Essential codimension

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, $a-b = [P:Q]\text{.}$