C*-Algebras

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.

Prove a converse to the following proposition: if $\mathcal{A}$ is an abelian $C^*$-algebra with maximal ideal space $\Sigma$ and $a \in \mathcal{A}$ is such that $\mathcal{A} = C^*(a)$ (the $C^*$-algebra generated by $a$ and the identity), then the map $\tau : \Sigma \to \sigma (a)$ defined by $\tau (h) = h(a)$ is a homeomorphism. Specifically, show: if $K$ is a compact subset of $\mathbb {C}$, $C(K)$ is a singly generated $C^*$-algebra.

If $\mathcal{A}$ is an abelian $C^*$-algebra with a finite number of $C^*$-generators $a_1, \dots , a_n$, then there is a compact subset $X$ of $\mathbb {C}^n$ and an isometric $*$-isomorphism $\rho : \mathcal{A} \to C(X)$ such that $\rho (a_k) = z_k$, $1 \leq k \leq n$, where $z_k(\lambda_1, \dots , \lambda_n) = \lambda_k$ (see Exercise VII.8.12).

If $X$ is a compact Hausdorff space, show that $X$ is totally disconnected if and only if $C(X)$ is the closed linear span of its projections ($\equiv$ hermitian idempotents).

Using the terminology of the preceding exercise, show that if $(X, \Omega , \mu )$ is a $\sigma$-finite measure space, the maximal ideal space of $L^\infty (X, \Omega , \mu )$ is totally disconnected.

If $\mathcal{A}$ is a $C^*$-algebra with identity and $a = a^*$, show that $\exp (ia) = u$ is unitary. Is the converse true?

Let $X$ be compact and fix a point $x_0$ in $X$. Let

Show that $\mathcal{A}$ is an abelian $C^*$-algebra with identity and find its maximal ideal space.

If $X$ is completely regular, then $C_b(X)$ is a $C^*$-algebra and its maximal ideal space is the Stone--Cech compactification of $X$.

Prove the uniqueness statement in the following proposition for the case that $\mathcal{A}$ is abelian: if $a \in \operatorname {Re} \mathcal{A}$ (the hermitian elements of $\mathcal{A}$), then there are unique positive elements $u, v$ in $\mathcal{A}$ such that $a = u - v$ and $uv = vu = 0$.

Prove: if $a \in \mathcal{A}_+$ and $n \geq 1$, there is a unique element $b$ in $\mathcal{A}_+$ such that $a = b^n$.

Let $A \in \mathcal{B}(L^2(0,1))$ be defined by $(Af)(t) = tf(t)$. Show that $A \geq 0$ and find $A^{1/n}$.

Let $(X, \Omega , \mu )$ be a $\sigma$-finite measure space, let $\phi \in L^\infty (X, \Omega , \mu )$, and define $M_\phi$ as the multiplication operator on $L^2(X, \Omega , \mu )$ given by $M_\phi f = \phi f$. Show that $M_\phi \geq 0$ if and only if $\phi (x) \geq 0$ a.e.$[\mu ]$. What is $M_\phi^{1/n}$? If $M_\phi \in \operatorname {Re} \mathcal{B}(\mathcal{H})$, find the positive and negative parts of $M_\phi$.

Find an example of a positive operator on a Hilbert space that has a nonhermitian square root.

If $a \in \operatorname {Re} \mathcal{A}$, show that $\left|a\right| \equiv (a^2)^{1/2} = a_+ + a_-$, where $a_+$ and $a_-$ are the positive and negative parts of $a$ (that is, the unique positive elements with $a = a_+ - a_-$ and $a_+ a_- = a_- a_+ = 0$).

If $a \in \mathcal{A}_+$, show that $x^* a x \in \mathcal{A}_+$ for every $x$ in $\mathcal{A}$.

If $a, b \in \mathcal{A}$, $0 \leq a \leq b$, and $a$ is invertible, then $b$ is invertible and $b^{-1} \leq a^{-1}$.

If $a, b \in \operatorname {Re} \mathcal{A}$, $a \leq b$, and $ab = ba$, then $f(a) \leq f(b)$ for every increasing continuous function $f$ on $\mathbb {R}$.

If $a \in \operatorname {Re} \mathcal{A}$ and $\left\| a\right\| \leq 1$, show that $a$ is the sum of two unitaries.

If $\alpha > 0$, define $f_\alpha : (-\alpha^{-1}, \infty ) \to \mathbb {R}$ by $f_\alpha (t) = t/(1 + \alpha t) = \alpha^{-1}[1 - (1 + \alpha t)^{-1}]$. Show:

If $a, b \in \mathcal{A}_+$ and $a \leq b$, show that $a^\beta \leq b^\beta$ for $0 \leq \beta \leq 1$. (Here $a^\beta = f(a)$ where $f(t) = t^\beta$.)

Give an example of a $C^*$-algebra $\mathcal{A}$ and positive elements $a, b$ in $\mathcal{A}$ such that $a \leq b$ but $b^2 - a^2 \notin \mathcal{A}_+$.

Let $\mathcal{A} = \mathcal{B}(l^2)$, let $a =$ the unilateral shift on $l^2$, and let $b = a^*$. Show that $\sigma (ab) \neq \sigma (ba)$.

Let $W \in \mathcal{B}(\mathcal{H})$ and show that the following statements are equivalent: (a) $W$ is a partial isometry (that is, an operator such that for $h$ in $(\ker W)^\perp$, $\left\| Wh\right\| = \left\| h\right\|$); (b) $W^*$ is a partial isometry; (c) $W^*W$ is a projection; (d) $WW^*$ is a projection; (e) $WW^*W = W$; (f) $W^*WW^* = W^*$.

If $W$ is a partial isometry, show that $W^*W$ is the projection onto the initial space of $W$ (that is, the space $(\ker W)^\perp$) and $WW^*$ is the projection onto the final space of $W$ (that is, the space $\operatorname {ran} W$).

If $W_1, W_2$ are partial isometries, define $W_1 \precsim W_2$ to mean that $W_1^* W_1 \leq W_2^* W_2$, $W_1 W_1^* \leq W_2 W_2^*$, and $W_2 h = W_1 h$ whenever $h$ is in the initial space of $W_1$. Show that $\precsim$ is a partial ordering on the set of partial isometries and that a partial isometry $W$ is a maximal element in this ordering if and only if either $W$ or $W^*$ is an isometry.

Using the terminology of the preceding exercise, show that the extreme points of ball $\mathcal{B}(\mathcal{H})$ are the maximal partial isometries. (See Exercise V.7.10.)

Find the polar decomposition of each of the following operators:

Let $A \in \mathcal{B}(\mathcal{H})$ such that $\ker A = (0)$ and $A \geq 0$ and define $S$ on $\mathcal{K} = \mathcal{H} \oplus \mathcal{H} \oplus \cdots$ by $S(h_1, h_2, \dots ) = (0, Ah_1, Ah_2, \dots )$. Find the polar decomposition of $S$, $S = W \left|S\right|$, and show that $S = \left|S\right| W$.

Show that the parts of the polar decomposition of a normal operator commute.

If $A \in \mathcal{B}(\mathcal{H})$, show that there is a positive operator $P$ and a partial isometry $W$ such that $A = PW$. Discuss the uniqueness of $P$ and $W$.

If $A$ is normal and invertible, show that the parts of the polar decomposition of $A$ belong to $C^*(A)$.

Give an example of a normal operator $A$ such that the partial isometry in the polar decomposition of $A$ does not belong to $C^*(A)$.

Show that for an arbitrary $C^*$-algebra $\mathcal{A}$ it is not necessarily true that $ab \in \mathcal{A}_+$ whenever $a$ and $b \in \mathcal{A}_+$. If, however, $a, b \in \mathcal{A}_+$ and $ab = ba$, then $ab \in \mathcal{A}_+$.

Complete the proof of the following proposition: if $X$ is compact and $I$ is a closed ideal of $C(X)$, then there is a closed subset $F$ of $X$ such that $I = \{ f \in C(X) : f(x) = 0 \text{ for all } x \text{ in } F\}$. Moreover, $C(X)/I$ is isometrically isomorphic to $C(F)$.

Show that $M_n(\mathbb {C})$ has no nontrivial ideals. Find all of the left ideals.

If $\alpha$ is an infinite cardinal number, let $I_\alpha = \{ A \in \mathcal{B}(\mathcal{H}) : \dim \operatorname {cl}(\operatorname {ran} A) \leq \alpha \}$. Show that $I_\alpha$ is a closed ideal in $\mathcal{B}(\mathcal{H})$.

Let $S$ be the unilateral shift on $l^2$. Show that $C^*(S) \supseteq \mathcal{B}_0(l^2)$ and $C^*(S)/\mathcal{B}_0(l^2)$ is abelian. Show that the maximal ideal space of $C^*(S)/\mathcal{B}_0(l^2)$ is homeomorphic to $\partial \mathbb {D}$.

If $V$ is the Volterra operator on $L^2(0,1)$, show that $C^*(V) = \mathbb {C} + \mathcal{B}_0(L^2(0,1))$.

If $\mathcal{A}$ is a $C^*$-algebra, $I$ is a closed ideal of $\mathcal{A}$, and $\mathcal{B}$ is a $C^*$-subalgebra of $\mathcal{A}$, show that the $C^*$-algebra generated by $I \cup \mathcal{B}$ is $I + \mathcal{B}$.

If $\mathcal{A}$ is a $C^*$-algebra and $I$ and $J$ are closed ideals in $\mathcal{A}$, show that $I + J$ is a closed ideal of $\mathcal{A}$.

Let $\mathcal{A}$ be a $C^*$-algebra with identity and let $\pi : \mathcal{A} \to \mathcal{B}(\mathcal{H})$ be a $*$-homomorphism [but don't assume that $\pi (1) = 1$]. Let $P_1 = \pi (1)$. Show that $P_1$ is a projection and $\mathcal{H}_1 \equiv P_1 \mathcal{H}$ reduces $\pi (\mathcal{A})$. If $\pi_1(a) = \pi (a) | \mathcal{H}_1$, show that $\pi_1: \mathcal{A} \to \mathcal{B}(\mathcal{H}_1)$ is a representation.

Show that the following representation is a cyclic representation and find all of the cyclic vectors: if $(X, \Omega , \mu )$ is a $\sigma$-finite measure space and $\mathcal{H} = L^2(\mu )$, then $\pi : L^\infty (\mu ) \to \mathcal{B}(\mathcal{H})$ defined by $\pi (\phi ) = M_\phi$ (the multiplication operator) is a representation.

Show that the following representation is a cyclic representation and find all the cyclic vectors: if $X$ is a compact space and $\mu$ is a positive Borel measure on $X$, then $\pi : C(X) \to \mathcal{B}(L^2(\mu ))$ defined by $\pi (f) = M_f$ is a representation.

If $X$ is compact and $\mu$ is a positive measure on $X$, let $\pi_\mu : C(X) \to \mathcal{B}(L^2(\mu ))$ be the representation defined by $\pi_\mu (f) = M_f$. If $\mu , \nu$ are positive measures on $X$ show that $\pi_\mu \oplus \pi_\nu$ is cyclic if and only if $\mu \perp \nu$. If $\mu \perp \nu$, then $\pi_\mu \oplus \pi_\nu$ is equivalent to $\pi_{\mu + \nu }$. Also, $\pi_\mu^{(n)}$ is not cyclic if $n \geq 2$.

Verify the statements in the following example. Let $X$ be a compact space and let $\{ \mu_n\}$ be a sequence of measures on $X$. For each $n$ let $\pi_n : C(X) \to \mathcal{B}(L^2(\mu_n))$ be defined by $\pi_n(f) = M_f$ on $L^2(\mu_n)$. Then $\pi = \bigoplus_n \pi_n$ is a representation. If the measures $\{ \mu_n\}$ are pairwise mutually singular, then $\pi$ is equivalent to the representation $f \mapsto M_f$ of $C(X) \to \mathcal{B}(L^2(\mu ))$, where $\mu = \sum_{n=1}^\infty \mu_n / 2^n \left\| \mu_n\right\|$.

If $\mathcal{A} = \mathbb {C} + \mathcal{B}_0(\mathcal{H})$ and $\pi : \mathcal{A} \to \mathcal{B}(\mathcal{H})$ is the identity representation, show that $\pi^{(\alpha )}$ is a cyclic representation.

Fix a Banach limit LIM on $l^\infty (\mathbb {N})$ and let $\mathcal{H}$ be a separable Hilbert space with an orthonormal basis $\{ e_n\}$. Define $f: \mathcal{B}(\mathcal{H}) \to \mathbb {C}$ by $f(T) = \operatorname {LIM}\{ \langle Te_n, e_n \rangle \}$. Show that $f$ is a state on $\mathcal{B}(\mathcal{H})$. If $\pi_f$ is the corresponding cyclic representation, show that $\ker \pi_f = \mathcal{B}_0(\mathcal{H})$. Hence $\pi_f$ induces a cyclic representation of $\mathcal{B}(\mathcal{H})/\mathcal{B}_0(\mathcal{H})$ that is isometric. Is $\mathcal{H}_f$ separable?

If $f$ is a positive linear functional on $\mathcal{A}$ and $\alpha \in (0, \infty )$, show that $\pi_f$ and $\pi_{\alpha f}$ are equivalent representations.

If $a \in \mathcal{A}$, then $a \geq 0$ if and only if $f(a) \geq 0$ for every state $f$.

If $a \in \mathcal{A}$ and $a \neq 0$, then there is a state $f$ on $\mathcal{A}$ such that $f(a) \neq 0$.

If $\mathcal{A}$ is a separable $C^*$-algebra and $\{ f_n\}$ is a countable weak* dense subset of $S_{\mathcal{A}}$ (the state space of $\mathcal{A}$), let $f = \sum_n 2^{-n} f_n$. Show that $\pi_f$ is an isometry.