Elementary Theory of Banach Algebras

Suppose $B(X)$ is the algebra of all bounded linear operators on the Banach space $X$, with

if $A, A_{1}$, and $A_{2} \in B(X)$. Prove that $B(X)$ is a Banach algebra.

Let $n$ be a positive integer, let $X$ be the space of all complex $n$-tuples (normed in any way, as long as the axioms for a normed linear space are satisfied), and let $B(X)$ be as in Exercise 1. Prove that the spectrum of each member of $B(X)$ consists of at most $n$ complex numbers. What are they?

Take $X = L^{2}(-\infty , \infty )$, suppose $\varphi \in L^{\infty }(-\infty , \infty )$, and let $M$ be the multiplication operator which takes $f \in L^{2}$ to $\varphi f$. Show that $M$ is a bounded linear operator on $L^{2}$ and that the spectrum of $M$ is equal to the essential range of $\varphi$ (Chap. 3, Exercise 19).

What is the spectrum of the shift operator on $\ell^{2}$? (See Sec. 17.20 for the definition).

Prove that the closure of an ideal in a Banach algebra is an ideal.

If $X$ is a compact Hausdorff space, find all maximal ideals in $C(X)$.

Suppose $A$ is a commutative Banach algebra with unit, which is generated by a single element $x$. This means that the polynomials in $x$ are dense in $A$. Prove that the complement of $\sigma (x)$ is a connected subset of the plane. Hint: If $\lambda \notin \sigma (x)$, there are polynomials $P_{n}$ such that $P_{n}(x) \rightarrow (x-\lambda e)^{-1}$ in $A$. Prove that $P_{n}(z) \rightarrow (z-\lambda )^{-1}$ uniformly for $z \in \sigma (x)$.

Suppose $\sum_{0}^{\infty }\left|c_{n}\right| < \infty , f(z) = \sum_{0}^{\infty } c_{n} z^{n},\left|f(z)\right| > 0$ for every $z \in \bar{U}$, and $1 / f(z) = \sum_{0}^{\infty } a_{n} z^{n}$. Prove that $\sum_{0}^{\infty }\left|a_{n}\right| < \infty$.

Prove that a closed linear subspace of the Banach algebra $L^{1}\left(\mathbb {R}^{1}\right)$ (see Sec. 9.19) is translation invariant if and only if it is an ideal.

Show that $L^{1}(T)$ is a commutative Banach algebra (without unit) if multiplication is defined by

Find all complex homomorphisms of $L^{1}(T)$, as in Theorem 9.23. If $E$ is a set of integers and if $I_{E}$ is the set of all $f \in L^{1}(T)$ such that $\hat{f}(n) = 0$ for all $n \in E$, prove that $I_{E}$ is a closed ideal in $L^{1}(T)$, and prove that every closed ideal in $L^{1}(T)$ is obtained in this manner.

The resolvent $R(\lambda , x)$ of an element $x$ in a Banach algebra with unit is defined as

for all complex $\lambda$ for which this inverse exists. Prove the identity

and use it to give an alternative proof of Theorem 18.5.

Let $A$ be a commutative Banach algebra with unit. The radical of $A$ is defined to be the intersection of all maximal ideals of $A$. Prove that the following three statements about an element $x \in A$ are equivalent:

Find an element $x$ in a Banach algebra $A$ (for instance, a bounded linear operator on a Hilbert space) such that $x^{n} \neq 0$ for all $n > 0$, but $\lim_{n \rightarrow 0}\left\| x^{n}\right\|^{1 / n} = 0$.

Suppose $A$ is a commutative Banach algebra with unit, and let $\Delta$ be the set of all complex homomorphisms of $A$, as in Sec. 18.16. Associate with each $x \in A$ a function $\hat{x}$ on $\Delta$ by the formula

$\hat{x}$ is called the Gelfand transform of $x$. Prove that the mapping $x \rightarrow \hat{x}$ is a homomorphism of $A$ onto an algebra $\hat{A}$ of complex functions on $\Delta$, with pointwise multiplication. Under what condition on $A$ is this homomorphism an isomorphism? (See Exercise 12).

Prove that the spectral radius $\rho (x)$ is equal to

Prove that the range of the function $\hat{x}$ is exactly the spectrum $\sigma (x)$.

If $A$ is a commutative Banach algebra without unit, let $A_{1}$ be the algebra of all ordered pairs $(x, \lambda )$, with $x \in A$ and $\lambda$ a complex number; addition and multiplication are defined in the "obvious" way, and $\left\| (x, \lambda )\right\| = \left\| x\right\| +\left|\lambda \right|$. Prove that $A_{1}$ is a commutative Banach algebra with unit and that the mapping $x \rightarrow (x, 0)$ is an isometric isomorphism of $A$ onto a maximal ideal of $A_{1}$. This is a standard embedding of an algebra without unit in one with unit.

Show that $H^{\infty }$ is a commutative Banach algebra with unit, relative to the supremum norm and pointwise addition and multiplication. The mapping $f \rightarrow f(\alpha )$ is a complex homomorphism of $H^{\infty }$, whenever $\left|\alpha \right| < 1$. Prove that there must be others.

Show that the set of all functions $(z-1)^{2} f$, where $f \in H^{\infty }$, is an ideal in $H^{\infty }$ which is not closed. Hint:

Suppose $\varphi$ is an inner function. Prove that $\left\{ \varphi f: f \in H^{\infty }\right\}$ is a closed ideal in $H^{\infty }$. In other words, prove that if $\left\{ f_{n}\right\}$ is a sequence in $H^{\infty }$ such that $\varphi f_{n} \rightarrow g$ uniformly in $U$, then $g / \varphi \in H^{\infty }$.