On Some Homological Properties of Hypergroup Algebras with Relation to Their Character Spaces

Salman Babayi,Seyedeh Fatemeh Shariati,Ralf Meyer,Amir Sahami,Mehdi Rostami

doi:10.1155/2022/4939971

Salman Babayi, Seyedeh Fatemeh Shariati + Show 3 more

Open Access

https://doi.org/10.1155/2022/4939971

Copy DOI

Abstract

In this paper, we study the notion of approximate biprojectivity and left φ -biprojectivity of some Banach algebras, where φ is a character. Indeed, we show that approximate biprojectivity of the hypergroup algebra L 1 K implies that K is compact. Moreover, we investigate left φ -biprojectivity of certain hypergroup algebras, namely, abstract Segal algebras. As a main result, we conclude that (with some mild conditions) the abstract Segal algebra B is left φ -biprojective if and only if K is compact, where K is a hypergroup. We also study the approximate biflatness and left φ -biflatness of hypergroup algebras in terms of amenability of their related hypergroups.

Highlights

Introduction and PreliminariesHypergroups are a suitable generalization of classical locally compact groups
Biprojectivity of some well-known Banach algebras associated to locally compact groups, such as group algebras and measure algebras, is studied in [4, 5]
A Banach algebra A is called approximately biprojective if there exists a net of continuous A-bimodule morphism from A into A∧ ⊗ A such that πA°ρα(a) ⟶ a for every a ∈ A, where πA: A∧ ⊗ A ⟶ A is the diagonal operator defined by πA(a ⊗ b) ab

Summary

Introduction and Preliminaries

Hypergroups are a suitable generalization of classical locally compact groups. In classical setting, the convolution of two point mass measures is a point mass measure, while in hypergroup structure, the convolution of two point mass measures is a probability measure with compact support. Hu et al in [11] defined the notion of left φ-contractibility for Banach algebras. For a locally compact group G, it is shown that left φ-contractibility of L1(G) (or M(G)) is equivalent to compactness of G ( eorem 6.1 in [15]) Motivated by these considerations, the first author defined the homological notion of left φ-biprojectivity for Banach algebras (see, e.g., [13]). In [2, 3], it is proved that every commutative or compact hypergroup has a unique (left) Haar measure. E existence and the uniqueness of left Haar measure on a general locally compact hypergroup were proved recently by Chapovsky in [15]. We study approximate biflatness and left φ-biflatness of some hypergroup algebras in terms of amenability of their related hypergroups

Approximate Biprojectivity and Left φ-Biprojectivity of Hypergroup Algebras

Approximate Biflatness and Left φ-Biflatness of Hypergroup Algebras