Local existence of 𝒦-sets, projective tensor products, and Arens regularity for 𝒜(ℰ₁+…+ℰ_{𝓃})

Abstract

and (iii) A(Y 1 + · · · + Yn) is isomorphic to the projective tensor product C(Y 1 )⊗ · · ·⊗C(Yn).This extends what was previously known for groups such as T or for the case n = 2 to the general locally compact abelian group. Old results concerning the local existence of Kronecker and Kp-sets are improved.

The Figa–Talamanca–Herz–Lebesgue Banach algebras $A_p^r(G)=A_p(G)\cap L^r(G)$

“…More results in this regard are obtained in this section, namely Theorems 4-7. These improve results of ours in [21] and powerful results of Colin Graham in [24].…”

“…Clearly Remark 10. Graham's result in [24] is a vast improvement of part of our corollary 6 in [20], where even extreme non Arens regularity is proved for A 2 (E), for symmetric sets E, and where the study of the Arens regularity of quotients of A p (G) was initiated.…”

“…We emphasize that our A p (G), coincides with A p (G)à la Herz [27]. It follows that A p (G) * is the set of convolution operators on L p , denoted in [27] [24]). Remark 1.…”

“…Thus any multiplicative linear functional η on A r p has a multiplicative extension to A p . Hence by [24], η = δ x for some x in G. [24]. Thus by (a),…”

“…is regular, since if K and F are a compact and a closed subset of G, respectively, such that coincides with that of A p , which, by [24], is G. Alternately note that…”