The Quasi-Product of Bounded Nets with Left/Right Approx. Identities

The Quasi-Product of Bounded Nets with Left/Right Approximate Identities

Lemma 1: Let $\mathfrak{A}$ be a normed algebra. a) If $\{ f(\mu) \}_{\mu \in M}$ is a bounded net in $\mathfrak{A}$ and $\{ e(\lambda) \}_{\lambda \in \Lambda}$ be a left approximate identity for $\mathfrak{A}$ then $\{ f (\mu) \circ e(\lambda) \}_{(\lambda, \mu) \in \Lambda \times M}$ is a left approximate identity for $\mathfrak{A}$. b) If $\{ f(\mu) \}_{\mu \in M}$ is a right approximate identity for $\mathfrak{A}$ and $\{ e(\lambda) \}_{\lambda \in \Lambda}$ is a bounded net then $\{ f (\mu) \circ e(\lambda) \}_{(\lambda, \mu) \in \Lambda \times M}$ is a right approximate identity for $\mathfrak{A}$.

For each $\lambda \in \Lambda$ and each $\mu \in M$, $f(\mu)$ and $e(\lambda)$ are elements in $\mathfrak{A}$ and $f(\mu) \circ e(\lambda)$ denotes the quasi-product of these two elements.

• Proof of a): Let $\Lambda \times M$ be the cartesian product of the sets directed sets $(\Lambda, \leq)$, $(M, \leq)$, where $\Lambda \times M$ is also directed such that if $(\lambda_1, \mu_1), (\lambda_2, \mu_2) \in \Lambda \times M$ is such that $(\lambda_1, \mu_1) \leq (\lambda_2, \mu_2)$ if and only if $\lambda_1 \leq \lambda_2$ and $\mu_1 \leq \mu_2$. (Here, there is some abuse of notation. Of course, the direction relation on $\Lambda$, $M$, and $\Lambda \times M$ is not necessarily the same, but we will use the same symbol, $\leq$, in all three regards.)

• Since $\{ f(\mu) \}_{\mu \in M}$ is a bounded net, there exists an $K > 0$ such that for every $\mu \in M$, $\| f(\mu) \| \leq K$.

• Let $a \in \mathfrak{A}$. Since $\{e(\lambda) \}_{\lambda \in \Lambda}$ is a left approximate identity for $\mathfrak{A}$ we have that $e(\lambda)a \to a$. So for $\frac{\epsilon}{1 + K} > 0$ there exists a $\lambda_0 \in \Lambda$ such that if $\lambda \geq \lambda_0$ then:

(1)

\begin{align} \quad \| e(\lambda)a - a \| < \frac{\epsilon}{1+K} \end{align}

• Take any $\mu_0 \in M$. Then if $(\lambda, \mu) \geq (\lambda_0, \mu_0)$ then $\mu \geq \mu_0$ and so:

(2)

\begin{align} \quad \| [f(\mu) \circ e(\lambda)]a - a \| &= \| [f(\mu) + e(\lambda) - f(\mu)e(\lambda)]a - a \| \\ &= \| f(\mu)a + e(\lambda)a - f(\mu)e(\lambda)a - a \| \\ &= \| [e(\lambda)a - a] + f(\mu)[a - e(\lambda)a] \| \\ & \leq \| e(\lambda)a - a \| + \| f(\mu) \| \| a - e(\lambda)a \| \\ & \leq [1 + \| f(\mu) \|] \| e(\lambda)a - a \| \\ & \leq [1 + K] \frac{\epsilon}{1 + K} \\ & < \epsilon \end{align}

• So $[f(\mu) \circ e(\lambda)]a \to a$ for every $a \in \mathfrak{A}$. Thus $f(\mu) \circ e(\lambda)$ is a left approximate identity for $\mathfrak{A}$. $\blacksquare$

• Proof of b) Since $\{ e(\lambda) \}_{\lambda \in \Lambda}$ is bounded there exists a $K > 0$ such that $\| e(\lambda) \| \leq K$ for all $\lambda \in \Lambda$.

• Let $a \in \mathfrak{A}$ and let $\epsilon > 0$ be given. Since $af(\mu) \to a$, for $\frac{\epsilon}{1 + K} > 0$ there exists a $\mu_0 \in M$ such that if $\mu \geq \mu_0$ then:

(3)

\begin{align} \quad \| af(\mu) - a \| < \epsilon \end{align}

• Take any $\lambda_0 \in \Lambda$. Then if $(\lambda, \mu) \geq (\lambda_0, \mu_0)$ then $\mu \geq \mu_0$ and so:

(4)

\begin{align} \quad \| a[f(\mu) \circ e(\lambda)] - a \| &= \| a[f(\mu) + e(\lambda) - f(\mu)e(\lambda)] - a \| \\ &= \| af(\mu) + ae(\lambda) - af(\mu)e(\lambda) - a \| \\ &= \| [af(\mu) - a] + [a - af(\mu)]e(\lambda) \| \\ &\leq \| af(\mu) - a \| + \| a - af(\mu) \| \| e(\lambda) \| \\ &\leq \| af(\mu) - a \| [1 + \| e(\lambda) \|] \\ &< \frac{\epsilon}{1 + K} [1 + K] \\ &< \epsilon \end{align}

• Therefore $a[f(\mu) \circ e(\lambda)] \to a$ for every $a \in \mathfrak{A}$ and so $\{ f(\mu) \circ e(\lambda) \}_{(\lambda, \mu) \in \Lambda \times M}$ is a right approximate identity for $\mathfrak{A}$. $\blacksquare$

Corollary 2: Let $\mathfrak{A}$ be a normed algebra. a) If $\{ e(\lambda) \}_{\lambda \in \Lambda}$ is a bounded left approximate identity for $\mathfrak{A}$ and $\{ f(\mu) \}_{\mu \in M}$ is a bounded net in $\mathfrak{A}$. Then $\{ f (\mu) \circ e(\lambda) \}_{(\lambda, \mu) \in \Lambda \times M}$ is a bounded left approximate identity for $\mathfrak{A}$. b) If $\{ f(\mu) \}_{\mu \in M}$ is a bounded right approximate identity for $\mathfrak{A}$ and $\{ e(\lambda) \}_{\lambda \in \Lambda}$ is a bounded net in $\mathfrak{A}$ then $\{ f (\mu) \circ e(\lambda) \}_{(\lambda, \mu) \in \Lambda \times M}$ is a bounded right approximate identity for $\mathfrak{A}$

• Proof: By lemma 1 we have that $\{ f(\mu) \circ e(\lambda) \}_{(\lambda, \mu) \in \Lambda \times M}$ is a left approximate identity for $\mathfrak{A}$. Since $\{ e(\lambda) \}_{\lambda \in \Lambda}$ is bounded there exists an $K_1 > 0$ such that $\| e(\lambda) \| \leq K_1$ for all $\lambda \in \Lambda$, and since $\{ f(\mu) \}_{\mu \in M}$ is bounded there exists an $K_2 > 0$ such that $\| f(\mu) \| \leq K_2$ for all $\mu \in M$.

• Let $K = K_2 + K_1 + K_2K_1$. Then for all $(\lambda, \mu) \in \Lambda \times M$ we have that:

(5)

\begin{align} \quad \| f(\mu) \circ e(\lambda) \| &= \| f(\mu) + e(\lambda) - f(\mu)e(\lambda) \| \\ & \leq \| f(\mu) \| + \| e(\lambda) \| + \| f(\mu) \| \| e(\lambda) \| \\ & \leq K_2 + K_1 + K_2K_1 \\ & \leq K \end{align}

• So $\{ f(\mu) \circ e(\lambda) \}_{(\lambda, \mu) \in \Lambda \times M}$ is a bounded left approximate identity for $\mathfrak{A}$. $\blacksquare$

Corollary 3: Let $\mathfrak{A}$ be a normed algebra. Let $\{ e(\lambda) \}_{\lambda \in \Lambda}$ be a bounded left approximate identity for $\mathfrak{A}$ and let $\{ f(\mu) \}_{\mu \in M}$ be a bounded right approximate identity for $\mathfrak{A}$. Then $\{ f(\mu) \circ e(\lambda) \}_{(\lambda, \mu) \in \Lambda \times M}$ is a bounded two-sided approximate identity for $\mathfrak{A}$.

• Proof: Since $\{ f(mu) \}_{\mu \in M}$ is a bounded net and $\{ e(\lambda) \}_{\lambda \in \Lambda}$ is a left approximate identity for $\mathfrak{A}$ we have by Lemma 1 and Corollary 2 that $\{ f(\mu) \circ e(\lambda) \}_{(\lambda, \mu) \in \Lambda \times M}$ is a bounded left approximate identity for $\mathfrak{A}$.

• Similarly, since $\{ e(\lambda) \}_{\lambda \in \Lambda}$ is a bounded net and $\{ f(\mu) \}_{\mu \in M}$ is a right approximate identity for $\mathfrak{A}$ we again have by Lemma 1 and Corollary 2 that $\{ f(\mu) \circ e(\lambda) \}_{(\lambda, \mu) \in \Lambda \times M}$ is a bounded right approximate identity for $\mathfrak{A}$.

• Thus $\{ f(\mu) \circ e(\lambda) \}_{(\lambda, \mu) \in \Lambda \times M}$ is a bounded two-sided approximate identity for $\mathfrak{A}$. $\blacksquare$