Math Problems Vault

Problem VII.1.1

Let $\mathcal{A}$ be an algebra that is also a Banach space and such that if $a \in \mathcal{A}$ , the maps $x \mapsto ax$ and $x \mapsto xa$ of $\mathcal{A} \to \mathcal{A}$ are continuous. Let $\mathcal{A}_1 = \mathcal{A} \times \mathbb {F}$ as in Proposition 1.3. If $a \in \mathcal{A}$ , define $L_a: \mathcal{A}_1 \to \mathcal{A}_1$ by $L_a(x, \xi ) = (ax + \xi a, 0)$ . Show that $L_a \in \mathcal{B}(\mathcal{A}_1)$ and if $\vertiii {a} = \left\| L_a\right\|$ , then $\vertiii {\cdot }$ is equivalent to the norm of $\mathcal{A}$ and $\mathcal{A}$ with $\vertiii {\cdot }$ is a Banach algebra.

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Problem VII.1.2

Complete the proof of the following proposition. If $\mathcal{A}$ is a Banach algebra without an identity, let $\mathcal{A}_1 = \mathcal{A} \times \mathbb {F}$ . Define algebraic operations on $\mathcal{A}_1$ by

\begin{align} (a, \alpha ) + (b, \beta ) & = (a + b, \alpha + \beta ); \\ \beta (a, \alpha ) & = (\beta a, \beta \alpha ); \\ (a, \alpha )(b, \beta ) & = (ab + \alpha b + \beta a, \alpha \beta ). \end{align}

Define $\left\| (a, \alpha )\right\| = \left\| a\right\| + \left|\alpha \right|$ . Then $\mathcal{A}_1$ with this norm and the algebraic operations defined above is a Banach algebra with identity $(0, 1)$ and $a \mapsto (a, 0)$ is an isometric isomorphism of $\mathcal{A}$ into $\mathcal{A}_1$ .

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Problem VII.1.3

Verify the statements made in Examples (1.4) through (1.9) and (1.11).

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Problem VII.1.4

(a)(b)(c)(d)

Let $G$ be a locally compact group.

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(a)

If $\phi \in C_c(G)$ and $\varepsilon > 0$ , show that there is an open neighborhood $U$ of $e$ in $G$ such that $\left\| \phi_x - \phi_y\right\| < \varepsilon$ whenever $xy^{-1} \in U$ . [Here $\phi_x(z) = \phi (xz)$ .]

(b)

Show that if $f \in L^p(G)$ , $1 \le p < \infty$ , and $\varepsilon > 0$ , there is an open neighborhood $U$ of $e$ in $G$ such that $\left\| f_x - f_y\right\|_p < \varepsilon$ whenever $xy^{-1} \in U$ .

(c)

Show that if $f \in L^1(G)$ and $g \in L^\infty (G)$ , $h(x) = \int f(xy^{-1})g(y) \, dm(y)$ defines a bounded continuous function $h: G \to \mathbb {F}$ .

(d)

If $f, g \in L^1(G)$ and $h$ is defined as in (c), show that $h \in L^1(G)$ .

Problem VII.1.5

Prove the following proposition. If $\{ \mathcal{A}_i\}$ is a collection of Banach algebras, then $\bigoplus_0 \mathcal{A}_i$ and $\bigoplus_\infty \mathcal{A}_i$ are Banach algebras, where $\bigoplus_0 \mathcal{A}_i \equiv \left\{ a \in \prod_i \mathcal{A}_i : \text{for all } \varepsilon > 0, \{ i : \left\| a(i)\right\| \ge \varepsilon \} \text{ is finite}\right\}$ .

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Problem VII.1.6

(a)(b)(c)

Let $\{ \mathcal{A}_i : i \in I\}$ be a collection of Banach algebras.

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(a)

Show that $\bigoplus_0 \mathcal{A}_i$ is a closed ideal of $\bigoplus_\infty \mathcal{A}_i$ .

(b)

Show that $\bigoplus_\infty \mathcal{A}_i$ has an identity if and only if each $\mathcal{A}_i$ has an identity.

(c)

Show that $\bigoplus_0 \mathcal{A}_i$ has an identity if $I$ is finite and each $\mathcal{A}_i$ has an identity.

Problem VII.1.7

If $X, Y$ are completely regular, show that $C_b(X) \oplus_\infty C_b(Y)$ is isometrically isomorphic to $C_b(X \oplus Y)$ , where $X \oplus Y$ is the disjoint union of $X$ and $Y$ .

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Problem VII.1.8

If $X$ and $Y$ are locally compact, show that $C_0(X) \oplus_\infty C_0(Y)$ is isometrically isomorphic to $C_0(X \oplus Y)$ .

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Problem VII.1.9

Let $\{ X_i : i \in I\}$ be a collection of locally compact spaces and let $X =$ the disjoint union of these spaces furnished with the topology $\left\{ U \subseteq X : U \cap X_i \text{ is open in } X_i \text{ for all } i\right\}$ . Show that $X$ is locally compact and $\bigoplus_0 C_0(X_i)$ is isometrically isomorphic to $C_0(X)$ .

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