The Positive Elements in a C*-Algebra

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 $ab \in \mathcal{A}_+$.