Invariant $\varphi$-means for abstract Segal algebras related to locally compact groups

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

“…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][18][19]). Recall that it is known that Δ(B) � φ| B : φ ∈ Δ(A) 􏼈 􏼉 (see Lemma 2.2 in [1]).…”

“…Replace (m α ) with (a 0 m α ), we may suppose that (m α ) is a bounded net in A such that m α a − φ(a)m α ⟶ 0 and φ(m α ) � 1, for all a ∈ A. However, 1a − φ(a)m α � φ m α 􏼁a − φ(a)m α � am α − φ(a)m α ⟶ 0, (18) for all a ∈ A. Let a be any element in kerφ and put it at above fact.…”

On left $\varphi$-biflat Banach algebras

“…A Banach algebra A is called left ϕ-amenable if there exists an element m ∈ A * * such that am = ϕ(a)m and ϕ(m) = 1 for every a ∈ A. Note that the Segal algebra S(G) is left ϕ-amenable if and only if G is amenable, for further information see [1], [8] and [7].…”

Left $ϕ$-biprojectivity of some Banach algebras

“…Following [9] notations, a Banach algebra A is left φ-contractible if there exists an element m ∈ A such that am = φ(a)m and φ(m) = 1, for every a ∈ A. For recent works about this concept reader referred to [1], [3] and [7].…”