Research Article On Some Homological Properties of Hypergroup Algebras with Relation to Their Character Spaces Amir Sahami , 1 Mehdi Rostami, 2 Seyedeh Fatemeh Shariati, 2 and Salman Babayi 3 1 Department of Mathematics Faculty of Basic Science, Ilam University, P.O. Box 69315-516, Ilam, Iran 2 Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran 3 Department of Mathematics, Faculty of Science, Urmia University, Urmia, Iran Correspondence should be addressed to Amir Sahami; a.sahami@ilam.ac.ir Received 25 October 2021; Revised 8 December 2021; Accepted 6 January 2022; Published 29 January 2022 Academic Editor: Ralf Meyer Copyright © 2022 Amir Sahami et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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. 1. 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. e study of hypergroups was initiated in the 1970s by Dunkl [1], Jewwet [2], and Spector [3], each of them in various axioms. However, in this paper, we will base our work on Jewett’s axioms in [2]. Biprojectivity is an important homological notion that arises naturally in Helemskii’s works in the 1980s; interested readers can refer to his comprehensive book [4]. Bipro- jectivity of some well-known Banach algebras associated to locally compact groups, such as group algebras and measure algebras, is studied in [4, 5]. Biprojectivity of the hypergroup algebra L 1 (K) is studied in [6]. As a generalization of this notion, Y. Zhang in [7] introduced the notion of approxi- mate biprojectivity. Indeed, a Banach algebra A is called approximately biprojective if there exists a net (ρ α ) of continuous A-bimodule morphism from A into A ∧ ⊗ A such that π A ° ρ α (a) ⟶ a for every a ∈ A, where π A : A ∧ ⊗ A ⟶ A is the diagonal operator defined by π A (a ⊗ b)� ab. For recent works about this concept, refer to [8]. roughout the paper, Δ(A) stands for the set of all nonzero multiplicative linear functionals on A.Kaniuthetal. [9] introduced the notion of left φ-amenable Banach alge- bras (φ ∈Δ(A)) as a generalization of the notion of ame- nable Banach algebras introduced by Johnson in [10]. A Banach algebra A is called left φ-amenable if every derivation D from A into X ∗ is inner, for every Banach A-bimodule X with the left module action a · x � φ(a)x for all a ∈ A and x ∈ X. Hu et al. in [11] defined the notion of left φ-contract- ibility for Banach algebras. Following [12], a Banach algebra A is called left φ-contractible, where φ ∈Δ(A), if there exists m ∈ A such that am � φ(a)m and φ(m)� 1, for every a ∈ A. For a locally compact group G, it is shown that left φ-contractibility of L 1 (G) (or M(G)) is equivalent to compactness of G (eorem 6.1 in [15]). Motivated by these considerations, the first author de- fined the homological notion of left φ-biprojectivity for Banach algebras (see, e.g., [13]). Here is the definition of his new notion. A Banach algebra A is called left φ-biprojective, Hindawi Journal of Mathematics Volume 2022, Article ID 4939971, 5 pages https://doi.org/10.1155/2022/4939971