seventh order functions.     

A Fredholm operator approach to Morita equivalence

GivenC*-algebrasA andB and an imprimitivityA-B-bimoduleX, we construct an explicit isomorphismX _*:K _i (A)K _i (B), whereK _i denotes the complexK-theory functors fori=0, 1. Our techniques do not require separability nor the existence of countable approximate identities. We thus extend to generalC*-algebras the result of Brown, Green, and Rieffel according to which, strongly Morita equivalentC*-algebras have isomorphicK-groups. The method employed includes a study of Fredholm operators on Hilbert modules.On leave from the University of São Paulo, Brazil.     

Hybrid bases for varieties of semigroups

We consider the lower part of the lattice of varieties of semigroups. We present finite bases of hybrid identities for the varieties of normal bands, commutative bands and abelian groups of finite exponent.The variety A_n,0 of abelian groups provides an example of a variety which has no finite base of hyperidentities (cf. [12]) but has a finite base of hybrid identities.     

Isotopes of prime (−1, 1)-and Jordan algebras

We deal with adjoint commutator and Jordan algebras of isotopes of prime strictly (−1, 1)-algebras. It is proved that a system of identities of the form [x ₁, x ₂, x ₂, x ₃,…, x _n ] for n = 2,.…, 5 is discernible on isotopes of prime (−1, 1)-algebras. Also it is shown that adjoint Jordan algebras for suitable isotopes of prime (−1, 1)-algebras may possess distinct sets of identities. In particular, isotopes of a prime Jordan monster have different sets of identities in general.     

An Identity of Ramanujan and the Representation of Integers as Sums of Triangular Numbers

Let k be a positive number and t _k(n) denote the number of representations of n as a sum of k triangular numbers. In this paper, we will calculate t _2k (n) in the spirit of Ramanujan. We first use the complex theory of elliptic functions to prove a theta function identity. Then from this identity we derive two Lambert series identities, one of them is a well-known identity of Ramanujan. Using a variant form of Ramanujan's identity, we study two classes of Lambert series and derive some theta function identities related to these Lambert series . We calculate t ₁₂(n), t ₁₆(n), t ₂₀(n), t ₂₄(n), and t ₂₈(n) using these Lambert series identities. We also re-derive a recent result of H. H. Chan and K. S. Chua [6] about t ₃₂(n). In addition, we derive some identities involving the Ramanujan function (n), the divisor function ₁₁(n), and t ₂₄(n). Our methods do not depend upon the theory of modular forms and are somewhat more transparent.     

Approximate amenability is not bounded approximate amenability

We obtain a new criterion sufficient for approximate amenability of Banach algebras, and an associated criterion sufficient for approximate amenability of _c0$c_0$-direct sums of approximately amenable Banach algebras. We use these criteria to give examples of Banach algebras which are approximately amenable, but not boundedly approximately amenable. Thus we answer a question which was open since the year 2004. This is, so to speak, the complementary result to one of our earlier ones in the paper [F. Ghahramani, C.J. Read, Approximate identities in approximate amenability, J. Funct. Anal. 262 (9) (2012) 3929–3945], where we gave examples of boundedly approximately amenable Banach algebras that are not boundedly approximately contractible.     

A Basis for the Graded Identities of the Matrix Algebra of Order Two over a Finite Field of Characteristic p≠2

Let K be a finite field of characteristic p>2, and let M₂(K) be the matrix algebra of order two over K. We describe up to a graded isomorphism the 2-gradings of M₂(K). It turns out that there are only two nonisomorphic nontrivial such gradings. Furthermore, we exhibit finite bases of the graded polynomial identities for each one of these two gradings. One can distinguish these two gradings by means of the graded polynomial identities they satisfy.     

Graded Polynomial Identities for Tensor Products by the Grassmann Algebra

Abstract

Let 𝕂 be a field of characteristic zero, and R be a G-graded 𝕂-algebra. We consider the algebra R ? E, then deduce its G × ?₂-graded polynomial identities starting from the G-graded polynomial identities of R. As a consequence, we describe a basis for the ? _n  × ?₂-graded identities of the algebras M _n (E). Moreover we give the graded cocharacter sequence of M ₂(E), and show that M ₂(E) is PI-equivalent to M _1,1(E) ? E. This fact is a particular case of a more general result obtained by Kemer.     

Graded central polynomials for the matrix algebra of order two

Let K be an infinite integral domain, and let A = M ₂(K) be the matrix algebra of order two over K. The algebra A can be given a natural \mathbbZ₂{\mathbb{Z}_2} -grading by assuming that the diagonal matrices are the 0-component while the off-diagonal ones form the 1-component. In this paper we study the graded identities and the graded central polynomials of A. We exhibit finite bases for these graded identities and central polynomials. It turns out that the behavior of the graded identities and central polynomials in the case under consideration is much like that in the case when K is an infinite field of characteristic 0 or p > 2. Our proofs are characteristic-free so they work when K is an infinite field, char K = 2. Thus we describe finite bases of the graded identities and graded central polynomials for M ₂(K) in this case as well.     

Convergence of Newton–Kantorovich Approximations to an Approximate Zero

In this paper, we prove an a posteriori and an a priori convergence theorem for Newton–Kantorovich approximations starting from an initial point x ₀. We apply these results to operators that are analytic at interior points of a closed ball centered at x ₀ and of radius R. We obtain some theorems on approximate zeros and on approximate zeros of second kind for these operators, which improve previous results.     

Approximation Numbers of Identity Operators between Symmetric Sequence Spaces

We prove asymptotic formulas for the behavior of approximation quantities of identity operators between symmetric sequence spaces. These formulas extend recent results of Defant, Masty o, and Michels for identities l_pⁿF_n with an n-dimensional symmetric normed space F_n with p-concavity conditions on F_n and 1p2. We consider the general case of identities E_nF_n with weak assumptions on the asymptotic behavior of the fundamental sequences of the n-dimensional symmetric spaces E_n and F_n. We give applications to Lorentz and Orlicz sequence spaces, again considerably generalizing results of Pietsch, Defant, Masty o, and Michels.     

Ideals of identities of representations of nilpotent lie algebras

Let L be a Lie algebra, nilpotent of class 2, over an infinite field K, and suppose that the centre C of L is one dimensional; such Lie algebras are called Heisenberg algebras. Let ρ:L→hom _KV be a finite dimensional representation of the Heisenberg algebra L such that ρ(C) contains non-singular linear transformations of V, and denote l(ρ) the ideal of identities for the representation ρ. We prove that the ideals of identities of representations containing I(ρ) and generated by multilinear polynomials satisfy the ACC. Let sl ₂(L) be the Lie algebra of the traceless 2×2 matrices over K, and suppose the characteristic of K equals 2. As a corollary we obtain that the ideals of identities of representations of Lie algebras containing that of the regular representation of sl ₂(K) and generated by multilinear polynomials, are finitely based. In addition we show that one cannot simply dispense with the condition of multilinearity. Namely, we show that the ACC is violated for the ideals of representations of Lie algebras (over an infinite field of characteristic 2) that contain the identities of the regular representation of sl ₂(K).