Semigroup Algebras Research Articles

The Bochner–Schoenberg–Eberlein Property for Totally Ordered Semigroup Algebras

The concepts of BSE property and BSE algebras were introduced and studied by Takahasi and Hatori in 1990 and later by Kaniuth and Ulger. This abbreviation refers to a famous theorem proved by Bochner and Schoenberg for \(L^1({\mathbb {R}})\), where \({\mathbb {R}}\) is the additive group of real numbers, and by Eberlein for \(L^1(G)\) of a locally compact abelian group G. In this paper we investigate this property for the Banach algebra \(L^p(S,\mu )\;(1\le p<\infty )\) where S is a compact totally ordered semigroup with multiplication \(xy=\max \{x,y\}\) and \(\mu \) is a regular bounded continuous measure on S. As an application, we have shown that \(L^1(S,\mu )\) is not an ideal in its second dual.

Left Equalizer Simple Semigroups

In this paper we characterize and construct semigroups whose right regular representation is a left cancellative semigroup. These semigroups will be called left equalizer simple semigroups. For a congruence $\varrho$ on a semigroup $S$, let ${\mathbb F}[\varrho]$ denote the ideal of the semigroup algebra ${\mathbb F}[S]$ which determines the kernel of the extended homomorphism of ${\mathbb F}[S]$ onto ${\mathbb F}[S/\varrho]$ induced by the canonical homomorphism of $S$ onto $S/\varrho$. We examine the right colons $({\mathbb F}[\varrho]:_r{\mathbb F}[S])=\{ a\in {\mathbb F}[S]:\ {\mathbb F}[S]a\subseteq {\mathbb F}[\varrho ]\}$ in general, and in that special case when $\varrho$ has the property that the factor semigroup $S/\varrho$ is left equalizer simple.

On character amenability of semigroup algebras

We study the character amenability of semigroup algebras. We work on general semigroups and certain semigroups such as inverse semigroups with a finite number of idempotents, inverse semigroups with uniformly locally finite idempotent set, Brandt and Rees semigroup and study the character amenability of the semigroup algebra l1(S) in relation to the structures of the semigroup S. In particular, we show that for any semigroup S, if ℓ1(S) is character amenable, then S is amenable and regular. We also show that the left character amenability of the semigroup algebra ℓ1(S) on a Brandt semigroup S over a group G with index set J is equivalent to the amenability of G and J being finite. Finally, we show that for a Rees semigroup S with a zero over the group G, the left character amenability of ℓ1(S) is equivalent to its amenability, this is in turn equivalent to G being amenable.

Cox rings of moduli of quasi-parabolic principal bundles and the K-Pieri rule

We study a toric degeneration of the Cox ring of the moduli of quasi-principal SLm(C) bundles on a marked projective line in the case where the parabolic data is chosen in the stabilizer of the highest weight vector in Cm or its dual representation ⋀m−1(Cm). The result of this degeneration is an affine semigroup algebra which is naturally related to the combinatorics of the K-Pieri rule from Kac–Moody representation theory. We find that this algebra is normal and Gorenstein, with a quadratic square-free Gröbner basis. This implies that the Cox ring is Gorenstein and Koszul for generic choices of markings, and generalizes results of Castravet, Tevelev and Sturmfels, Xu. Along the way we describe a relationship between the Cox ring and a classical invariant ring studied by Weyl.

Arens regularity of certain weighted semigroup algebras and countability

In this paper, among other things, we show that for a large class of semigroups, the Arens regularity of the weighted semigroup algebra \(\ell _1(S,\omega )\) implies the countability of S. This generalizes the main result of Craw and Young (Q J Math 25:351–358, 1974), stating that every countable semigroup admits a weight \(\omega \) for which \(\ell _1(S,\omega )\) is Arens regular and no uncountable group admits such a weight.

Group algebras and semigroup algebras defined by permutation relations of fixed length

Let H be a subgroup of Sym n, the symmetric group of degree n. For a fixed integer l ≥ 2, the group G presented with generators x1, x2,…,xn and with relations xi1xi2⋯xil = xσ(i1) xσ(i2)⋯xσ(il), where σ runs through H, is considered. It is shown that G has a free subgroup of finite index. For a field K, properties of the algebra K[G] are derived. In particular, the Jacobson radical 𝒥(K[G]) is always nilpotent, and in many cases the algebra K[G] is semiprimitive. Results on the growth and the Gelfand–Kirillov dimension of K[G] are given. Further properties of the semigroup S and the semigroup algebra K[S] with the same presentation are obtained, in case S is cancellative. The Jacobson radical is nilpotent in this case as well, and sufficient conditions for the algebra to be semiprimitive are given.

Module and Hochschild cohomology of certain semigroup algebras

We study the relation between the module and Hochschild cohomology groups of Banach algebras.We show that, for every commutative Banach A-A-bimodule X and every k ∈ N, the seminormed spaces HAk (A,X*) and Hk(A /J,X*) are isomorphic, where J is a specific closed ideal of A. As an example, we show that, for an inverse semigroup S with the set of idempotents E, where l1(E) acts on l1(S) by multiplication on the right and trivially on the left, the first module cohomology \(H_{{\ell ^1}\left( E \right)}^1\) (l1(S), l1(GS)(2n+1)) is trivial for each n ∈ N, where GS is the maximal group homomorphic image of S. Also, the second module cohomology \(H_{{\ell ^1}\left( E \right)}^2\) (l1(S), l1(GS)(2n+1)) is a Banach space.

Simplicity, primitivity and semiprimitivity of étale groupoid algebras with applications to inverse semigroup algebras

This paper studies simplicity, primitivity and semiprimitivity of algebras associated to étale groupoids. Applications to inverse semigroup algebras are presented. The results also recover the semiprimitivity of Leavitt path algebras and can be used to recover the known primitivity criterion for Leavitt path algebras.

The operator algebra generated by the translation, dilation and multiplication semigroups

The weak operator topology closed operator algebra on L 2 ( R ) generated by the one-parameter semigroups for translation, dilation and multiplication by e i λ x , λ ≥ 0 , is shown to be a reflexive operator algebra, in the sense of Halmos, with invariant subspace lattice equal to a binest. This triple semigroup algebra, A p h , is antisymmetric in the sense that A ph ∩ A p h ⁎ = C I , it has a nonzero proper weakly closed ideal generated by the finite-rank operators, and its unitary automorphism group is R . Furthermore, the 8 choices of semigroup triples provide 2 unitary equivalence classes of operator algebras, with A ph and A p h ⁎ being chiral representatives.

Twisted Semigroup Algebras

We study 2-cocycle twists, or equivalently Zhang twists, of semigroup algebras over a field ${\mathbb K}$ . If the underlying semigroup is affine, that is abelian, cancellative and finitely generated, then $\mathsf {Spec}~{\mathbb K}[S]$ is an affine toric variety over ${\mathbb K}$ , and we refer to the twists of ${\mathbb K}[S]$ as quantum affine toric varieties. We show that every quantum affine toric variety has a “dense quantum torus”, in the sense that it has a localization isomorphic to a quantum torus. We study quantum affine toric varieties and show that many geometric regularity properties of the original toric variety survive the deformation process.

Azumaya Semigroup Algebras

A semigroup is called almost idempotent-free if it has exactly one nonzero \({\mathcal {J}}\)-class containing idempotents. It is mainly investigated when almost idempotent-free semigroup algebras are Azumaya algebras. Necessary and sufficient conditions for semigroup algebras of an almost idempotent-free semigroup with certain properties to be Azumaya are obtained.

Module derivations on semigroup algebras

We show that for an inverse semigroup S with the set idempotents E acting on S trivially from left and by multiplication from right, any bounded module derivation from \(\ell ^1(S)\) to \(({\ell ^1(S)}/{J})^*=J^{\perp }\) is inner, where J is the closed ideal generated by elements of the form \(\delta _{set}-\delta _{st}\) with \(s,t\in S\) and \(e\in E\).

Prime Ideals in Algebras Determined by Submonoids of Nilpotent Groups

The prime spectrum of the semigroup algebra K[S] of a submonoid S of a finitely generated nilpotent group is studied via the spectra of the monoid S and of the group algebra K[G] of the group G of fractions of S. It is shown that the classical Krull dimension of K[S] is equal to the Hirsch length of the group G provided that G is nilpotent of class two. This uses the fact that prime ideals of S are completely prime. An infinite family of prime ideals of a submonoid of a free nilpotent group of class three with two generators which are not completely prime is constructed. They lead to prime ideals of the corresponding algebra. Prime ideals of the monoid of all upper triangular n × n matrices with non-negative integer entries are described and it follows that they are completely prime and finite in number.

Witt vector rings and the relative de Rham Witt complex

In this paper we develop a novel approach to Witt vector rings and to the (relative) de Rham Witt complex. We do this in the generality of arbitrary commutative algebras and arbitrary truncation sets. In our construction of Witt vector rings the ring structure is obvious and there is no need for universal polynomials. Moreover a natural generalization of the construction easily leads to the relative de Rham Witt complex. Our approach is based on the use of free or at least torsion free presentations of a given commutative ring R and it is an important fact that the resulting objects are independent of all choices. The approach via presentations also sheds new light on our previous description of the ring of p-typical Witt vectors of a perfect Fp-algebra as a completion of a semigroup algebra. We develop this description in different directions. For example, we show that the semigroup algebra can be replaced by any free presentation of R equipped with a linear lift of the Frobenius automorphism. Using the result in Appendix A by Umberto Zannier we also extend the description of the Witt vector ring as a completion to all F‾p-algebras with injective Frobenius map.

The Algebra of Semigroups of Sets

We study the algebra of semigroups of sets (i.e. families of sets closed under finite unions) and its applications. For each $n &gt; 1$ we produce two finite nested families of pairwise different semigroups of sets consisting of subsets of $\mathsf{R}^n$ without the Baire property.

Singular braids and partial permutations

We introduce and study the partial singular braid monoid PSBn, a monoid that contains both the inverse braid monoid IBn and the singular braid monoid SBn. Our main results include a characterization of Green’s relations, a presentation in terms of generators and relations, and a proof that PSBn embeds in the semigroup algebra C[IBn].

Ample semigroups and Frobenius algebras

We prove that the semigroup algebra of an ample semigroup \(S\) over a field is Frobenius if and only if \(S\) is a finite inverse semigroup.

Approximate biprojectivity of certain semigroup algebras

In this paper, we investigate the notion of approximate biprojectivity for semigroup algebras and for some Banach algebras related to semigroup algebras. We show that $$\ell ^{1}(S)$$ is approximately biprojective if and only if $$\ell ^{1}(S)$$ is biprojective, provided that $$S$$ is a uniformly locally finite inverse semigroup. Also for a Clifford semigroup $$S$$ , we show that approximate biprojectivity of $$\ell ^{1}(S)^{**}$$ gives pseudo-amenability of $$\ell ^{1}(S)$$ . We give a class of Banach algebras related to semigroup algebras which is not approximately biprojective.

Boundary quotients of C*-algebras of right LCM semigroups

We study C*-algebras associated to right LCM semigroups, that is, semigroups which are left cancellative and for which any two principal right ideals are either disjoint or intersect in another principal right ideal. If P is such a semigroup, its C*-algebra admits a natural boundary quotient Q(P). We show that Q(P) is isomorphic to the tight C*-algebra of a certain inverse semigroup associated to P, and thus is isomorphic to the C*-algebra of an étale groupoid. We use this to give conditions on P which guarantee that Q(P) is simple and purely infinite, and give applications to self-similar groups and Zappa–Szép products of semigroups.

Module amenability of restricted semigroup algebras under module actions

In this article, weshow that module amenability with the canonical action of restricted semigroup algebra l1r (S) and semigroup algebra l1(Sr) are equivalent, where Sr is the restricted semigroup of associated to the inverse semigroup S. We use this to give a characterization of module amenability of restricted semigroup algebra l1r (S) with the canonical action, where S is a Clifford semigroup.