Bounded multiplier algebras arising from Fock representation associated to semigroups

Abstract

In this article, we attempt to introduce the "Multiplier algebra" associated to the Fock representation that arising from the left-cancellative semigroup $\mathcal{S}$ (denoted by $M(\mathcal{S})$) by adopting the concept of multiplier algebra of a $C^*$-algebra. Then, we investigate the basic properties and examples of the multiplier algebras. In order to make sense of multiplier algebra, we establish two key results of the multiplier algebras. We demonstrate that $M(\mathcal{S})$ is an unital Banach algebra if $\mathcal{S}$ is a left-cancellative semigroup. In the consideration, $G$ is a group, we demonstrate that $M(G)$ is a $C^*$-algebra. We illustrate that the associated multiplier algebras $M(\mathbb{Z}_{+}), M(\mathbb{Z}^2_+)$ are identified with respective Hardy algebras $H^{\infty}(\mathbb{D})$ and $H^{\infty}(\mathbb{D}^2)$ for $\mathcal{S}=\mathbb{Z}_{+}, \mathbb{Z}^2_{+}.$ Next, we discuss that multiplier algebra associated to the free semigroup $\mathcal{S}=\mathbb{F}^+_{n}$. We clearly show that the well-known non-commutative Hardy algebra $\mathbb{F}_{n}^{\infty}$ (introduced and thoroughly studied by G. Popescu) and the multiplier algebra $M(\mathbb{F}_{n}^+)$ are isometrically isomorphic. Finally, using the operator space technique, we have demonstrated an intriguing result that $M(\mathcal{S})$ is an operator algebra (specifically, thanks to celebrated Blecher-Ruan-Sinclair theorem).