Representations of a semigroup over a Hilbert space

Question

A representations of a discrete semigroup $S$ with an involution $\star$ over a Hilbert space $H$ is a semigroup homomorphism $\varphi : S \to B(H)$ that preserve the involution. That is, for any $a, b\in S$

1. $\varphi(ab)=\varphi(a)\varphi(b)$
2. $\varphi(a^{\star})=\varphi(a)^{\star}$

There is a construction that represent any inverse semigroup over a Hilbert space which I found from this paper, but do not understand some parts of it. Let me rephrase it here as I understood:

It start with the $\star$-group-algebra $\mathbb{C}[S]=\left\{\sum_{\text{finite}}az_a : a\in S, z_a\in\mathbb{C}\right\}$ whose objects are formal finite linear combinations and addition, scalar multiplication, product, $\star$ operations defined in most obvious ways. Then completion of $\mathbb{C}[S]$ under $l_2$ norm $\left\Vert \sum_{}az_a\right\Vert_2=\sqrt{\sum_{} |z_a|^2}$ makes it into a Hilbert space denoted by $l^2(S).$ So we have a chain of inclusions $S\hookrightarrow \mathbb{C}[S]\hookrightarrow l^2(S)$ and we have a representation $\varphi : S\to B(l^2(S))$ given by $$\varphi(a)(b) = \begin{cases} ba, & \text{if } aa^{\star}\ge b^{\star}b\\ 0, & \text{otherwise}\end{cases}$$ and extended linearly and continuously. Now my questions are:

1. Is this the construction mentioned in the third page of this paper?
2. Is $l^2(S)$ same as the $C^{\ast}$-algebra mentioned in the first page as $C^{\ast}(S)$?
3. Where can I read about this construction with more details?

Answer 1

1. Regarding your description of Duncan and Paterson's regular representation (which they call $\lambda $ and you are calling $\varphi $), I think you should define $\varphi (a)(b)=ab$, as opposed to $ba$, and require $a^*a\geq bb^*$, rather than $aa^{*}\ge b^{*}b$. Notice that with your definition you get an anti-representation, that is, $\varphi $ reverses multiplication in the sense that $\varphi (a_1a_2) = \varphi (a_2) \circ \varphi (a_1)$.

2. $l^2(S)$ is not a C*-algebra. It is a Hilbert space. $C^*(S)$ is the envelopping C$^*$-algebra of $l^1(S)$.

3. Paterson's book [1] has a lot more on C*-algebras of inverse semigroups. You might also want to to check [2], where a different construction is made, and which is more suitable to certain important applications, such as the Cuntz-Krieger algebras.

[1] Paterson, Alan L. T., Groupoids, inverse semigroups, and their operator algebras, Progress in Mathematics (Boston, Mass.). 170. Boston, MA: Birkhäuser. xvi, 274 p. (1999). ZBL0913.22001.

[2] Exel, Ruy, Inverse semigroups and combinatorial (C^*)-algebras, Bull. Braz. Math. Soc. (N.S.) 39, No. 2, 191-313 (2008). ZBL1173.46035.