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

Authors: Amir Sahami; Mehdi Rostami; Seyedeh Fatemeh Shariati; Salman Babayi
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, we study the notion of approximate biprojectivity and leftφ-biprojectivity of some Banach algebras, where φ is a character. Indeed, we show that approximate biprojectivity of the hypergroup algebra L1K 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.


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## Body

## 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. The 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]. Biprojectivity 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 L1K is studied in [6]. As a generalization of this notion, Y. Zhang in [7] introduced the notion of approximate biprojectivity. Indeed, a Banach algebra A is called approximately biprojective if there exists a net ρα of continuous A-bimodule morphism from A into A⊗∧A such that πA°ραa⟶a for every a∈A, where πA:A⊗∧A⟶A is the diagonal operator defined by πAa⊗b=ab. For recent works about this concept, refer to [8].Throughout the paper,ΔA stands for the set of all nonzero multiplicative linear functionals on A. Kaniuth et al. [9] introduced the notion of left φ-amenable Banach algebras φ∈ΔA as a generalization of the notion of amenable 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=φax for all a∈A and x∈X.Hu et al. in [11] defined the notion of left φ-contractibility for Banach algebras. Following [12], a Banach algebra A is called left φ-contractible, where φ∈ΔA, if there exists m∈A such that am=φam and φm=1, for every a∈A. For a locally compact group G, it is shown that left φ-contractibility of L1G (or MG) is equivalent to compactness of G (Theorem 6.1 in [15]).Motivated by these considerations, the first author defined the homological notion of leftφ-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, where φ∈ΔA, if there exists a bounded linear map ρ:A⟶A⊗∧A such that(1)ρab=a⋅ρb=φbρa,φ°πA°ρa=φa,a,b∈A.The right case can be similarly defined.The first and the second authors in [13] explained the relation between left φ-contractibility and left φ-biprojectivity. They proved that left φ-contractibility implies left φ-biprojectivity, and the converse is valid if A is commutative or has a left approximate identity.We give some brief backgrounds on hypergroups and their related algebras and establish our notation; for details, see [14]. Let K be a locally compact Hausdorff space and MK denote the set of all bounded complex Radon measures on K, where the norm of each measure μ∈MK is the total variation μ. Also, we denote by CK the set of all nonempty compact subsets of K and equipped this space with the Michael topology, that is, the topology generated by the sets(2)UV,W≔L∈CK:L∩V≠∅,L⊆W,for open subsets V and W in K.The spaceK is called a hypergroup if there is a convolution ∗:MK×MK⟶MK, an involution x↦x- on K, and an element e∈K (called the identity element) such that the following holds:(i)
MK,∗ is a Banach algebra.(ii)
δx∗δy is a probability measure with compact support.(iii)
The mapx,y↦δx∗δy is continuous from K×K into MK equipped with the weak ∗ topology.(iv)
The mapx,y↦suppδx∗δy is continuous from K×K into CK.(v)
For eachx∈K, δx∗δe=δe∗δx=δx.(vi)
The mappingx↦x- is a homeomorphism on K of period 2 and δx∗δy-=δy-∗δx- and e∈suppδx∗δy if and only if x=y- for each x,y∈K. Here, the measure μ- is given by μ-E=μE- for all Borel subsets E⊆K.A hypergroupK is called commutative if δx∗δy=δy∗δx for all x,y∈K. In [2, 3], it is proved that every commutative or compact hypergroup has a unique (left) Haar measure. The existence and the uniqueness of left Haar measure on a general locally compact hypergroup were proved recently by Chapovsky in [15]. Throughout this paper, we assume that K is a hypergroup with a unique Haar measure ωK. That is,(3)∫kfx∗ydωky=∫KfydωKy,x∈K,for every Borel measurable function f on K. Then, L1K,⋅1 with the involution f∗x=fx-¯ and the convolution(4)f∗gx=∫Kfygy-∗xdωKy,is a Banach ∗-algebra, where fx∗y=∫Kfdδx∗δy for every x,y∈K. A nonzero bounded continuous function χ:K⟶ℂ is called a character on K if χx∗y=χxχy for every x,y∈K. The set of all characters on K will be denoted by ΥbK, i.e.,(5)ΥbK=χ∈CbK:χ≠0,χx∗y=χxχyx,y∈K.Forχ∈ΥbK, define φχ on L1K via(6)φχf=∫Kχs¯fsdωKs.It may be observed thatφχ∈ΔL1K, and note that there is at least one character on L1K, namely, the augmentation character φ1. Further, if K is commutative, it is well-known that there is no other character on L1K; that is, ΔL1K=φχ:χ∈ΥbK. For more details, see Section 2.2 in [3].In the present paper, we show that approximate biprojectivity of hypergroup algebraL1K implies that K is compact. After that, we study left φ-biprojectivity of general abstract Segal algebras with respect to the hypergroup algebra L1K. As a result (with a mild condition), we conclude that the abstract Segal algebra B is left φ-biprojective if and only if K is compact. We also study approximate biflatness and left ϕ-biflatness of some hypergroup algebras in terms of amenability of their related hypergroups.


## 2. Approximate Biprojectivity and Leftφ-Biprojectivity of Hypergroup Algebras

Recall that ifA is a Banach algebra and I is a closed two-sided ideal in A, then for each φ∈ΔA such that I⊆kerφ, the map φ¯a+I=φa is a character on A/I.Proposition 1.
LetA be a Banach algebra, φ∈ΔA, and let I⊆kerφ be a closed two-sided ideal of A such that AI¯=I. If A is left φ-biprojective, then A/I is left φ¯-biprojective.Proof. SinceA is left φ-biprojective, there is a bounded linear map ρ:A⟶A⊗∧A such that a⋅ρb=ρab=φbρa and φ°πA°ρa=φa for every a∈A. Consider the quotient map q:A⟶A/I. It may be noted that(7)idA⊗q°ρat=idA⊗qφtρa=0a∈A,t∈I.By the assumptionAI¯=I, we have idA⊗q°ρ|I=0. Thus, idA⊗q°ρ can be dropped on A/I. Define η:A/I⟶A/I⊗∧A/I by η=q⊗idA/I°idA⊗q°ρ. It may be observed that(8)φ¯°πA/I°ηa+I=φa=φ¯a+Ia∈A.Moreover,(9)ηa+Ib+I=q⊗idA/I°idA⊗q°ρab=q⊗idA/I°idA⊗q°a⋅ρb=a⋅q⊗idA/I°idA⊗q°ρb=a+I⋅ηb+I,and also(10)ηa+Ib+I=q⊗idA/I°idA⊗q°ρab=q⊗idA/I°idA⊗q°φbρa=a⋅q⊗idA/I°idA⊗q°ρa=φ¯b+Iηa+I.Hence,A/I is left φ¯-biprojective.Corollary 1.
LetA be a Banach algebra with a left approximate identity and φ∈ΔA. If I⊆kerφ is a closed two-sided ideal of A and A is left φ-biprojective, then A/I is left φ¯-biprojective.In the group case, it is well-known that the group algebraL1K is biprojective if and only if G is compact (see, for example, [4]). In the similar way, in Theorem 3.1 in [6], it is shown that if the hypergroup algebra L1K is biprojective, then K is compact. In the following, we give a generalization of this result and characterize approximate biprojectivity of the hypergroup algebra L1K.Theorem 1.
LetK be a locally compact hypergroup. If the hypergroup algebra L1K is approximately biprojective, then K is compact.Proof. LetL1K be approximately biprojective. Since L1K has a bounded approximate identity (Theorem 1.6.15 in [3]), the hypergroup algebra L1K is left φ-contractible for every φ∈ΔL1K (Theorem 3.9 in [20]). Consider the augmentation character on L1K via(11)φ1f=∫KfdωK.So, there existsm∈L1K satisfying f∗m=φ1fm and φ1m=1 for every f∈L1K. Choose f∈L1K such that φ1f=1. It is worth noting that(12)δx∗m=φ1fδx∗m,=δx∗φ1fm,=δx∗f∗m,=δx∗f∗m,=φ1δx∗fm.Sinceδx-∗ωK=ωK, we have φ1δx∗f=φ1f. Thus, δx∗m=m. It follows that m is a constant function. Therefore, Tex translation failed. The latter implies that K is compact (page 40 in [3]).We study leftϕ-biprojectivity of abstract Segal algebras. So, we start with definition of abstract Segal algebras.Definition 1.
LetA be a Banach algebra with norm ⋅A. We say that a Banach algebra B with norm ⋅B is an abstract Segal algebra with respect to A if(i)
B is a dense left ideal in A(ii)
There existsM>0 such that bA≤MbB for every b∈B(iii)
There existsC>0 such that abB≤CaAbB for every a∈A and b∈B
Moreover, we say thatB is a symmetric abstract Segal algebras if B is a two-sided ideal of A and there exists C>0 such that baB≤CaAbB for every a∈A and b∈B.
For more details about abstract Segal algebras, see [16]. We note that there are some abstract Segal algebras which are not symmetric. Homological and cohomological properties of abstract Segal algebras have been studied in many papers (see, for example, [17–19]). Recall that it is known that ΔB=φ|B:φ∈ΔA (see Lemma 2.2 in [1]). It is worthwhile to mention that Essmaili et al. in [6] studied right φ-biprojectivity (they called it condition Q) of Banach algebras associated with a hypergroup, especially symmetric Segal algebras. As a main result, they showed that if K is a commutative hypergroup and B is a symmetric abstract Segal algebra with respect to L1K, then B is right φ|B-biprojective if and only if K is compact, where φ∈ΔL1K. In the following, using Theorem 1, we extend the left version of (Corollary 3.11 in [6]).Theorem 2.
LetK be a locally compact hypergroup, φχ∈ΔL1K, and let B be an abstract Segal algebra with respect to L1K which possess a left approximate identity. Then, the following statements are equivalent:(i)
B is left φχ|B-biprojective(ii)
K is compactProof. Suppose thatB is left φχ|B-biprojective. Applying Proposition 1 in [18], B is left φχ|B-contractible. By Proposition 2.5 in [1], L1K is left φχ-contractible. By similar argument as in the proof of Theorem 1, K is compact.Conversely, letK be compact with the normalized Haar measure ωK (page 40 in [3]). Then, for each χ∈ΥbK, we have(13)∫KχxdωKx=∫KλdωKx=λωK=λ∞,for some λ>0. So, χ∈L1K. Put m=χ/λ2. We claim that f∗m=φχfm and φχm=1 for every f∈L1K. To see this,(14)f∗χx=∫Kfyχy-∗xdωKy=∫Kfyχ-yχxdωKy=φχfχx,and also(15)φχχ=∫Kχ-xχxdωKx=∫Kχx2dωKx=λ2ωK=λ2.So,L1K is left φχ-contractible. Using Proposition 2.5 in [1] again, B is left φχ|B-contractible. Hence, B is left φχ|B-biprojective (Lemma 1 in [18]).


## 3. Approximate Biflatness and Leftφ-Biflatness of Hypergroup Algebras

In this section, we study approximate biflatness and leftφ-biflatness of some algebras related to a hypergroup.We remind that a Banach algebraA is called approximately biflat if there exists a net ρα of A-bimodule morphisms from A⊗∧A∗ into A∗ such that ρα°πA∗⟶W∗OTidA∗. Here, W∗OT stands for the weak ∗ operator topology (see [20]).Following [21], a locally compact hypergroup K is called left amenable if there exists a left invariant mean on LUCK. That is, a bounded linear functional μ:LUCK⟶ℂ such that μ1=μ=1 and μlxf=μf for all f∈LUCK and x∈K.Proposition 2.
LetK be a locally compact hypergroup and let B be an abstract Segal algebra with respect to L1K which possess a left approximate identity. If B is approximately biflat, then K is left amenable.Proof. Suppose thatB is approximately biflat and fix a character φ∈ΔB. Since B has a left approximate identity, by Proposition 2. 4 in [22], B is left φ-amenable. Now, Proposition 2.3 in [1] follows that L1K is left φ-amenable. Applying Theorem 3.5 in [14], we conclude that K is left amenable.Corollary 2.
LetK be a locally compact hypergroup and let B be an abstract Segal algebra with respect to L1K which possess a left approximate identity. If B∗∗ is approximately biflat, then K is left amenable.Proof. Fix a characterφ∈ΔL1K. Since B has a left approximate identity, similar to the arguments as in the proof of (Theorem 2.2 in [23]), approximately biflatness of B∗∗ follows that B is left φ|B-amenable. It deduces that L1K is left φ-amenable. Thus, by Theorem 3.5 in [14], K is left amenable.LetA be a Banach algebra and φ∈ΔA. Then, A is called left φ-biflat if there exists a bounded linear map ρ:A⟶A⊗∧A∗∗ such that(16)ρab=a⋅ρb=φbρa,φ˜°πA∗∗°ρa=φa,a,b∈A.Here,φ˜:A∗∗⟶ℂ is a unique extension of ϕ which is defined by φ˜FFφ for all F∈A∗∗. The right case can be defined similarly. For further information, see [24].Theorem 3.
LetK be a locally compact hypergroup, φχ∈ΔL1K, and let B be an abstract Segal algebra with respect to L1K which possess a left approximate identity. The following statements are equivalent:(i)
B is left φ-biflat(ii)
K is left amenable(iii)
B∗∗ is left φ-biflatProof. (i)⇒ (ii) Let B be left φ-biflat. Since B has a left approximate identity, by Lemma 2.1 in [21], B is left φ-amenable. It follows that L1K is left φ-amenable. Now, applying Theorem 3.5 in [14], we conclude that K is left amenable.(ii)⇒ (iii) Let K be a left amenable hypergroup. Then, by Theorem 3.5 in [14], L1K is left φ-amenable. Now, by Proposition 2.3 in [1], B is left φ-amenable. Using Proposition 3.4 in [13], B∗∗ is left φ-amenable and then Lemma 2.3 in [25] implies that B∗∗ is left φ-biflat.(iii)⇒ (i) Suppose that B∗∗ is left φ-biflat. Then, by Theorem 2.2 in [21], B is left φ-biflat.Suppose that1≤p<∞ and K is a locally compact hypergroup. Set SpK=L1K∩LpK with the norm(17)fS=f1+fp.SpK becomes an abstract Segal algebra with respect to L1K. It is known that SpK possess a left approximate identity (see [2]).The following corollary is an easy consequence of Proposition2 and Corollary 2.Corollary 3.
LetK be a locally compact hypergroup and 1≤p<∞. If SpK∗∗ (or SpK) is approximately biflat, then K is left amenable.Example 1.
In this example, we give a Banach algebra which is leftφ-biflat but it is not right φ-biflat.
LetA be a Banach space with dimA>1 and φ∈A∗ be a nonzero functional such that φ=1. Define a multiplication in A via a∗b=φab, for all a,b∈A. It is easy to see that A is a Banach algebra and ΔA=φ. We denote the unitization of A with A♯. It is known that A is a closed ideal in A♯. Moreover, φ has an extension to A♯, that is, φe:A♯⟶ℂ which is given by φea+λe=φa+λ for all a∈A and λ∈ℂ. We claim that A♯ is left φe-biflat but it is not right φe-biflat. To see this, we know that A is left φ-amenable, applying Lemma 3.2 in [13], it follows that A♯ is left φe-amenable. It gives that A♯ is left φe-biflat. Now, assume conversely that A♯ is right φe-biflat. Since A♯ is unital, A♯∗∗ has an element m=ρe such that ma=φam and φ˜°πAe∗∗m=1, for all a∈A. Following the similar arguments as in Theorem 1.4 in [13], we have a bounded net mα in Ae such that mαa−φamα⟶0 and φmα=1, for all a∈Ae. Pick an element a0 in A such that φa0=1. Replace mα with a0mα, we may suppose that mα is a bounded net in A such that mαa−φamα⟶0 and φmα=1, for all a∈A. However,(18)1a−φamα=φmαa−φamα=amα−φamα⟶0,for all a∈A. Let a be any element in kerφ and put it at above fact. It follows that kerφ=0. So, φ is an isomorphism. Therefore, dimA=dimℂ=1 which is a contradiction.



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