The Discrete Semigroup Algebra of a Semigroup S

The Discrete Semigroup Algebra of a Semigroup S

Let $S$ be a semigroup. That is, $S$ is a set with an associative binary operation defined on $S$. Let $\alpha$ be a real-valued function on $S$ such that $\alpha(x) > 0$ for every $x \in S$, and such that for every $x, y \in S$:

(1)
\begin{align} \quad \alpha(x \cdot y) \leq \alpha (x) \alpha (y) \end{align}

Let:

(2)
\begin{align} \quad \ell^1(S, \alpha) = \left \{ f : S \to \mathbb{C} \biggr | \sum_{x \in S} |f(s)| \alpha(s) < \infty \right \} \end{align}

For each $f, g \in \ell^1(S, \alpha)$ and all $a \in \mathbf{F}$ we define addition $f + g$, scalar multiplication $a f$, and convolution multiplication $f * g$ for each $x \in S$ by:

(3)
\begin{align} \quad (f + g)(x) &= f(x) + g(x) \\ \quad (af)(x) &= af(x) \\ \quad (f * g)(x) &= \sum_{tu = x} f(t)g(u) \quad \mathrm{where} \quad (f * g)(x) = 0 \: \mathrm{if} \: tu = x \: \mathrm{has \: no \: solutions} \end{align}

Define a function $\| \cdot \| : \ell^1(S, \alpha) \to [0, \infty)$ for each $f \in \ell^1(S, \alpha)$ by:

(4)
\begin{align} \quad \| f \| = \sum_{x \in S} |f(x)| \alpha(x) \end{align}
Proposition 1: Let $S$ be a semigroup and let $\alpha$ be a positive real-valued function on $S$. Then $\ell^1(S, \alpha)$ is a Banach algebra.
  • Proof: It is easy to verify that $\ell^1(S, \alpha)$ with the operations of addition and scalar multiplication, as well as the norm $\| \cdot \|$ is a Banach space. We will show that with the additional operation $*$ of convolution multiplication that $\ell^1(S, \alpha)$ is an algebra and then we will show that $\| \cdot \|$ is an algebra norm on $\ell^1(S, \alpha)$.
  • 1. Showing that $(f * g) * h = f * (g * h)$: Let $f, g, h \in \ell^1(S, \alpha)$. Then for every $x \in S$:
(5)
\begin{align} \quad [(f* g) * h](x) &= \sum_{tu = x} (f * g)(t)h(u) \\ &= \sum_{tu = x} \left [ \sum_{vw = t} f(v)g(w) \right ] h(u) \\ &= \sum_{tu = x} \sum_{vw = t} f(v)g(w)h(u) \\ &= \sum_{vwu = x} f(v)g(w)h(u) \quad (*) \end{align}
  • And also:
(6)
\begin{align} \quad [f * (g * h)](x) &= \sum_{ut = x} f(u)(g * h)(t) \\ &= \sum_{ut = x} f(u) \left [\sum_{vw = t} g(v)h(w) \right ] \\ &= \sum_{ut = x} \sum_{vw = t} f(u)g(v)h(w) \\ &= \sum_{uvw = x} f(u)g(v)h(w) \quad (**) \end{align}
  • These two sums are same by change of variables, and so we see that $(f * g) * h = f * (g * h)$.
  • 2. Showing that $f * (g + h) = f * g + f * h$: Let $f, g, h \in \ell^1(S, \alpha)$. Then for every $x \in S$:
(7)
\begin{align} \quad [f * (g + h)](x) = \sum_{tu = x} f(t)(g + h)(u) = \sum_{tu = x} [f(t)g(u) + f(t)h(u)] = \sum_{tu = x} f(t)g(u) + \sum_{tu = x} f(t)h(u) = (f * g)(x) + (f * h)(x) \end{align}
  • Therefore $f * (g + h) = f * g + f * h$.
  • 3. Showing that $(a f) * g = a (f * g) = f * (ag)$: Let $f, g \in \ell^1(S, \alpha)$ and let $a \in \mathbf{F}$. Then for every $x \in S$ we have that:
(8)
\begin{align} \quad [(af) * g](x) = \sum_{tu = x} af(t)g(u) = a \sum_{tu = x} f(t)g(u) = [a (f * g)](x) \end{align}
  • And also:
(9)
\begin{align} \quad [a(f * g)](x) = a \sum_{tu = x} f(t)g(u) = \sum_{tu = x} f(t) a g(u) = [f * (ag)](x) \end{align}
  • Therefore $(af) * g = a (f * g) = f * (ag)$.
  • Hence $\ell^1(S, \alpha)$ is an algebra over $\mathbf{F}$. We now show that $\| \cdot \|$ is a norm on $\ell^1(S, \alpha)$.
  • 1. Showing that $\| f \| = 0$ if and only if $f = 0$: Suppose that $\| f \| = 0$. Then $\displaystyle{\sum_{x \in S} |f(x)|\alpha(x) = 0}$. Since $\alpha(x) > 0$ for all $x \in S$, this implies that $|f(x)| = 0$ for all $x \in S$ and so $f = 0$. On the other hand, if $f = 0$ then $\displaystyle{\sum_{x \in S} |f(x)|\alpha(x) = 0}$, i.e., $\| f \| = 0$.
  • 2. Showing that $\| a f \| = |a| \| f \|$ for all $f \in \ell^1(S, \alpha)$ and all $a \in \mathbf{F}$: Let $f \in \ell^1(S, \alpha)$ and let $a \in \mathbf{F}$. Then:
(10)
\begin{align} \quad \| a f \| = \sum_{x \in S} |af(x)| \alpha(x) = \sum_{x \in S}|a||f(x)| \alpha(x) = |a| \sum_{x \in S} |f(x)| \alpha(x) = |a| \| f \| \end{align}
  • 3. Showing that $\| f + g \| \leq \| f \| + \| g \|$: Let $f, g \in \ell^1(S, \alpha)$. Then:
(11)
\begin{align} \quad \| f + g \| = \sum_{x \in S}|f(x) + g(x)|\alpha(x) \leq \sum_{x \in S} [|f(x)| + |g(x)|]\alpha(x) = \sum_{x \in S} [|f(x)| \alpha(x) + |g(x)| \alpha(x)] = \sum_{x \in S} |f(x)| \alpha(x) + \sum_{x \in S} |g(x)| \alpha(x) = \| f \| + \| g \| \end{align}
  • 4. Showing that $\| f * g \| \leq \| f \| \| g \|$: Let $f, g \in \ell^1(S, \alpha)$. Then using the fact that $\alpha(tu) \leq \alpha(t) \alpha(u)$, we have that::
(12)
\begin{align} \quad \| f * g \| = \sum_{x \in S} |(f * g)(x)| \alpha(x) = \sum_{x \in S} \left | \sum_{tu = x} f(t)g(u) \right | \alpha(x) \leq \sum_{x \in S} \left [ \sum_{tu = x} |f(t)g(u)| \right ] \alpha(x) = \sum_{tu \in S} |f(t)||g(u)| \alpha(tu) &\leq \sum_{tu \in S} |f(t)||g(u)|\alpha(t) \alpha(u) \\ &\leq \left [ \sum_{t \in S} |f(t)|\alpha(t) \right ] \left [ \sum_{u \in S} |f(u)| \alpha(u) \right ] \\ &\leq \| f \| \| g \| \end{align}
  • Therefore $\| \cdot \|$ is an algebra norm on $\ell^1(S, \alpha)$ and thus, $(\ell^1(S, \alpha), \| \cdot \|)$ is a normed algebra. $\blacksquare$
Definition: Let $S$ be a semigroup and let $\alpha$ be a positive real-valued function on $S$. Then $\ell^1(S, \alpha)$ with the norm $\| \cdot \| : \ell^1(S, \alpha) \to [0, \infty)$ defined for each $f \in \ell^1(S, \alpha)$ by $\displaystyle{\| f \| = \sum_{x \in S} |f(x)| \alpha(x)}$ is a Banach algebra, and if $\alpha(x) = 1$ for all $x \in S$ then $\ell^1(S, 1)$ is called the Discrete Semigroup Algebra of $S$.

Note that if $S$ is a group, then the discrete semigroup algebra of $S$ is the same as the discrete group algebra of $S$.

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