Segal Algebras on Non-Abelian Groups

Abstract

Abstract. Let Sl(G) be a Segal algebra on a locally compact group. The central functions of S '(G) are dense in the center of L\G). S\G) has central approximate units iff G G [SIN]. This is a generalization of a result of Reiter on the one hand and of Mosak on the other hand. The proofs depend on the structure theorems of [S/JV]-and [W]-groups. In the second part some new examples of Segal algebras are constructed. A locally compact group is discrete or Abelian iff every Segal algebra is rightinvariant. As opposed to the results, the proofs are not quite obvious.

Biflatness and Pseudo-Amenability of Segal Algebras

“…we obtain a net of positive norm-one functions in L 1 (G) which by (3.1) satisfies Proof If statement (i) holds, then L 1 (G) is amenable and therefore biflat [4, Theorem 2.9.65], and S 1 (G) has a central approximate identity (e λ ) λ which is bounded in L 1 (G) [21]. Hence, (e 2 λ ) λ is also an approximate identity for S 1 (G), so (ii) is a consequence of Proposition 2.5.…”

Approximate and pseudo-amenability of various classes of Banach algebras

“…Since G is SIN, it follows from the results of [20] that S(G) has a central approximate identity (e i ) which is bounded in the L 1 -norm.…”

On Pseudo‐contractibility and Ultra Central Approximate Identity of Some Semigroup Algebras

“…In fact, (π A (m α )) is a central approximate identity (approximate identity) with respect to the notion pseudo-contractibility (pseudo-amenability), respectively. We have to remind that for a locally compact group G, the Segal algebra S(G) has a central approximate identity if and only if G is a SIN group (see [3]).…”