Banach Algebras and Spectral Theory

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 $N-M \rightarrow \infty$.

Prove Proposition 11.2.

State and prove an analogue of Wiener's lemma 11.6 for power series that converge absolutely on the closed unit disc.

If $X$ is a compact Hausdorff space, show that there is a natural one-to-one correspondence between closed subsets of $X$ and closed ideals of $C(X)$.

Prove that the polynomials are dense in the polydisc algebra $A\left(U^{n}\right)$. (See Theorem 11.7). Suggestion: If $f \in A\left(U^{n}\right), 0 < r < 1$, and $f_{r}$ is defined by $f_{r}(z) = f(r z)$, then $f_{r}$ is the sum of an absolutely (hence uniformly) convergent multiple power series on $\bar{U}^{n}$.

Suppose $A$ is a commutative Banach algebra, $x \in A$, and $f$ is holomorphic in some open set $\Omega \subset \mathcal{C}$ that contains the range of $\hat{x}$. Prove that there exists $y \in A$ such that $\hat{y} = f \circ \hat{x}$, that is, such that $h(y) = f(h(x))$ for every complex homomorphism $h$ of $A$. Prove that $y$ is uniquely determined by $x$ and $f$ if $A$ is semisimple.

Suppose $A$ and $B$ are commutative Banach algebras, $B$ is semisimple, $\psi : A \rightarrow B$ is a homomorphism whose range is dense in $B$, and $\alpha : \Delta_{B} \rightarrow \Delta_{A}$ is defined by

Prove that $\alpha$ is a homeomorphism of $\Delta_{B}$ onto a compact subset of $\Delta_{A}$. [The fact that $\psi (A)$ is dense in $B$ implies that $\alpha$ is one-to-one and that the topology of $\Delta_{B}$ is the weak topology induced by the Gelfand transforms of the elements $\psi (x)$, for $x \in A$.

Let $A$ be the disc algebra, let $B = C(K)$, where $K$ is an arc in the unit disc, and let $\psi$ be the restriction mapping of $A$ into $B$. This example shows that $\alpha \left(\Delta_{B}\right)$ may be a proper subset of $\Delta_{A}$, even if $\psi$ is one-to-one.

Find an example in which $\psi (A) = B$ but $\alpha \left(\Delta_{B}\right) \neq \Delta_{A}$.

In Example $11.13(b)$ it was asserted that $\hat{A} \neq C(\Delta )$. Find several proofs of this.

Which properties of Lebesgue measure are used in Example 11.13(f)? Can Lebesgue measure be replaced by any positive measure, without changing any of the results?

Supply the details for the last paragraph in Example 11.13(f).

Let $C^{\prime }$ be the algebra of all continuously differentiable complex functions on the unit interval $[0,1]$, with pointwise multiplication, normed by

Let $A$ be the disc algebra. Associate to each $f \in A$ a function $f^{*} \in A$ by the formula

Then $f \rightarrow f^{*}$ is an involution on $A$.

Show that commuting idempotents have distance $\geq 1$. Explicitly, if $x^{2} = x$, $y^{2} = y, x y = y x$ for some $x$ and $y$ in a Banach algebra, then either $x = y$ or $\left\| x-y\right\| \geq 1$. Show that this may fail if $x y \neq y x$.

If $x y = y x$ for some $x$ and $y$ in a Banach algebra, prove that

Let $t$ be a large positive number, and define a norm on $\varphi^{2}$ by

Let $A$ be the algebra of all complex 2-by-2 matrices, with the corresponding operator norm:

For $y \in A$, let $y^{*}$ be the conjugate transpose of $y$. Consider a fixed $x \in A$, namely,

Prove the following statements.

A complex function $\phi$, defined on $\mathbb {R}^{n}$, is said to be positive-definite if

for every choice of $x_{1}, \ldots , x_{r}$, in $\mathbb {R}^{n}$ and for every choice of complex numbers $c_{1}, \ldots , c_{r}$.

Let $\Delta$ be the maximal ideal space of a commutative Banach algebra $A$. Call a closed set $\beta \subset \Delta$ an $A$-boundary if the maximum of $\left|\hat{x}\right|$ on $\Delta$ equals its maximum on $\beta$, for every $x \in A$. (Trivially, $\Delta$ is an $A$-boundary).

Prove that the intersection $\partial_{A}$ of all $A$-boundaries is an $A$-boundary. $\partial_{A}$ is called the Shilov boundary of $A$. The terminology is suggested by the maximum modulus property of holomorphic functions. For instance, when $A$ is the disc algebra, then $\partial_{A}$ is the unit circle, which is the topological boundary of $\Delta$, the closed unit disc.

Outline of proof: Show first that there is an $A$-boundary $\beta_{0}$ which is minimal in the sense that no proper subset of $\beta_{0}$ is an $A$-boundary. (Partially order the collection of $A$-boundaries by set inclusion, etc). Then pick $h_{0} \in \beta_{0}$, pick $x_{1}, \ldots , x_{n} \in A$ with $\hat{x}_{i}\left(h_{0}\right) = 0$, and put

Since $\beta_{0}$ is minimal, there exists $x \in A$ with $\left\| \hat{x}\right\|_{\infty } = 1$ and $\left|\hat{x}(h)\right| < 1$ on $\beta_{0}-V$. If $y = x^{m}$ and $m$ is sufficiently large, then $\left|\hat{x}_{i} \hat{y}\right| < 1$ on $\beta_{0}$, for all $i$. Hence $\left\| \hat{x}_{i} \hat{y}\right\|_{\infty } < 1$. Conclude from this first that $\left|\hat{y}(h)\right| = \left\| \hat{y}\right\|_{\infty }$ only in $V$, hence that $V$ intersects every $A$-boundary $\beta$, and finally that $h_{0} \in \beta$. Thus $\beta_{0} \subset \beta$, and $\beta_{0} = \partial_{A}$.

Suppose $A$ is a Banach algebra, $m$ is an integer, $m \geq 2, K < \infty$, and

for every $x \in A$. Show that there exist constants $K_{n} < \infty$, for $n = 1,2,3, \ldots$, such that

(This extends Theorem 11.12).

Suppose $\left\{ \omega_{n}\right\} (-\infty < n < \infty )$ are positive numbers such that $\omega_{0} = 1$ and

for all integers $m$ and $n$. Let $A = A\left\{ \omega_{n}\right\}$ be the set of all complex functions $f$ on the integers for which the norm

is finite. Define multiplication in $A$ by

Let $\Delta$ be the maximal ideal space of $L^{\infty }(m)$, as in Section 11.13. Show that

Let $L^{\infty }(m)$ be as above and prove: If $f_{n} \in L^{\infty }(m)$ and $f_{n} \rightarrow 0$ in the weak topology of $L^{\infty }(m)$, then $\int_{0}^{1}\left|f_{n}\right|^{p} d m \rightarrow 0$ for every $p \in (0, \infty )$. Show, by constructing an example, that the converse is false.

Prove the following partial converse of Theorem 11.31: If $F$ is a bounded linear functional on a $B^{*}$-algebra $A$, and $\left\| F\right\| = F(0) = 1$, then $F$ is positive.

Suggestion: Choose $x \in A, \left\| x\right\| \leq 1$, put $F\left(x x^{*}\right) = \alpha +\beta i$,

for $-\infty < t < \infty$. Use Theorem 11.28 to show that $\sigma \left(x x^{*}\right) \subset [0,1]$ and that therefore

Proceed as in Lemma 5.26.

In $\mathbb {C}^{2}$, let $K_{1}$ consist of all points $\left(e^{i \theta }, e^{-i \theta }\right)$, and let $K_{2}$ consist of the points $\left(e^{i \theta }, e^{i \theta }\right), 0 \leq \theta \leq 2 \pi$. Of these circles, show that $K_{1}$ is polynomially convex but $K_{2}$ is not. How about $K_{3} = \left\{ (\cos \theta , \sin \theta ): 0 \leq \theta \leq \pi \right\}$?

Show that a 3-by-3 matrix $M$ commutes with

Deduce from this that centralizers (see Section 11.21) need not be commutative.

The completion of an inner product space is a Hilbert space. Make this statement more precise, and prove it. (See the proof of Theorem 12.40 for an application).

Suppose $N$ is a positive integer, $\alpha \in \mathcal{C}, \alpha^{N} = 1$, and $\alpha^{2} \neq 1$. Prove that every Hilbert space inner product satisfies the identities

and