Elementary Properties and Examples

Verify the statements made in the following examples.

• If $\mathcal{H}$ is a Hilbert space, $\mathcal{A} = \mathcal{B}(\mathcal{H})$ is a $C^*$-algebra where for each $A$ in $\mathcal{B}(\mathcal{H})$, $A^* =$ the adjoint of $A$.

• If $\mathcal{H}$ is a Hilbert space, $\mathcal{B}_0(\mathcal{H})$ is a $C^*$-subalgebra of $\mathcal{B}(\mathcal{H})$, though $\mathcal{B}_0(\mathcal{H})$ does not have an identity if $\mathcal{H}$ is infinite dimensional.

• If $X$ is a compact space, $C(X)$ is a $C^*$-algebra where $f^*(x) = \overline{f(x)}$ for $f$ in $C(X)$ and $x$ in $X$.

• If $(X, \Omega , \mu )$ is a $\sigma$-finite measure space, $L^\infty (X, \Omega , \mu )$ is a $C^*$-algebra where the involution is defined as in the preceding part.

• If $X$ is locally compact but not compact, $C_0(X)$ is a $C^*$-algebra without identity.

Let $\mathcal{A} = \{ f \in C(\operatorname {cl} \mathbb {D}): f \text{ is analytic in } \mathbb {D}\}$ and for $f$ in $\mathcal{A}$ define $f^*$ by $f^*(z) = \overline{f(\bar{z})}$. Show that $\mathcal{A}$ is a Banach algebra, $f^* \in \mathcal{A}$ when $f \in \mathcal{A}$, and $\left\| f^*\right\| = \left\| f\right\|$, but $\mathcal{A}$ is not a $C^*$-algebra.

If $\{ \mathcal{A}_i : i \in I\}$ is a collection of $C^*$-algebras, show that $\bigoplus_\infty \mathcal{A}_i$ and $\bigoplus_0 \mathcal{A}_i$ are $C^*$-algebras.

Let $X$ be a locally compact space and let $\mathcal{A}$ be a $C^*$-algebra. If $C_b(X, \mathcal{A}) =$ the collection of bounded continuous functions from $X \to \mathcal{A}$, show that $C_b(X, \mathcal{A})$ is a $C^*$-algebra. Let $C_0(X, \mathcal{A}) =$ all of the continuous functions $f: X \to \mathcal{A}$ such that for every $\varepsilon > 0$, $\{ x \in X : \left\| f(x)\right\| \geq \varepsilon \}$ is compact. Show that $C_0(X, \mathcal{A})$ is a $C^*$-algebra.