Multiplicative Linear Functionals on a Banach Algebra

Multiplicative Linear Functionals on a Banach Algebra

Definition: Let $X$ be a Banach algebra over $\mathbb{C}$. A linear functional $f : X \to \mathbb{C}$ is said to be Multiplicative if $f$ is not the zero functional and $f(xy) = f(x)f(y)$ for all $x, y \in X$. The Set of All Multiplicative Linear Functionals on $X$ is denoted $\Phi_X$. The Set of All Multiplicative Linear Functionals on $X$ AND the Zero Linear Functional is denoted $\Phi_X^{\infty}$.

Note that a multiplicative linear functional is simply a nonzero homomorphism of $X$ to $\mathbb{C}$.

We first prove some elementary properties of multiplicative linear functionals.

Proposition 1: Let $X$ be a Banach algebra over $\mathbb{C}$. If $f : X \to \mathbb{C}$ is a multiplicative linear functional then $f$ is bounded and $\| f \| \leq 1$.
  • Proof: Let $f$ be a multiplicative linear functional. We will show that for all $x \in X$ with $\| x \| < 1$ we have that $f(x) \neq 1$.
  • Suppose instead that there exists an $x \in X$ with $\| x \| < 1$ such that $f(x) = 1$. Let $\displaystyle{y = \sum_{n=1}^{\infty} x^n}$. Observe that $\displaystyle{\sum_{n=1}^{\infty} \| x^n \| \leq \sum_{n=1}^{\infty} \| x \|^n < \infty}$ since $\| x \| < 1$. Since $X$ is a Banach space, the absolute convergence of $y$ implies that $y \in X$. Furthermore observe that:
(1)
\begin{align} \quad x +xy = x + x \sum_{n=1}^{\infty} x^n = x + \sum_{n=2}^{\infty} x^n = \sum_{n=1}^{\infty} x^n = y \end{align}
  • Therefore $y = x + xy$. Applying $f$ to both sides of this equation yields:
(2)
\begin{align} \quad f(y) = f(x + xy) = f(x) + f(x)f(y) = 1 + f(y) \end{align}
  • Which is a contradiction. So for all $x \in X$ with $\| x \| < 1$ we have that $f(x) \neq 1$.
  • We now show that if $x \in X$ is such that $\| x \| < 1$ then $|f(x)| \leq 1$. Let $x \in X$ with $\| x \| < 1$ and suppose that $|f(x)| > 1$. Then $\| x \| < |f(x)|$. Consider the vector $\frac{x}{f(x)}$. Then:
(3)
\begin{align} \quad \left \| \frac{x}{f(x)} \right \| = \frac{\| x \|}{|f(x)|} < 1 \end{align}
  • So by the early proof we must have that:
(4)
\begin{align} \quad 1 = f \left ( \frac{x}{f(x)} \right ) \neq 1 \end{align}
  • Again a contradiction. Thus if $x \in X$ is such that $\| x \| < 1$ then $|f(x)| \leq 1$.
  • Finally, if $x \in X$ is such that $\| x \| \leq 1$. To see why, let $0 < a < 1$. Then $\| ax \| < 1$ and so $a|f(x)| |f(ax)| \leq 1$. Thus $|f(x)| \leq \frac{1}{a}$ for all $0 < a < 1$ implying that $|f(x)| \leq 1$. So:
(5)
\begin{align} \quad \| f \| = \sup_{\| x \| \leq 1} \{ |f(x)| \} \leq 1 \quad \blacksquare \end{align}
Proposition 2: Let $X$ be a Banach algebra over $\mathbb{C}$ with unit. If $f : X \to \mathbb{C}$ is a multiplicative linear functional then $f(1) = 1$ and so $\| f \| = 1$.
  • Proof: We have that $1 = 1 \cdot 1$ and so:
(6)
\begin{align} \quad f(1) = f(1 \cdot 1) = f(1)f(1) \end{align}
  • So either $f(1) = 0$ or $f(1) = 1$. If $f(1) = 0$ then for all $x \in X$ we have that $f(x) = f(1 \cdot x) = f(1)f(x) = 0$, which cannot happen since $f$ is not the zero functional. Thus $f(1) = 1$. Thus by proposition 1 we have that:
(7)
\begin{align} \quad \| f \| = \sup_{\| x \| \leq 1} \{ |f(x)| \} = 1 \end{align}
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