On Hulanicki and Barnes lemmas for $${\varvec {p}} $$-Banach algebras
Authors :- PA Dabhi, KB Solanki
Publication :- Proceedings-Mathematical Sciences, 2022
We prove Hulanicki’s lemma for p-Banach algebras which is as follows: let 0<p≤1, and let A be a p-Banach ∗-algebra, S be a ∗-subalgebra of A, and let T be a faithful ∗-representation of A on a Hilbert space H satisfying ∥Tx∥pop=limn→∞∥xn∥1n for all x=x∗∈S. If A has a unit e, then assume in addition that Te=I, the identity operator in B(H). If x=x∗∈S, then the spectrum of x in A is same as the spectrum of Tx in B(H). Let (G, d) be a metric space with the counting measure μ satisfying some growth conditions. Let ω(x,y)=(1+d(x,y))δ for some 0<δ≤1 and 0<p≤1. Let Apω be the collection of kernels K on G×G satisfying max{supx∑y|K(x,y)|pω(x,y)p,supy∑x|K(x,y)|pω(x,y)p}<∞. Each K∈Apω defines a bounded linear operator on ℓ2(G). If in addition, ω satisfies the weak growth condition, then we show that Apω is inverse closed in B(ℓ2(G)) which is Barnes’ lemma for p-Banach algebras. We shall also discuss inverse-closedness of the p-Banach algebra of weighted p-summable sequences over Z2d with the twisted convolution as an application of these results.