Invariant vector means and complementability of Banach spaces in their second duals

Let X be a Banach space. Fix a torsion-free commutative and cancellative semigroup S whose torsion-free rank is the same as the density of $$X^{**}$$ X ∗ ∗ . We then show that X is complemented in $$X^{**}$$ X ∗ ∗ if and only if there exists an invariant mean $$M:\ell _\infty (S,X)\rightarrow X$$ M : ℓ ∞ ( S , X ) → X . This improves upon previous results due to Bustos Domecq (J Math Anal Appl 275(2):512–520, 2002), Kania (J Math Anal Appl 445:797–802, 2017), Goucher and Kania (Studia Math 260:91–101, 2021).

Connes Amenability of $l^1$-Munn Algebras

In this paper, we study Connes amenability of $l^1$-Munn algebras. We apply this results to semigroup algebras. We show that for a weakly cancellative semigroup $S$ with finite idempotents, amenability and Connes amenability are equivalent.

A finite cancellative semigroup is a group

We prove the proposition addressed in the title of this paper.

Complete Heyting algebra-valued convergence semigroups

Considering a complete Heyting algebra H, we introduce a notion of stratified H-convergence semigroup. We develop some basic facts on the subject, besides obtaining conditions under which a stratified H-convergence semigroup is a stratified H-convergence group. We supply a variety of natural examples; and show that every stratified H-convergence semigroup with identity is a stratified H-quasiuniform convergence space. We also show that given a commutative cancellative semigroup equipped with a stratified H-quasi-unifom structure satisfying a certain property gives rise to a stratified H-convergence semigroup via a stratified H-quasi-uniform convergence structure.

THE MULTIPLIER ALGEBRA OF A BEURLING ALGEBRA

For a discrete abelian cancellative semigroup $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}S$ with a weight function $\omega $ and associated multiplier semigroup $M_\omega (S)$ consisting of $\omega $-bounded multipliers, the multiplier algebra of the Beurling algebra of $(S,\omega )$ coincides with the Beurling algebra of $M_\omega (S)$ with the induced weight.

Inverse Semigroup C*-Algebras Associated with Left Cancellative Semigroups

To each discrete left cancellative semigroup S one may associate an inverse semigroup Il(S), often called the left inverse hull of S. We show how the full and reduced C*-algebras of Il(S) are related to the full and reduced semigroup C*-algebras for S, recently introduced by Li, and give conditions ensuring that these algebras are isomorphic. Our picture provides an enhanced understanding of Li's algebras.

CANCELLATIVE AND MALCEV PRESENTATIONS FOR FINITE REES INDEX SUBSEMIGROUPS AND EXTENSIONS

It is known that, for semigroups, the property of admitting a finite presentation is preserved on passing to subsemigroups and extensions of finite Rees index. The present paper shows that the same holds true for Malcev, cancellative, left-cancellative and right-cancellative presentations. (A Malcev (respectively, cancellative, left-cancellative, right-cancellative) presentation is a presentation of a special type that can be used to define any group-embeddable (respectively, cancellative, left-cancellative, right-cancellative) semigroup.)

Connes-amenability of bidual and weighted semigroup algebras

We investigate the notion of Connes-amenability, introduced by Runde in [10], for bidual algebras and weighted semigroup algebras. We provide some simplifications to the notion of a $\sigma WC$-virtual diagonal, as introduced in [13], especially in the case of the bidual of an Arens regular Banach algebra. We apply these results to discrete, weighted, weakly cancellative semigroup algebras, showing that these behave in the same way as ${ C}^*$-algebras with regards Connes-amenability of the bidual algebra. We also show that for each one of these cancellative semigroup algebras $l^1(S,\omega)$, we have that $l^1(S,\omega)$ is Connes-amenable (with respect to the canonical predual $c_0(S)$) if and only if $l^1(S,\omega)$ is amenable, which is in turn equivalent to $S$ being an amenable group, and the weight satisfying a certain restrictive condition. This latter point was first shown by Grønnbæk in [6], but we provide a unified proof. Finally, we consider the homological notion of injectivity, and show that here, weighted semigroup algebras do not behave like ${ C}^*$-algebras.

Baer-Levi semigroups of partial transformations

Let X be an infinite set and suppose א0 ≤ q ≤ |X|. The Baer-Levi semigroup on X is the set of all injective ‘total’ transformations α: X → X such that |X\Xα| = q. It is known to be a right simple, right cancellative semigroup without idempotents, its automorphisms are “inner”, and some of its congruences are restrictions of Malcev congruences on I(X), the symmetric inverse semigroup on X. Here we consider algebraic properties of the semigroup consisting of all injective ‘partial’ transformations α of X such that |X\Xα| = q: in particular, we descried the ideals and Green's relations of it and some of its subsemigroups.