[Paper Review] Module Biprojective and Module Biflat Banach Algebras

Abasalt Bodaghi, Massoud Amini|ArXiv.org|Dec 20, 2009

Advanced Operator Algebra Research16 references11 citations

TL;DR

This paper introduces module biprojective and module biflat Banach algebras—generalizations of classical biprojectivity and biflatness for Banach algebras that are modules over another Banach algebra. It establishes that for an inverse semigroup $S$ with upward directed idempotents $E$, the semigroup algebra $\ell^1(S)$ is module biprojective as an $\ell^1(E)$-module if and only if the group image $S/\approx$ is finite, and module biflat if and only if $S$ is amenable.

ABSTRACT

In this paper we define module biprojctivity and module biflatness for a Banach algebra which is a Banach module over another Banach algebra with compatible actions, and find their relation to classical biprojectivity and biflatness. As a typical example, We show that for an inverse semigroup $S$ with an upward directed set of idempotents $E$, the semigroup algebra $ \ell ^{1}(S)$, as an $\ell ^{1}(E)$-module, is module biprojective if and only if an appropriate group homomorphic image of $S$ is finite. Also we show that $\ell ^{1}(S)$ is module biflat if and only if $S$ is amenable.

Motivation & Objective

• To extend the classical notions of biprojectivity and biflatness to Banach algebras that are modules over another Banach algebra with compatible actions.
• To investigate the relationship between module biprojectivity, module biflatness, and module amenability/super-amenability in the context of inverse semigroup algebras.
• To characterize module biprojective and module biflat structures for $\ell^1(S)$, where $S$ is an inverse semigroup with upward directed idempotents.
• To clarify the connection between module properties and structural features of the semigroup, such as finiteness of the group image $S/\approx$ and amenability of $S$.
• To generalize known results on biflatness and biprojectivity of semigroup algebras to the module-theoretic setting.

Proposed method

• Define module biprojectivity and module biflatness for a Banach algebra $\mathcal{A}$ that is a Banach module over another Banach algebra $\mathfrak{A}$ with compatible actions.
• Use the quotient algebra $\mathcal{A}/J$, where $J$ is the closed ideal generated by $\alpha \cdot (ab) - (ab) \cdot \alpha$ for $a \in \mathcal{A}, \alpha \in \mathfrak{A}$, to relate module properties to classical properties.
• Apply results from module amenability and module super-amenability (from prior work) to establish equivalences for $\ell^1(S)$ as an $\ell^1(E)$-module.
• Utilize the isomorphism $\ell^1(S)/J \cong \ell^1(S/\approx)$, where $S/\approx$ is the group image of $S$, to reduce the problem to group algebra properties.
• Leverage known characterizations of amenability and finiteness in inverse semigroups and their group images to derive module conditions.
• Use the fact that $\ell^1(S) \widehat{\otimes}_{\ell^1(E)} \ell^1(S)$ is a commutative $\ell^1(E)$-module to analyze module module tensor products.

Experimental results

Research questions

• RQ1 When is a Banach algebra $\mathcal{A}$ that is a module over another Banach algebra $\mathfrak{A}$ module biprojective?
• RQ2 When is such an algebra module biflat, and how does this relate to classical biprojectivity and biflatness?
• RQ3 For the semigroup algebra $\ell^1(S)$, where $S$ is an inverse semigroup with upward directed idempotents $E$, when is $\ell^1(S)$ module biprojective as an $\ell^1(E)$-module?
• RQ4 When is $\ell^1(S)$ module biflat as an $\ell^1(E)$-module?
• RQ5 How do the module properties of $\ell^1(S)$ relate to the finiteness of the group image $S/\approx$ and the amenability of $S$?

Key findings

• For an inverse semigroup $S$ with upward directed idempotents $E$, $\ell^1(S)$ is module biprojective as an $\ell^1(E)$-module if and only if the group image $S/\approx$ is finite.
• The same algebra $\ell^1(S)$ is module biflat as an $\ell^1(E)$-module if and only if $S$ is amenable.
• If $S/\approx$ is infinite, then $\ell^1(S)$ is not biprojective, even if it is module biprojective.
• If $\ell^1(S)$ is biflat, then $S$ must be amenable, showing that classical biflatness implies the amenability condition.
• For the bicyclic semigroup $\mathcal{C}$, $\ell^1(\mathcal{C})$ is module biflat but not module biprojective, since $\mathcal{C}/\approx \cong \mathbb{Z}$ is infinite.
• For the semigroup $\mathbb{N}$ under maximum operation, $\ell^1(\mathbb{N})$ is module biprojective and module biflat (since $\mathbb{N}/\approx$ is trivial), but not biprojective or biflat under convolution due to non-uniform local finiteness.