Bounded multiplier algebras arising from Fock representation associated to semigroups

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).