Banach Algebras

Use the identity $(x y)^{n} = x(y x)^{n-1} y$ to prove that $x y$ and $y x$ always have the same spectral radius.

Prove that every finite-dimensional $A$ is isomorphic to an algebra of matrices. Hint: The proof of Theorem 10.2 shows that every $A$ is isomorphic to a subalgebra of $\mathscr {P}(A)$. Conclude that $x y = e$ implies $y x = e$ if $\operatorname {dim} A < \infty$.

Let $A_{0}$ and $A_{1}$ be the algebras of all complex 2-by-2 matrices of the form

Prove that every two-dimensional complex algebra $A$ with unit $e$ is isomorphic to one of these and that $A_{0}$ is not isomorphic to $A_{1}$. Hint: Show that $A$ has a basis $\left\{ e, a\right\}$ in which $a^{2} = \lambda e$ for some $\lambda \in \mathbb {C}$. Distinguish between the cases $\lambda = 0, \lambda \neq 0$. Show that there exists a three-dimensional noncommutative Banach algebra.

Let $A$ be the algebra of all complex functions $f$ on $\left\{ 1,2,3, \ldots \right\}$ which are 0 except at finitely many points, with pointwise addition and multiplication and norm

Show that multiplication is left-continuous (hence also right-continuous, since $A$ is commutative) but not jointly continuous. (Adjunction of a unit element, as suggested in Section 10.1, would make no difference). Show, in fact, that there is a sequence $\left\{ f_{n}\right\}$ in $A$ so that $\left\| f_{n}\right\| \rightarrow 0$ but $\left\| f_{n}^{2}\right\| \rightarrow \infty$, as $n \rightarrow \infty$.

Let $C^{2} = C^{2}([0,1])$ be the space of all complex functions on [0, 1] whose second derivative is continuous. Choose $a > 0, b > 0$, and define

Show that this makes $C^{2}$ into a Banach space, for every choice of $a, b$, but that the Banach algebra axioms hold if and only if $2 b \leq a^{2}$. (For necessity, consider $x$ and $x^{2}$).

What happens if the process of adjoining a unit (described in Section 10.1) is applied to an algebra $A$ which already has a unit? Clearly, the result cannot be an algebra $A_{1}$ with two units. Explain.

Suppose that $\Omega$ is open in $C$ and that $f: \Omega \rightarrow A$ and $\varphi : \Omega \rightarrow C$ are holomorphic. Prove that $\varphi f: \Omega \rightarrow A$ is holomorphic.

Another proof that $\sigma (x) \neq \varnothing$ can be based on Liouville's theorem 3.32 and the fact that $(\lambda e-x)^{-1} \rightarrow 0$ as $\lambda \rightarrow \infty$. Complete the details.

Call $x \in A$ a topological divisor of zero if there is a sequence $\left\{ y_{n}\right\}$ in $A$, with $\left\| y_{n}\right\| = 1$, such that

Find the spectrum of the operator $T \in \mathscr {B}\left(\ell^{2}\right)$ given by

Suppose $K = \left\{ \lambda \in \mathscr {C}: 1 \leq \left|\lambda \right| \leq 2\right\}$. Put $f(\lambda ) = \lambda$. Let $A$ be the smallest closed subalgebra of $C(K)$ that contains 1 and $f$. Let $B$ be the smallest closed subalgebra of $C(K)$ that contains $f$ and $1 / f$. Describe the spectra $\sigma_{A}(f)$ and $\sigma_{B}(f)$.

Suppose $X$ is a Banach space, $T \in \mathscr {B}(X)$ is compact, and $\left\| T^{n}\right\| \geq 1$ for all $n \geq 1$. Prove that the point spectrum of $T$ is not empty.

Let $X = C([0,1])$ and define $T \in \mathscr {B}(X)$ by

Show that $\sigma_{p}(T) = \varnothing$. Hence $f\left(\sigma_{p}(T)\right) = \varnothing$ for every $f$, but if $f \equiv 0$, then $\tilde{f}(T) =$ 0 ; hence

The extra hypotheses in (c) and (d) of Theorem 10.33 are thus needed.

Suppose that the spectrum of some $x \in A$ is not connected. Prove that $A$ contains a nontrivial idempotent 2. (This is defined in Exercise 2).

Prove also that $A = A_{0} \oplus A_{1}$, where

Suppose $\Omega$ is open in $\mathcal{C}, \alpha$ is an isolated boundary point of $\Omega , f: \Omega \rightarrow X$ is a holomorphic $X$-valued function (where $X$ is some complex Banach space), and there is a smallest nonnegative integer $n$ such that

is bounded as $\lambda \rightarrow \alpha$. If $n > 0$, then $f$ is said to have a pole of order $n$ at $\alpha$.

Let $S_{R}$ be the right shift, acting on $\ell^{2}$, as in Exercise 2. Let $\left\{ c_{n}\right\}$ be a sequence of complex numbers such that $c_{n} \neq 0$ but $c_{n} \rightarrow 0$ as $n \rightarrow \infty$. Define $M \in \mathscr {B}\left(\ell^{2}\right)$ by

and define $T \in \mathscr {B}\left(\ell^{2}\right)$ by $T = M S_{R}$.

Suppose $x \in A, x_{n} \in A$, and $\lim x_{n} = x$. Suppose $\Omega$ is an open set in $\mathscr {C}$ that contains a component of $\sigma (x)$. Prove that $\sigma \left(x_{n}\right)$ intersects $\Omega$ for all sufficiently large $n$. (This strengthens Theorem 10.20). Hint: If $\sigma (x) \subset \Omega \cup \Omega_{0}$, where $\Omega_{0}$ is an open set disjoint from $\Omega$, consider the function $f$ that is 1 in $\Omega , 0$ in $\Omega_{0}$.

Let $C_{R}$ be the algebra of all real continuous functions on $[0,1]$, with the supremum norm. This satisfies all requirements of a Banach algebra, except that the scalars are now real.

Suppose $A = C(T)$, the algebra of all continuous complex functions on the unit circle $T$, with the supremum norm. Show that two invertible members of $C(T)$ are in the same coset of $G_{1}$ if and only if they are homotopic mappings of $T$ into the set of all nonzero complex numbers. Deduce from this that $G / G_{1}$ is isomorphic to the additive group of the integers. (The notation is as in Theorem 10.34).

Suppose $A = M(R)$, the convolution algebra of all complex Borel measures on the real line; see (e) of Example 10.3. Supply the details in the following proof that $G / G_{1}$ is uncountable: If $\alpha \in R$, let $\delta_{\alpha }$ be the unit mass concentrated at $\alpha$. Assume $\delta_{\alpha } \in G_{1}$. Then $\delta_{\alpha } = \exp \left(\mu_{\alpha }\right)$ for some $\mu_{\alpha } \in M(R)$; hence, for $-\infty < t < \infty$,

where $k$ is an integer. Since $\hat{\mu }_{\alpha }$ is a bounded function, $\alpha = 0$. Thus $\delta_{0}$ is the only $\delta_{\alpha }$ in $G_{1}$. No coset of $G_{1}$ in $G$ contains therefore more than one $\delta_{\alpha }$.

As regards the invariant subspace problem, described in the introduction to Theorem 10.35, explain why the problem is

Let $S_{R}$ be the right shift on $\ell^{2}$, as in Exercise 2. Prove that 0 is the only compact $T \in \mathscr {B}\left(\ell^{2}\right)$ that commutes with $S_{R}$. Hint: If $T \neq 0$ then

does not tend to 0 when $N-M \rightarrow \infty$.