Section2Essential codimension
¶A fundamental tool we use throughout is the notion of essential codimension due to Brown, Douglas and Fillmore ([5] Remark 4.9). It associates an integer to a pair of projections \(P,Q\) whose difference is compact by means of the Fredholm operator \(QP: P\Hil \to Q\Hil\text{.}\)
Definition2.1
Given a pair of projections \(P,Q\) whose difference is compact, the essential codimension of \(P\) in \(Q\text{,}\) denoted \([P:Q]\text{,}\) is the integer defined by
\begin{equation*}
[P:Q] :=
\begin{cases}
\trace P-\trace Q & \text{if}\ \trace P,\trace Q \lt \infty, \\[0.5em]
\ind(V^{*}W) & \text{if}\ \trace(P) = \trace(Q) = \infty,\ \text{where} \\
& W^{*}W = V^{*}V = I, WW^{*} = P, VV^{*} = Q. \\[0.4em]
\end{cases}
\end{equation*}
Equivalently, essential codimension maybe be defined as
\begin{equation*}
[P:Q] := \ind(QP), \quad\text{where}\ QP : P\Hil \to Q\Hil.
\end{equation*}
Several simple properties of essential codimension which we use are collated here for reference. Proofs can be found in, for example, Proposition 2.2 of [7]. Each property can be derived from standard facts about Fredholm index.
Proposition2.2
Let \(P_1,P_2\) and \(Q_1,Q_2\) each be mutually orthogonal pairs of projections with the property that \(P_j-Q_j\) is compact for \(j=1,2\text{.}\) Suppose also that \(R_1\) is a projection for which \(Q_1-R_1\) is compact. Then
- \([P_1:Q_1] = -[Q_1:P_1]\)
- \([P_1:Q_1] + [P_2:Q_2] = [P_1+P_2:Q_1+Q_2]\)
- \([P_1:R_1] = [P_1:Q_1] + [Q_1:R_1]\)
The original result of Brown, Douglas and Fillmore ([5] Remark 4.9) characterizes when projections can be conjugated by a unitary which is a compact perturbation of the identity. More specifically, they proved that there is a unitary \(U = I+K\) with \(K\) compact which conjugates \(P,Q\) if and only if \(P-Q\) is compact and their essential codimension is zero. The next proposition comes from Proposition 2.7(ii) of [14] and extends the Brown–Douglas–Fillmore result verbatim to an arbitrary proper operator ideal \(\mathcal{J}\text{,}\) where \(\mathcal{J}\) is two-sided but not necessarily norm-closed. Herein, \(\mathcal{J}\) will always denote a proper operator ideal.
Proposition2.3
If \(P,Q\) are projections and \(\mathcal{J}\) is a proper operator ideal, then \(Q = UPU^{*}\) for some unitary \(U = I+K\) with \(K \in \mathcal{J}\) if and only if \(P-Q \in \mathcal{J}\) and \([P:Q] = 0\text{.}\)
The following proposition is a reformulation of Proposition 2.8 of [14] for the case when the ideal is the Hilbert–Schmidt class \(\mathcal{C}_2\text{.}\) This proposition relates the Kadison integer to essential codimension in the following manner. If \(P\) is a projection with diagonal \((d_n)\) and \(a,b\) are as in Theorem 1.1 with \(a+b \lt \infty\text{,}\) then, by choosing \(Q\) to be the projection onto \(\spans \{ e_n \mid d_n \ge \frac{1}{2} \}\text{,}\) Proposition 2.4 guarantees \(P-Q\) is Hilbert–Schmidt (a fact which was known to Arveson) and that \(a-b = [P:Q]\text{.}\)
Proposition2.4
Suppose \(P,Q\) are projections. Then \(P-Q\) is Hilbert–Schmidt if and only if in some (equivalently, every) orthonormal basis \(\{e_n\}_{n=1}^{\infty}\) which diagonalizes \(Q\text{,}\) the diagonal \((d_n)\) of \(P\) satisfies \(a+b \lt \infty\text{,}\) where
\begin{equation*}
a := \sum_{e_n \in Q^{\perp}\Hil} d_n = \trace(Q^{\perp}PQ^{\perp}) \quad\text{and}\quad b := \sum_{e_n \in Q\Hil} (1-d_n) = \trace(Q-QPQ).
\end{equation*}
Whenever \(P-Q\) is Hilbert–Schmidt, \(a-b = [P:Q]\text{.}\)