Module operator virtual diagonals on the Fourier algebra of an inverse semigroup

For an amenable inverse semigroup S with the set of idempotents E and a minimal idempotent, we explicitly construct a contractive and positive module operator virtual diagonal on the Fourier algebra A(S), as a completely contractive Banach algebra and operator module over \(\ell ^1(E)\). This generalizes a well known result of Zhong-Jin Ruan on operator amenability of the Fourier algebra of a (discrete) group Ruan (Am J Math 117:1449–1474, 1995).     

Module character amenability of Banach algebras

In this paper we introduce the notion of module character amenable Banach algebras and show that they possess module character virtual (approximate) diagonals. As a basic example, we show that for an inverse semigroup S with the set of idempotents E, the semigroup algebra ? ¹(S) is module character amenable as an ? ¹(E)-module if only if S is amenable.     

Module Amenability for Semigroup Algebras

We extend the concept of amenability of a Banach algebra A to the case that there is an extra \A-module structure on A, and show that when S is an inverse semigroup with subsemigroup E of idempotents, then A = \ell¹(S) as a Banach module over \A = \ell¹(E) is module amenable if and only if S is amenable. When S is a discrete group, \ell¹(E) = and this is just the Johnsons theorem.     

Super module amenability of inverse semigroup algebras

In this paper we compare the notions of super amenability and super module amenability of Banach algebras, which are Banach modules over another Banach algebra with compatible actions. We find conditions for the two notions to be equivalent. In particular, we study arbitrary module actions of l ¹(E _S ) on l ¹(S) for an inverse semigroup S with the set of idempotents E _S and show that under certain conditions, l ¹(S) is super module amenable if and only if S is finite. We also study the super module amenability of l ¹(S)^?? and module biprojectivity of l ¹(S), for arbitrary actions.     

Realizing irreducible semigroups and real algebras of compact operators

Let B(X) be the algebra of bounded operators on a complex Banach space X. Viewing B(X) as an algebra over R, we study the structure of those irreducible subalgebras which contain nonzero compact operators. In particular, irreducible algebras of trace-class operators with real trace are characterized. This yields an extension of Brauer-type results on matrices to operators in infinite dimensions, answering the question: is an irreducible semigroup of compact operators with real spectra realizable, i.e., simultaneously similar to a semigroup whose matrices are real?     

Biflatness of ℓ<Superscript>1</Superscript>-Semilattice Algebras

We show that if L is a semilattice then the ℓ¹-convolution algebra of L is biflat precisely when L is "uniformly locally finite". Our proof technique shows in passing that if this convolution algebra is biflat then it is isomorphic as a Banach algebra to the Banach space ℓ¹(L) equipped with pointwise multiplication. At the end we sketch how these techniques may be extended to prove an analogous characterisation of biflatness for Clifford semigroup algebras.     

Universal Jensen＇s Equations in Banach Modules over a C^＊-Algebra and Its Unitary Group

In this paper, we prove the generalized Hyers-Ulam-Rassias stability of universal Jensen‘s equations in Banach modules over a unital C^*-algebra. It is applied to show the stability of universal Jensen‘s equations in a Hilbert module over a unital C^*-algebra. Moreover, we prove the stability of linear operators in a Hilbert module over a unital C^*-algebra.     

Amenability constants for semilattice algebras

For any finite commutative idempotent semigroup S, a semilattice, we show how to compute the amenability constant of its semigroup algebra ℓ ¹(S). This amenability constant is always of the form 4n+1. We then show that these give lower bounds to amenability constants of certain Banach algebras graded over semilattices. We also give example of a commutative Clifford semigroups G _n whose semigroup algebras ℓ ¹(G _n ) admit amenability constants of the form 41+4(n−1)/n. We also show there is no commutative semigroup whose semigroup algebra has an amenability constant between 5 and 9. N. Spronk’s research was supported by NSERC Grant 312515-05.     

Approximate Amenability of Certain Semigroup Algebras

In the present paper, it is shown that a left cancellative semigroup S (not necessarily with identity) is left amenable whenever the Banach algebra ℓ¹(S) is approximately amenable. It is also proved that if S is a Brandt semigroup over a group G with an index set I, then ℓ¹(S) is approximately amenable if and only if G is amenable. Moreover ℓ¹(S) is amenable if and only if G is amenable and I is finite. For a left cancellative foundation semigroup S with an identity such that for every M_a(S)-measurable subset B of S and s ∈ S the set sB is M_a(S)-measurable, it is proved that if the measure algebra M_a(S) is approximately amenable, then S is left amenable. Concrete examples are given to show that the converse is negative.     

On module cohomology of the Fourier algebra of an inverse semigroup

Semigroup Forum - For an inverse semigroup S with the set of idempotents E, we find necessary and sufficient conditions for the Fourier algebra A(S) to be module amenable, module character...     

Algebras of almost periodic functions with Bohr-Fourier spectrum in a semigroup: Hermite property and its applications

It is proved that the unital Banach algebra of almost periodic functions of several variables with Bohr-Fourier spectrum in a given additive semigroup is an Hermite ring. The same property holds for the Wiener algebra of functions that in addition have absolutely convergent Bohr-Fourier series. As applications of the Hermite property of these algebras, we study factorizations of Wiener-Hopf type of rectangular matrix functions and the Toeplitz corona problem in the context of almost periodic functions of several variables.     

Radicals of algebras graded by cancellative linear semigroups

We consider algebras over a field of characteristic zero, and prove that the Jacobson radical is homogeneous in every algebra graded by a linear cancellative semigroup. It follows that the semigroup algebra of every linear cancellative semigroup is semisimple.

     

Coupled maps and analytic function spaces

We consider ergodic properties of weakly coupled analytic and expanding circle maps. For weak enough coupling a natural ergodic measure exists and exhibits exponent decay of time correlations. The marginal densities of the natural measure are analytic. A spatial decay of correlations (e.g. polynomial) in these densities may arise from a similar spatial decay of the couplings. The space of couplings and observables is a Banach algebra of analytic functions of infinitely many variables. This algebra acts upon a Banach module of complex measures with analytic marginal densities. A Perron-Frobenius type operator acts on the Banach module and we apply a re-summation technique to derive uniform bounds for this operator. Explicit bounds are calculated for some examples.     

Homological properties of modules over semigroup algebras

Let S be a semigroup. In this paper we investigate the injectivity of ?¹(S) as a Banach right module over ?¹(S). For weakly cancellative S this is the same as studying the flatness of the predual left module c₀(S). For such semigroups S, we also investigate the projectivity of c₀(S). We prove that for many semigroups S for which the Banach algebra ?¹(S) is non-amenable, the ?¹(S)-module ?¹(S) is not injective. The main result about the projectivity of c₀(S) states that for a weakly cancellative inverse semigroup S, c₀(S) is projective if and only if S is finite.     

Dual Clifford modules

A relationship between the space dual to the realification of a module over a Clifford algebra and the realification of the dual of this module is established. This relationship is completely analogous to a similar classical relationship between the space dual to the realification of a complex vector space and the realification of the dual of the given space. On the basis of this relationship, questions concerning extension of linear functionals on Clifford modules, in particular, a counterpart of the Hahn–Banach theorem, are considered.     

Quiver approach to some ordered semigroup algebras

The aim of this paper is to research the relation among generalized path algebras, pseudo-admissible ordered semigroup algebras and path algebras over algebras. First, the Gabriel theorem for contract pseudo-admissible ordered semigroup algebras is given. Second, a family of pseudo-admissible ordered semigroups is mixed together via the method of generalized path algebras to construct a new pseudo-admissible ordered semigroup as the extended version. Finally, we characterize the path algebra over an algebra in two ways through normal and non-normal generalized path algebras, respectively, over a field.     

Factorizations in numerical semigroup algebras

We study a numerical semigroup ring as an algebra over another numerical semigroup ring. The complete intersection property of numerical semigroup algebras is investigated using factorizations of monomials into minimal ones. The goal is to study whether a flat rectangular algebra is a complete intersection. Along this direction, special types of algebras generated by few monomials are worked out in detail.     

具预警功能的三部件并联可修复系统的可靠性分析

研究了具有预警功能的三部件并联且有两个热备器的可修复系统,系统可转化为一个Banch空间上的抽象Cauchy问题.首先利用半群理论证明了该系统强解的存在唯一性,其次运用了代数方法证明了0是系统算子的简单本征值,最后对系统和不含预警功能的系统的可用度进行了比较分析.