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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.

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 \(n\text{.}\)

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

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.

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)).