Section3The Kadison theorem and some applications
We begin by using the tools developed in Section §2 to identify the integer \(a-b\) in Kadison's theorem, that is, to prove Theorem 1.3.
By ((1.2))
\begin{equation*}
a-b = \tr\big( \projtwo{}\projone{}\projtwo{}- \projtwo{} + \projtwo{}^\perp \projone{} \projtwo{}^\perp \big) = \tr\big(\projtwo{}(\projone{}-\projtwo{})\projtwo{}+ \projtwo{}^\perp (\projone{}-\projtwo{})\projtwo{}^\perp\big)
\end{equation*}
and by ((1.1)), \(\projtwo{}(\projone{}-\projtwo{})\projtwo{} \in \mathcal L^1\) and \(\projtwo{}^\perp (\projone{}-\projtwo{}) \projtwo{}^\perp \in \mathcal L^1\text{.}\) Thus by Proposition 2.8, \(\projone{}-\projtwo{}\in \mathcal L^2\) and \([\projone{}:\projtwo{}]= \tr( \projtwo{}(\projone{}-\projtwo{})\projtwo{}+\projtwo{}^\perp(\projone{}-\projtwo{})\projtwo{}^\perp).\)
As a first consequence of Kadison's theorem and of the work in Section §2, we observe that if the diagonal of a projection \(\projone{}\) clusters sufficiently fast around \(0\) and \(1\) (that is, if \(a+b\lt \infty\text{,}\) or, equivalently, if \(\projone{}-\projtwo{} \in \mathcal L^2\)), then one can “read” from the diagonal the essential codimension \([\projone{}:\projtwo{}]\text{.}\) But what if \(a+b= \infty\text{?}\)
If \(a=\infty\) and \(b \lt \infty\text{,}\) from \(\projtwo{}^\perp(\projone{}-\projtwo{})\projtwo{}^\perp \in \mathcal L^1\) we can deduce that \(\projone{} \wedge \projtwo{}^\perp\) is finite and \(s\in \mathcal L^2\text{,}\) and hence from \(\projtwo{}(\projone{}-\projtwo{})\projtwo{}\not \in \mathcal L^1\) it follows that \(\projone{}^\perp \wedge \projtwo{}\) is infinite. Similarly, if \(a \lt \infty\) and \(b=\infty\) then \(\projone{} \wedge \projtwo{}^\perp\) is infinite. In either case \((\projone{},\projtwo{})\) is not a Fredholm pair and in particular, \(\projone{}-\projtwo{}\not \in \K\text{.}\)
Less trivial is the case when \(a=b=\infty\) and \(\projone{}-\projtwo{}\in \K \setminus \mathcal L^2\text{,}\) as we see from the following proposition. We first need to introduce some notation. Next, given two sequences \(\xi\) and \(\eta\) of non-negative numbers converging to \(0\text{,}\) with \(\xi^*\) and \(\eta^*\) their monotone non-increasing rearrangements, we say that \(\xi\) is majorized by \(\eta\) (\(\xi\prec \eta\)) if \(\sum_{j=1}^n\xi^*_j\le \sum_{j=1}^n\eta^*_j\) for all \(n\text{.}\)
Proposition3.1
Suppose \(\projone{},\projtwo{}\) are projections with \(\projone{}-\projtwo{} \in \K \setminus \mathcal{L}^2\text{.}\) Then there exists a projection \(\projone{}'\) such that \(\projone{}'-\projtwo{} \in \K\text{,}\) \([\projone{}':\projtwo{}] \not= [\projone{}:\projtwo{}]\) and there is an orthonormal basis \(\{e_n\}\) that diagonalizes \(\projtwo{}\) such that \(E(\projone{})=E(\projone{}')\text{.}\)
By Proposition 2.7 (i), \(\projone{}-\projtwo{} \in \K\) implies that \(\projone{} \wedge \projtwo{}^\perp\) and \(\projone{}^\perp \wedge \projtwo{}\) are both finite. Thus \(\projone{}_0-\projtwo{}_0 = (\projone{}-\projtwo{}) + (\projone{} \wedge \projtwo{}^\perp - \projone{}^\perp \wedge \projtwo{}) \in \K \setminus \mathcal{L}^2\text{.}\) It suffices to prove the proposition for projections in generic position because then we simply set \(\projone{}' := \projone{} \wedge \projtwo{} + \projone{} \wedge \projtwo{}^\perp + \projone{}_0'\) for the general case. So to simplify notation, assume henceforth that \(\projone{},\projtwo{}\) are in generic position and have the form as in ((2.5)). In particular, \(\projtwo{}\sim \projtwo{}^\perp\sim 1\) and by Lemma 2.5, \([\projone{}:\projtwo{}]=0\text{.}\)
Next, choose a rank one projection \(r'\le \projtwo{}^{\perp}\) and let \(r:=\projtwo{}^{\perp} - r'\text{.}\) After identifying \(B(\h) \simeq B(\projtwo{}\h \oplus r\h \oplus \mathbb{C})\) with \(M_2(\projtwo{}B(\h)\projtwo{}) \oplus \mathbb{C}\) via a partial isometry taking \(\projtwo{}\h \to r\h\text{,}\) consider the projection
\begin{equation*}
\tilde{\projone{}} :=
\begin{pmatrix}
c^2 & cs & 0 \\
cs & s^2 & 0 \\
0 & 0 & 1 \\
\end{pmatrix}.
\end{equation*}
Note that \(\tilde{\projone{}} - (0 \oplus 0 \oplus 1)\) and \(\projtwo{}\) are in generic position relative to their join, and hence their essential codimension is zero. This implies that \([\tilde{\projone{}},\projtwo{}] = 1 \not= 0 = [\projone{},\projtwo{}]\text{.}\)
Now, choose an orthonormal basis \(\{e_n\}\) that diagonalizes \(\projtwo{}\text{,}\) \(E\) its corresponding conditional expectation, and let \(\xi\) be the diagonal sequence of \(s^2\text{.}\) Then under natural notations we have
\begin{equation*}
E(\projone{})= E_\projtwo{}(\projone{})\oplus E_{\projtwo{}^\perp}(\projone{})
\end{equation*}
and furthermore, \(E_\projtwo{}(\projone{})=E_\projtwo{}(\tilde{\projone{}})= E_\projtwo{}(c^2)\) and \(E_{\projtwo{}^\perp}(\projone{})=E_{\projtwo{}}(s^2)= \diag \xi\text{.}\)
Since \(\projone{}-\projtwo{} \in \K \setminus \mathcal{L}^2\text{,}\) by Proposition 2.7 we have that \(s^2 \in \K \setminus \mathcal{L}^1\text{,}\) that is \(\xi\to 0\) but \(\xi\not \in \ell^1\text{.}\) By the Schur–Horn theorem for compact operators [20] \(\xi\) is majorized by the eigenvalues sequence \(\lambda(s^2)\) of the operator \(s^2\) and hence,
\begin{equation*}
\xi\prec \lambda(s^2)\prec \lambda
\begin{pmatrix}
s^2 & 0 \\
0 & 1 \\
\end{pmatrix}.
\end{equation*}
Now \( \begin{pmatrix} s^2 & 0 \\ 0 & 1 \\ \end{pmatrix} \) is a positive compact operator with zero kernel belonging to \(\B(\projtwo{}^\perp \h)\text{.}\) Hence by [20] there is a unitary \(u\in \B(\projtwo{}^\perp \h)\) such that
\begin{equation*}
\diag \xi= E_{\projtwo{}^\perp\h}\big (u
\begin{pmatrix}
s^2 & 0 \\
0 & 1 \\
\end{pmatrix}
u^*\big).
\end{equation*}
Let \(u':= 1\mid _{\projtwo{}\h}\oplus u\) and \(\projone{}':= u'(\tilde \projone{})u'^*\text{.}\) Then
\begin{equation*}
E(\projone{}')= E_\projtwo{}(c^2)\oplus \diag \xi = E(\projone{}).
\end{equation*}
Since \(u'\projtwo{}u'^{*} = \projtwo{}\) we have
\begin{equation*}
[\projone{}':\projtwo{}] = [u\tilde{\projone{}}u^{*}:u\projtwo{}u^{*}] = [\tilde{\projone{}}:\projtwo{}] \not= [\projone{},\projtwo{}]. \qedhere
\end{equation*}
As a second application of Theorem 1.3 and of the techniques used to prove it, we will consider a recent work by Bownik and Jasper [6]. Based on Kadison's characterization of diagonals of projections, Bownik and Jasper characterized the diagonals of selfadjoint operators with finite spectrum and in a key part of their analysis they too encountered an index obstruction similar to the one in Theorem 1.1 (ii). Following their notations, if \(z\in \B(\h)\) is a selfadjoint operator with finite spectrum we let \(\sigma(z)= \{a_j\}_{j=-m}^{n+r}\) and \(\projone{}_j= \chi_{\{a_j\}}(z)\) be the spectral projection corresponding to the eigenvalue \(a_j\text{,}\) so that
\begin{equation*}
z= \sum _{j=-m}^{n+r}a_j \projone{}_j.
\end{equation*}
For ease of notations perform if necessary a transformation so to have
\begin{equation*}
\tr(\projone{}_j)\lt \infty \text{ for }j\lt 0 \text { and } j>n+1, \quad a_0=0, \text{ and } a_{n+1}=1.
\end{equation*}
Let \(\{e_n\}\) be an orthonormal basis, \(\{d_n\}\) be the diagonal of \(z\) with respect to that basis and let as in Theorem 1.1,
\begin{equation*}
a = \sum_{d_n \le 1/2 } d_n
\qquad\text{and}\qquad
b = \sum_{d_n > 1/2 } (1-d_n),
\end{equation*}
Then their Theorem 4.1, which is a key component of the necessity part of their characterization, states that
Theorem3.2[6]
If \(a+b\lt \infty\) then
- \(\tr(\projone{}_j) \lt \infty\) for \(0\lt j\lt n+1\text{;}\)
- \(a-b - \sum_{j\ne n+1} a_j\tr(\projone{}_j) \in \mathbb Z.\)
Here of course we use the convention that \(0\cdot \infty=0\) and so \(a_0\tr(\projone{}_0)=0\) whether \(\tr(\projone{}_0)\) is finite or not.
We will present an independent proof of this result and at the same time identify the integer in (ii) proving that if we set \(\projtwo{}\) as in Theorem 1.1 to be the projection on \(\overline{\spans}\{e_j\mid\ d_j> 1/2 \}\text{,}\) then
\begin{equation}
a-b - \sum_{j\ne n+1} a_j\tr(\projone{}_j)= [\projone{}_{n+1}:\projtwo{}].
\tag{3.1}
\end{equation}
First we need an extension to positive elements of the equivalence of (i) and (ii) in Proposition 2.8.
Lemma3.3
Let \(\mathcal J\) be a proper ideal, \(x\in \B(\h)_+\) a positive contraction, and \(\projtwo{}\in \B(\h)\) a projection.
- If \(\projtwo{}-\projtwo{}x\projtwo{}\in \mathcal J\) and \(\projtwo{}^\perp x \projtwo{}^\perp\in \mathcal J\text{,}\) then \(x-\projtwo{}\in \mathcal J^{ 1/2 }\) and \(x\chi_{[0, \eps]}(x)\in \mathcal J\) for every \(0\lt \eps \lt 1\text{.}\)
- Assume that \(x\) is a projection or that \(\mathcal J\) is idempotent (i.e., \(\mathcal J= \mathcal J^2\)). If \(x-\projtwo{}\in \mathcal J^{ 1/2 }\) and \(x\chi_{[0, \eps]}(x)\in \mathcal J\) for some \(0\lt \eps \lt 1\text{,}\) then \(\projtwo{}-\projtwo{}x\projtwo{}\in \mathcal J\) and \(\projtwo{}^\perp x \projtwo{}^\perp\in \mathcal J\text{.}\)
(i). Since \(\projtwo{}^\perp x \projtwo{} x \projtwo{}^\perp\le \projtwo{}^\perp x^2 \projtwo{}^\perp\le \projtwo{}^\perp x \projtwo{}^\perp\in \mathcal J\text{,}\) it follows that \(\projtwo{} x \projtwo{}^\perp\) and \(\projtwo{}^\perp x \projtwo{}\) belong to \(\mathcal J^{ 1/2 }\text{.}\) But then
\begin{equation*}
x-\projtwo{} = (\projtwo{}x\projtwo{}- \projtwo{}) + \projtwo{}^\perp x \projtwo{}^\perp + \projtwo{} x \projtwo{}^\perp+ \projtwo{}^\perp x \projtwo{}\in \mathcal J^{ 1/2 }.
\end{equation*}
Let \(\eps>0\) and let \(x_\eps:= x\chi_{[0, \eps]}(x)\text{.}\) Then \(0\le \projtwo{}^\perp x_\eps \projtwo{}^\perp\le \projtwo{}^\perp x \projtwo{}^\perp\in \mathcal J\text{,}\) whence \(\projtwo{}^\perp x_\eps \projtwo{}^\perp\in \mathcal J\text{.}\) Furthermore,
\begin{equation*}
1-x\ge (1-x) \chi_{[0, \eps]}(x)\ge (1-\eps) \chi_{[0, \eps]}(x)\ge (1-\eps)x_\eps
\end{equation*}
and hence \(\projtwo{}-\projtwo{}x\projtwo{}\ge (1-\eps)\projtwo{}x_\eps \projtwo{}.\) Thus \(\projtwo{}x_\eps \projtwo{}\in \mathcal J\) and since
\begin{equation*}
0\le x_\eps \le 2\big(\projtwo{}x_\eps \projtwo{} + \projtwo{}^\perp x_\eps \projtwo{}^\perp)\in \mathcal J
\end{equation*}
it follows that \(x_\eps\in \mathcal J\text{.}\)
(ii). The case when \(x\) is a projection is given by ((2.12)). Assume then that \(\mathcal J\) is idempotent. Then \(x-\projtwo{}\in \mathcal J^{ 1/2 }= \mathcal J\) implies \(\projtwo{}-\projtwo{}x\projtwo{}=-\projtwo{}(x-\projtwo{})\projtwo{}\in \mathcal J\text{.}\) Furthermore, \((x-\projtwo{})^2= x^2-x\projtwo{}-\projtwo{}x+\projtwo{}\) hence \(\projtwo{}^\perp x^2 \projtwo{}^\perp= \projtwo{}^\perp (x-\projtwo{})^2 \projtwo{}^\perp\in \mathcal J\text{.}\) Then
\begin{equation*}
\projtwo{}^\perp (x-x_\eps) \projtwo{}^\perp\le \frac{1}{\eps}\projtwo{}^\perp (x-x_\eps)^2 \projtwo{}^\perp\le \frac{1}{\eps}\projtwo{}^\perp x^2 \projtwo{}^\perp \in \mathcal J
\end{equation*}
and hence
\begin{equation*}
\projtwo{}^\perp x \projtwo{}^\perp = \projtwo{}^\perp (x-x_\eps) \projtwo{}^\perp + \projtwo{}^\perp x_\eps \projtwo{}^\perp \in \mathcal J. \qedhere
\end{equation*}
Notice that if \(\mathcal J\) is not idempotent and \(k\in \mathcal J^{ 1/2 }_+\setminus \mathcal J\) is a positive contraction, then \(x:=1-k\) and \(\projtwo{}:=1\) satisfy both hypotheses of Lemma 3.3 (ii) but \(k=\projtwo{}-\projtwo{}x\projtwo{}\not \in \mathcal J\text{.}\)
Now we can proceed with the proof of Theorem 3.2 and ((3.1)).
Set \(x = \sum_{j=1}^{n+1} a_j \projone{}_j\text{.}\) Then \(0 \le x \le 1\) and
\begin{equation*}
z-x = \sum_{j=-m}^{-1}a_j \projone{}_j+\sum_{j=n+2}^{n+r}a_j \projone{}_j \quad\text{has finite rank.}
\end{equation*}
As in ((1.2)) we have that
\begin{equation*}
a= \tr(\projtwo{}^{\perp} x \projtwo{}^{\perp}) + \tr(\projtwo{}^\perp (z-x) \projtwo{}^\perp) = \tr(\projtwo{}^\perp z \projtwo{}^\perp)
\end{equation*}
and
\begin{equation*}
b= \tr(\projtwo{}-\projtwo{}x\projtwo{}) - \tr(\projtwo{}(z-x)\projtwo{}) = \tr( \projtwo{}- \projtwo{}z\projtwo{}),
\end{equation*}
hence \(\projtwo{} (z-\projtwo{}) \projtwo{} \in \mathcal L^1\) , \(\projtwo{}^\perp(z-\projtwo{})\projtwo{}^\perp \in \mathcal L^1\text{,}\) and
\begin{equation}
a-b = \tr\big( \projtwo{}(z-\projtwo{})\projtwo{}+ \projtwo{}^\perp (z-\projtwo{})\projtwo{}^\perp \big).
\tag{3.2}
\end{equation}
We also have \(\projtwo{}(x-\projtwo{})\projtwo{}\in \mathcal L^1\) and \(\projtwo{}^\perp (x-\projtwo{}) \projtwo{}^\perp \in \mathcal L^1\text{,}\) hence by Lemma 3.3, it follows that \(x-\projtwo{}\in \mathcal L^2\) and
\begin{equation*}
\sum_{j=1}^n a_j \projone{}_j= x\chi_{[0, a_n]}(x) \in \mathcal L^1.
\end{equation*}
But then \(x-\projone{}_{n+1}= \sum_{j=1}^n a_j \projone{}_j\) has finite rank and in particular, \(\tr(\projone{}_j)\lt \infty\) for \(0\lt j \lt n+1\text{,}\) thus proving (i). As a consequence,
\begin{equation*}
\projone{}_{n+1}-\projtwo{}= \projone{}_{n+1}-x+ x-\projtwo{}\in \mathcal L^2
\end{equation*}
and hence by Proposition 2.8,
\begin{equation}
[\projone{}_{n+1}:\projtwo{}]= \tr\big(\projtwo{}(\projone{}_{n+1}-\projtwo{})\projtwo{}+ \projtwo{}^\perp(\projone{}_{n+1}-\projtwo{})\projtwo{}^\perp\big).
\tag{3.3}
\end{equation}
Furthermore, \(y:= z-\projone{}_{n+1}= \sum_{j\ne n+1} a_j \projone{}_j \) has finite rank and in particular is in \(\mathcal L^1\text{,}\) so that
\begin{equation}
\sum_{j\ne n+1} a_j\tr(\projone{}_j)= \tr(y)= \tr(\projtwo{}y\projtwo{}+\projtwo{}^\perp y \projtwo{}^\perp).
\tag{3.4}
\end{equation}
Finally from ((3.2)) and ((3.3)),
\begin{align*}
a-b & = \tr\big( \projtwo{}(\projone{}_{n+1}-\projtwo{})\projtwo{}+ \projtwo{}^\perp(\projone{}_{n+1}-\projtwo{})\projtwo{}^\perp +\projtwo{}y\projtwo{}+ \projtwo{}^\perp y \projtwo{}^\perp \big)\\
& =[\projone{}_{n+1}:\projtwo{}]+ \tr(y).
\end{align*}
Thus by ((3.4)), \(a-b- \sum_{j\ne n+1} a_j\tr(\projone{}_j)= [\projone{}_{n+1}:\projtwo{}]\in \mathbb Z\text{.}\)