Subsection2.1Fundamental results
In this paper \(\h\) denotes a separable infinite dimensional complex Hilbert space and \(\K\) the \(C^{\ast}\)-algebra of compact operators on \(\h\text{.}\)
The notion of essential codimension of two projections was first introduced in ([8]).
Definition2.1[8]
Given projections \(\projone{},\projtwo{} \in\B(\h)\) for which \(\projone{}-\projtwo{} \in \K(\h)\text{,}\) the essential codimension \([\projone{}:\projtwo{}]\) of \(\projone{}\) and \(\projtwo{}\) is defined by:
\begin{equation*}
[\projone{}:\projtwo{}] :=
\begin{cases}
\tr(\projone{})-\tr(\projtwo{}) & \tr(\projone{}) \lt \infty, \tr(\projtwo{}) \lt \infty \\
\idx(v^{*}w) & \tr(\projone{})= \tr(\projtwo{})= \infty, w,\,v\ \text{isometries},\ ww^*= \projone{}, vv^*=\projtwo{}. \\
\end{cases}
\end{equation*}
This definition depends on the fact that \((v^*w)^*(v^*w)= 1+ w^*(\projtwo{}-\projone{})w\) and similarly for \(w^*v\text{.}\) Thus setting \(\pi\) to be the projection onto the Calkin algebra, we see that \(\pi(v^*w)\) is unitary and hence \(v^*w\) is Fredholm. Also, if \(\tilde w \) and \(\tilde v\) are another pair of isometries with ranges \(\projone{}\) and \(\projtwo{}\) respectively, then
\begin{equation*}
\idx(v^*w)= \idx( v^*\tilde v \tilde v^* \tilde w \tilde w^* w)= \idx(\tilde v^*\tilde w)
\end{equation*}
since \(w^*\tilde w\) and \(v^*\tilde v\) are unitaries. This shows that \([\projone{}:\projtwo{}]\) does not depend on the choice of the isometries \(w\) and \(v\text{.}\)
Some properties of the essential codimension were presented without proof in [7] and a complete exposition can be found in [9], together with an interesting application to liftability of projections in the corona algebra of, among others, \(C([0,1])\otimes \K\text{.}\)
Independently, and without reference to essential codimension, Avron, Seiler, and Simon defined in [4] the more general notion of Fredholm pairs of projections.
Definition2.2[4]
A pair of projections \((\projone{}, \projtwo{})\) in \(\B(\h)\) is said to be Fredholm if \(\projtwo{}\mid_{\projone{}\h}\) is a Fredholm operator as an element of \(\B(\projone{}\h, \projtwo{}\h)\text{.}\) The index of the pair is defined to be \(\idx(\projtwo{}\mid_{\projone{}\h})\text{.}\)
Notice that if \(v\in \B(\h_4,\h_3)\) and \(w\in \B(\h_1,\h_2)\) are unitaries, then
\begin{equation}
g\in \B(\h_2,\h_3) \ \text { is Fredholm } \ \Leftrightarrow \ v^*gw\in \B(\h_1, \h_4) \ \text{ is Fredholm.}
\tag{2.1}
\end{equation}
and then
\begin{equation}
\idx(v^*gw)= \idx g.
\tag{2.2}
\end{equation}
Thus if \(w\) and \(v\) are isometries with ranges \(\projone{}\) and \(\projtwo{}\) respectively, then
\begin{align}
\projtwo{}\mid_{\projone{}\h}\in \B(\projone{}\h, \projtwo{}\h)\text { is Fredholm }
& \ \Leftrightarrow \ v^*w = v^*\mid_{\projone{}\h}w\in \B(\h) \text { is Fredholm}\tag{2.3}\\
& \ \Leftrightarrow \ v^*\mid_{\projone{}\h}\in \B(\projone{}\h, \h) \text { is Fredholm}\tag{2.4}
\end{align}
Recall that \(v^{*}w\) is Fredholm if and only if \(\pi(v^{*}w)\) is invertible. We have seen above that if \(\projone{}-\projtwo{}\in \K\text{,}\) then \(\pi(v^{*}w)\) is unitary and hence \(v^*w\) is Fredholm, that is \((\projone{},\projtwo{})\) is a Fredholm pair and by ((2.2)), \([\projone{}:\projtwo{}]= \idx(\projtwo{}\mid_{\projone{}\h})\text{.}\) For consistency we will henceforth write \([\projone{}:\projtwo{}]:=\idx(\projtwo{}\mid_{\projone{}\h})\) whenever \((\projone{},\projtwo{})\) is a Fredholm pair even when \(\projone{}-\projtwo{}\not \in \K\text{.}\)
Soon after [4], W. Amrein, K. Sinha [1] realized that the proofs in [4] could be considerably simplified by reducing to the case of projections in generic position. This notion was first introduced by Dixmier [14] (he called them in “position p”) and independently by Krein, Kranosleskii and Milman [22], and further studied by Davis [13], Halmos[16] (he called them “generic pairs”), and others.
Definition2.3
Two projections \(\projone{}, \projtwo{}\in \B(\h)\) are said to be in generic position if
\begin{equation*}
\projone{} \wedge \projtwo{} = \projone{} \wedge \projtwo{}^{\perp} = \projone{}^{\perp} \wedge \projtwo{} = \projone{}^{\perp} \wedge \projtwo{}^{\perp}=0.
\end{equation*}
When just the first three projections are zero, the pair \(\projone{},\projtwo{}\) is in generic position in \(\B((\projone{}\vee \projtwo{})\h)\) and when there is no risk of confusion we will simply call them in generic position. For the readers' convenience we will collect here some results on projections in generic position. Good references can be found in the texts of Strătilă [23], Takesaki [24] and in the article of Amrein and Sinha [1].
Theorem2.4
Let \(\projone{}, \projtwo{}\in \B(\h)\) be two projections.
- Then the projections \(\projone{}_0:= \projone{}- \projone{}\wedge \projtwo{}- \projone{}\wedge \projtwo{}^\perp\) and \(\projtwo{}_0:= \projtwo{} - \projone{}\wedge \projtwo{}- \projone{}^\perp\wedge \projtwo{}\) are in generic position (in the Hilbert space \((\projone{}_0\vee \projtwo{}_0)\h\text{.}\))
Suppose \(\projone{}, \projtwo{}\) are in generic position and let \(N:=\{\projone{},\projtwo{}\}''\) be the von Neumann algebra generated by them. Then
- \(N_\projtwo{} (=(\projtwo{}N\projtwo{}\mid_{\projtwo{}\h})\) is masa of \(N\) and \(N\) can be identified with \(M_2(N_\projtwo{})\text{.}\)
- There are positive injective contractions \(c\) and \(s\) in \(N_\projtwo{}\) with \(c^2+s^2=1\) (the identity operator of \(N_\projtwo{}\)) such that \(\projone{}\) and \(\projtwo{}\) can be identified with
\begin{equation}
\projtwo{} =
\begin{pmatrix}
1 & 0 \\
0 & 0 \\
\end{pmatrix}
\quad\text{and}\quad
\projone{} =
\begin{pmatrix}
c^2 & cs \\
cs & s^2 \\
\end{pmatrix}.
\tag{2.5}
\end{equation}
- \(c=(\projtwo{}\projone{}\projtwo{})^{ 1/2 }\mid_{\projtwo{}\h}\text{,}\) \(s=(\projtwo{}\projone{}^\perp \projtwo{})^{ 1/2 }\mid_{\projtwo{}\h}\text{,}\) and \(\|s\|=\|\projone{}-\projtwo{}\|\text{.}\)
In this section we will often use the representation ((2.5)) without further reference. Notice that for projections in generic position, the equality \(\|s\|=\|\projone{}-\projtwo{}\|\) ([23], see also [14] and [24]) follows also from the identity:
\begin{equation}
\projone{}-\projtwo{}=
\begin{pmatrix}
-s^2 & cs \\
cs & s^2 \\
\end{pmatrix}
=
\begin{pmatrix}
-s & c \\
c & s \\
\end{pmatrix}
\begin{pmatrix}
s & 0 \\
0 & s \\
\end{pmatrix}
=
\begin{pmatrix}
s & 0 \\
0 & s \\
\end{pmatrix}
\begin{pmatrix}
-s & c \\
c & s \\
\end{pmatrix}
\tag{2.6}
\end{equation}
Notice also that if we set \( v: = \frac{1}{\sqrt{2}} \begin{pmatrix} (1-s)^{ 1/2 } & (1+s)^{ 1/2 } \\ (1+s)^{ 1/2 } & -(1-s)^{ 1/2 } \\ \end{pmatrix} \), then \(v=v^*\) is unitary and
\begin{equation}
\projone{}-\projtwo{}= v
\begin{pmatrix}
s & 0 \\
0 & -s \\
\end{pmatrix}
v^*.
\tag{2.7}
\end{equation}
It is well known that projections in generic position are unitarily equivalent in \(N\text{,}\) and over the years various authors (e.g., [4], [9], [14], [21]) have constructed different unitaries in \(N\) implementing the equivalence. We will use the following unitary:
\begin{equation}
\projone{}=u\projtwo{}u^*\quad \text{for the unitary }u:=
\begin{pmatrix}
c & -s \\
s & c \\
\end{pmatrix}.
\tag{2.8}
\end{equation}
As shown by Amrein and Sinha in [1], reduction to generic position makes the analysis of Fredholm pairs simpler and more transparent. For the convenience of the readers, we will provide here a short self-contained presentation of the main results on Fredholm pairs that we will need in the sequel, completing and generalizing results obtained in [1], [3], [4], [9]. The starting point is the analysis of the case when the projections are in generic position. Recall that Fredholm operators are characterized by being invertible modulo the compact operators.
Lemma2.5
Let \(\projone{},\projtwo{}\) be projections in generic position. Then the following are equivalent.
- \(\projone{}\mid_{\projtwo{}\h}\) is a Fredholm operator
- \(c\) is invertible
- \(\projone{}\mid_{\projtwo{}\h}\) is invertible
- \(\|\projone{}-\projtwo{}\|\lt 1\)
- \(\|\projone{}-\projtwo{}\|_{ess}\lt 1\text{.}\)
If these conditions are satisfied, then \(\idx(\projone{}\mid_{\projtwo{}\h})=0\text{.}\)
(i)\(\ \Rightarrow\ \)(ii). Since \(|\projone{}\mid_{\projtwo{}\h}|=c\) is injective, and since \(\projone{}\mid_{\projtwo{}\h}\) has closed range, by the Inverse Mapping Theorem \(\projone{}\mid_{\projtwo{}\h}\) is invertible and hence so is \(c\text{.}\)
(ii)\(\ \Rightarrow\ \)(iii). Immediate since \(\projone{}\mid_{\projtwo{}\h}= \begin{pmatrix}c^2\\cs\end{pmatrix} \) and \((1, c^{-1}s)= \begin{pmatrix}c^2\\cs\end{pmatrix}^{-1}\text{.}\)
(iii) \(\ \Rightarrow\ \) (i). Obvious.
(ii)\(\ \Leftrightarrow\ \)(iv). Immediate since \(\|\projone{}-\projtwo{}\|=\|s\|\lt 1\) and \(c\) is invertible iff \(c^2\ge \delta 1\) for some \(\delta>0\) iff \(s^2\le (1-\delta)1\) for some \(\delta>0\text{.}\)
(iv)\(\ \Leftrightarrow\ \)(v). By ((2.7)), \(\|\projone{}-\projtwo{}\|_{ess}= \|s\|\) and since \(s\) is positive and injective this implies that \(\|\projone{}-\projtwo{}\|=\|s\|\lt1\text{.}\) The other direction is trivial.
The equivalence of (i) and (iv) and the fact that then the index is zero were obtained in [1]. Using this lemma it is now easy to obtain a characterization of Fredholm pairs also when the projections are not in generic position.
Proposition2.6
Let \(\projone{}, \projtwo{}\) be projections in \(\B(\h)\text{.}\) Then \((\projone{},\projtwo{})\) is a Fredholm pair if and only if \(\|\projone{}-\projtwo{}\|_{ess} \lt 1\) and then \(\projone{}\wedge \projtwo{}^\perp\) and \(\projone{}^{\perp} \wedge \projtwo{}\) are finite and
\begin{equation*}
[\projone{}:\projtwo{}]= \tr(\projone{}\wedge \projtwo{}^\perp)-\tr(\projone{}^{\perp} \wedge \projtwo{}).
\end{equation*}
In the notations of Theorem 2.4 and ((2.6)) and ((2.7)) we have
\begin{equation*}
\projone{}= (\projone{}\wedge \projtwo{} + \projone{}\wedge \projtwo{}^\perp ) \oplus \projone{}_0\text{ and }
\projtwo{}= (\projone{}\wedge \projtwo{}+ \projone{}^{\perp} \wedge \projtwo{}) \oplus \projtwo{}_0
\end{equation*}
and hence
\begin{equation}
\projone{}-\projtwo{}= (\projone{}\wedge \projtwo{}^\perp- \projone{}^{\perp} \wedge \projtwo{}) \oplus (\projone{}_0-\projtwo{}_0).
\tag{2.9}
\end{equation}
As in [1], it is easy to see that \((\projone{},\projtwo{})\) is a Fredholm pair if and only if both \(\big(\projone{}\wedge \projtwo{}+ \projone{}\wedge \projtwo{}^\perp, \projone{}\wedge \projtwo{}+\projone{}^{\perp}\wedge \projtwo{}\big)\) and \((\projone{}_0,\projtwo{}_0)\) are Fredholm pairs, and then the index of \((\projone{},\projtwo{})\) is the sum of the indices of the other two pairs. It is obvious that the first pair is Fredholm if and only if both \(\projone{} \wedge \projtwo{}^{\perp}\) and \(\projone{}^{\perp}\wedge \projtwo{}\) are finite and then its index is \(\tr(\projone{} \wedge \projtwo{}^{\perp})-\tr(\projone{}^{\perp}\wedge \projtwo{})\text{.}\) By Lemma 2.5, the pair \((\projone{}_0,\projtwo{}_0)\) is Fredholm if and only if \(\|\projone{}_0-\projtwo{}_0\|_{ess}\lt 1\) and then its index is zero. Thus to conclude the proof one just notices that \(\projone{}-\projtwo{}\) is a finite rank perturbation of \(\projone{}_0-\projtwo{}_0\) and hence it has the same essential norm.
The implication that if \((\projone{},\projtwo{})\) is a Fredholm pair then \(\projone{} \wedge \projtwo{}^{\perp}\) and \(\projone{}^\perp \wedge \projtwo{}\) are finite and the formula for the index of the pair was obtained in [1]. The necessity and sufficiency of the condition \(\|\projone{}-\projtwo{}\|_{ess} \lt 1\) (albeit not expressed in terms of essential norm) was only implicit in [4], and was obtained explicitly and with more generality in [5].
We consider now the cases when the difference \(\projone{}-\projtwo{}\) belongs to some proper operator ideal \(\mathcal J\text{.}\)
Proposition2.7
Let \(\mathcal J\) be an operator ideal and \(\projone{},\projtwo{} \in B(\h)\) be projections. Then
- \(\projone{}-\projtwo{} \in \mathcal J\) if and only if \(\projone{}^\perp \wedge \projtwo{}\) and \(\projone{} \wedge \projtwo{}^\perp\) are finite and \(s \in \mathcal J\) (where \( \projone{}_0:= \projone{}- \projone{}\wedge \projtwo{}- \projone{} \wedge \projtwo{}^\perp = \begin{pmatrix} c^2 & cs \\ cs & s^2 \\ \end{pmatrix} \) as by ((2.5))).
- If \(\projone{}-\projtwo{}\in \mathcal J\text{,}\) then \([\projone{}:\projtwo{}]=0\) if and only if there is a unitary \(u\in 1+\mathcal J\) such that \(u\projtwo{}u^*=\projone{}\text{.}\)
(i). By ((2.9)) we see that \(\projone{}-\projtwo{}\in \mathcal J\) if and only if \(\projone{} \wedge \projtwo{}^\perp \in \mathcal J\text{,}\) \(\projone{}^\perp \wedge \projtwo{}\in \mathcal J\) and \(\projone{}_0-\projtwo{}_0\in \mathcal J\text{.}\) By ((2.7)), the last condition holds if and only if \(s \in \mathcal J\text{,}\) and the conclusion then follows from the fact that a projection belongs to a proper ideal if and only if it is finite.
(ii). First notice that from part (i) and Proposition 2.6 it follows that
\begin{equation}
\projone{}-\projtwo{}\in \mathcal J \text{ and } [\projone{}:\projtwo{}]=0\ \Leftrightarrow\ \tr (\projone{} \wedge \projtwo{}^\perp) = \tr (\projone{}^\perp \wedge \projtwo{})\lt \infty\text{ and }s\in \mathcal J.
\tag{2.10}
\end{equation}
Therefore if there is a unitary in \(1+ \mathcal J\) such that \(u\projtwo{}u^*=\projone{}\) , then it is immediate to see that \(\projone{}-\projtwo{}\in \mathcal J\) and \([\projone{}:\projtwo{}]=0\text{.}\)
Conversely, assume that \(\projone{}-\projtwo{}\in \mathcal J\) and \([\projone{}:\projtwo{}]=0\text{.}\) Then by ((2.10)) \(s\in \mathcal J\text{,}\) \(\projone{}^\perp \wedge \projtwo{} + \projone{} \wedge \projtwo{}^\perp\) is finite, and \(\tr (\projone{} \wedge \projtwo{}^\perp) = \tr (\projone{}^\perp \wedge \projtwo{})\) and hence \(\projone{} \wedge \projtwo{}^\perp \sim \projone{}^\perp \wedge \projtwo{}\text{.}\) Then there is a unitary \(u_1\) on \(\big(\projone{} \wedge \projtwo{}^\perp + \projone{}^\perp \wedge \projtwo{})\h\) such that \(u_1(\projone{}^\perp \wedge \projtwo{}) u_1^*= \projone{} \wedge \projtwo{}^\perp\text{.}\)
Let \(u_2:=\begin{pmatrix}c&-s\\s&c\end{pmatrix}\text{.}\) Then by ((2.8)), \(u_2\) is a unitary on \((\projone{}_0\vee \projtwo{}_0)\h)\) and \(\projone{}_0=u_2\projtwo{}_0u_2^*\text{.}\) Furthermore,
\begin{equation*}
u_2-1\mid_{(\projone{}_0\vee \projtwo{}_0)\h} = \begin{pmatrix}c-1&-s\\s&c-1\end{pmatrix}\in \mathcal J
\end{equation*}
because
\begin{equation*}
c-1= -s^2\big(c+1\big)^{-1}\in \mathcal J^2\subset \mathcal J.
\end{equation*}
Let \(u_3\) be the identity on \(\big(\projone{} \wedge \projtwo{}^\perp + \projone{}^\perp \wedge \projtwo{} + \projone{}_0\vee \projtwo{}_0\big)^\perp\text{.}\) Then \(u:= u_1\oplus u_2\oplus u_3\) is unitary and \(u\projtwo{}u^*=\projone{}\text{.}\) Since \(u_1\) has finite rank, we may conclude that \(u-1\in \mathcal J\text{.}\)
Property (ii) in the above proposition was obtained for \(\mathcal J= \K\) in [9].
Using Proposition 2.7 (i) we obtain an independent proof of the first part of Theorem 1.2 which extends it to arbitrary proper operator ideals, and establishes the sufficiency of the conditions listed.
Proposition2.8
Let \(\mathcal J\) be a proper ideal of \(\B(\h)\) and let \(\projone{},\projtwo{}\) be projections in \(\B(\h)\text{.}\) Then the following are equivalent:
- \(\projone{}-\projtwo{}\in \mathcal J\)
- \(\projtwo{}(\projone{}-\projtwo{})\projtwo{}\in \mathcal J^2\) and \(\projtwo{}^\perp(\projone{}-\projtwo{})\projtwo{}^\perp\in \mathcal J^2\)
- The projections \(\projone{} \wedge \projtwo{}^\perp\) and \(\projone{}^\perp \wedge \projtwo{}\) are finite and at least one the conditions \(\projtwo{}(\projone{}-\projtwo{})\projtwo{}\in \mathcal J^2\) and \(\projtwo{}^\perp(\projone{}-\projtwo{})\projtwo{}^\perp\in \mathcal J^2\) holds.
Furthermore if \(\projone{}-\projtwo{}\in \mathcal L^2\text{,}\) then
\begin{equation*}
[\projone{}:\projtwo{}]= \tr( \projtwo{}(\projone{}-\projtwo{})\projtwo{}+\projtwo{}^\perp(\projone{}-\projtwo{})\projtwo{}^\perp).
\end{equation*}
From ((2.9)) we see that
\begin{equation}
\projtwo{}(\projone{}-\projtwo{})\projtwo{}=-\projone{}^\perp \wedge \projtwo{} \oplus
\begin{pmatrix}
-s^2 & 0 \\
0 & 0 \\
\end{pmatrix}
\text{ and } \projtwo{}^\perp(\projone{}-\projtwo{})\projtwo{}^\perp=\projone{} \wedge \projtwo{}^\perp \oplus
\begin{pmatrix}
0&0\\
0&s^2\\
\end{pmatrix}.
\tag{2.11}
\end{equation}
Thus \(\projtwo{}(\projone{}-\projtwo{})\projtwo{}\in \mathcal J^2\) (resp., \(\projtwo{}^\perp(\projone{}-\projtwo{})\projtwo{}^\perp\in \mathcal J^2\)) if and only if \(\projone{}^\perp \wedge \projtwo{}\) is finite and \(s\in \mathcal J\) (resp., \(\projone{} \wedge \projtwo{}^\perp\) is finite and \(s\in \mathcal J\)). By Proposition 2.7 (i) it is now obvious that the conditions (i)-(iii) are equivalent.
Assume now that \(\projone{}-\projtwo{}\in \mathcal L^2\text{,}\) then by the implication (i) \(\Rightarrow \) (ii) we see that \(\projtwo{}(\projone{}-\projtwo{})\projtwo{}\in \mathcal L^1\) and \(\projtwo{}^\perp(\projone{}-\projtwo{})\projtwo{}^\perp\in \mathcal L^1\text{.}\) Then by ((2.11))
\begin{equation*}
\tr( \projtwo{}(\projone{}-\projtwo{})\projtwo{}+\projtwo{}^\perp(\projone{}-\projtwo{})\projtwo{}^\perp)= \tr (\projone{} \wedge \projtwo{}^\perp) - \tr (\projone{}^\perp \wedge \projtwo{})
\end{equation*}
because
\begin{equation*}
\tr \begin{pmatrix}s^2&0\\0&0\end{pmatrix}= \tr \begin{pmatrix}0&0\\0&s^2\end{pmatrix}\lt \infty
\end{equation*}
as \(s^2\in \mathcal L^1\text{.}\) The conclusion then follows from Proposition 2.6.
The equivalence of just (i) and (ii) follows immediately from the identity
\begin{equation*}
(\projone{}-\projtwo{})^2= - \projtwo{}(\projone{}-\projtwo{})\projtwo{} + \projtwo{}^\perp (\projone{}-\projtwo{})\projtwo{}^\perp
\end{equation*}
whence
\begin{equation}
((\projone{}-\projtwo{})_{+})^2=\projtwo{}^{\perp}(\projone{}-\projtwo{})\projtwo{}^{\perp}\quad\text{and}\quad ((\projone{}-\projtwo{})_{-})^2-=-\projtwo{}(\projone{}-\projtwo{})\projtwo{}.
\tag{2.12}
\end{equation}
To conclude this survey, we observe that every Fredholm operator can be associated in a natural way to a Fredholm pair of projections \((\projone{},\projtwo{})\) so that the index of the operator equals the index of the pair. To this end, consider any Fredholm operator \(x : \h \to \mathcal{K}\) and scale \(x\) to have norm 1. After choosing an arbitrary infinite, co-infinite projection \(\projtwo{}\) and identifying \(\mathcal{K}\) with \(\projtwo{}\h\text{,}\) we have the following proposition.
Proposition2.9
Suppose \(x : \h \to \projtwo{}\h\) is a contraction with \(\projtwo{}\) an infinite, co-infinite projection. Then \(x\) can be completed to an isometry \(w: \h \to \h\) (i.e., \(x = \projtwo{}w\)), and for any such completion, if \(x\) is Fredholm then \((\projone{} := ww^{*}, \projtwo{})\) is a Fredholm pair with \([\projone{}:\projtwo{}] = \idx x\text{.}\)
Consider the defect operator \((1-x^{*}x)^{ 1/2 }\) and a partial isometry \(v\) taking \(R_{1-x^{*}x}\) to \(vv^{*} \le \projtwo{}^{\perp}\text{.}\) Then define \(w = x + v(1-x^{*}x)^{ 1/2 }\text{.}\) Since \(v^{*}x = 0 = x^{*}v\text{,}\) a simple computation shows \(w^{*}w = 1\text{,}\) which establishes that \(x\) can be completed to an isometry.
Now suppose \(w\) is any such completion, and hence \(x = \projtwo{}w\text{.}\) Define \(\projone{} := ww^{*}\) and notice \(x = \projtwo{}\projone{}w = (\projtwo{}\mid_{\projone{}\h})w\) when the operators are viewed on the appropriate spaces. Then from ((2.1)) and ((2.2)), if \(x\) is Fredholm, so is \(\projtwo{}\mid_{\projone{}\h}\) and
\begin{equation*}
\idx x = \idx(\projtwo{}\mid_{\projone{}\h})= [\projone{}:\projtwo{}]. \qedhere
\end{equation*}
Subsection2.2Breuer Fredholm
As mentioned in the introduction, the essential codimension/relative index of projections has found its main application in the study of spectral flows in Fredholm modules. However, in many cases of interest the Fredholm modules are with respect to a semifinite von Neumann algebra (see [5], [10], [12], [11]). The following short summary may be of interest to the reader.
Let \(M\) be a semifinite von Neumann algebra with separable predual (but not necessarily a factor), \(\tau\) a faithful semifinite normal trace, and let \(\mathcal{J}_\tau(M)\) the ideal of \(\tau\)-compact operators
\begin{equation*}
\mathcal{J}_\tau(M):=\overline{ \spans \{x\in \M_+\mid \tau(x)\lt \infty\}} \quad \text {(norm closure).}
\end{equation*}
Let \(\pi: M\to M/ \mathcal{J}_\tau(M)\) be the canonical quotient map and let \(\|x\|_{ess}:=\|\pi(x)\|\) be the essential norm. Then and element \(x\in M\) is called \(\tau\)-Breuer Fredholm (also called just \(\tau\)-Fredholm) if \(\pi(x)\) is invertible. A necessary and sufficient condition is that \(\tau(N_x)\lt \infty\) (where \(N_x\) is the projection on the kernel of \(x\)) and that there exists a projection \(e\in M\) with \(\tau(e)\lt \infty\) such that \((1-e)\h \subset x\h\text{.}\) Then the index is defined as
\begin{equation*}
\idx(x)= \tau(N_x)-\tau(N_{x^*})\in \mathbb R
\end{equation*}
and satisfies the expected properties of an index, but of course it is no longer integer valued.
The original definition by Breuer was given in terms of the ideal \(\mathcal{J} (M)\) of compact operators on \(M\) which received considerable attention over the years,
\begin{equation*}
\mathcal{J} (M):=\overline{ \spans \{x\in \M_+\mid R_x \text { is finite}\}} \quad \text {(norm closure).}
\end{equation*}
When \(M\) is a factor and hence has a unique trace (up to normalization), the notions of \(\tau\)-Breuer–Fredholm and Breuer–Fredholm coincides, but for global algebras they do not and so their theory had to be partially re-derived in [5].
With these definitions almost all of the results listed here for \(\B(\h)\) hold with the same statements and mostly with the same proofs. So we will briefly list here only the properties that fail or that require a different proof.
Proposition 2.6 holds with the same statements and a natural modification of the proof of [4] in the case that \(M\) is a factor, but required more work for the general case [5] less the trace condition which is only relevant when \(M\) is a factor.
It is still true that if \(\projone{},\projtwo{}\in M\) are in generic position and form a Fredholm pair then \([\projone{}:\projtwo{}]=0\text{,}\) but contrary to Lemma 2.5, we can have \(\|\projone{}-\projtwo{}\|=1\) and \(g\) and \(c\) are only invertible modulo \(\mathcal{J}_\tau(M)\) as the following example shows.
Let \(M\) be a type II\(_\infty\) factor. Let \(\projtwo{}\sim \projtwo{}^\perp\sim1\) be a projection in \(M\) which we decompose into a sum \(\sum _1^\infty \projtwo{}_n=\projtwo{}\) of mutually orthogonal finite projections such that \(\tau\big(\sum _1^\infty \projtwo{}_{2n}\big) \lt \infty\text{.}\) Let
\begin{align*}
c & := \sum_{1}^\infty (\sqrt{\frac{1}{2n}}\projtwo{}_{2n}+ \sum_{0}^\infty \sqrt{1-\frac{1}{2n+1}}\projtwo{}_{2n+1}\\
s & := \sum_{1}^\infty \sqrt{1-\frac{1}{2n}}\projtwo{}_{2n}+ \sum_{0}^\infty \sqrt{\frac{1}{2n+1}}\projtwo{}_{2n+1}\\
\projtwo{} & =
\begin{pmatrix}
1 & 0 \\
0 & 0 \\
\end{pmatrix}
\quad\text{and}\quad
\projone{} =
\begin{pmatrix}
c^2 & cs \\
cs & s^2 \\
\end{pmatrix}.
\end{align*}
Then \(\projtwo{}\) and \(\projone{}\) are in generic position and \(\|\projone{}-\projtwo{}\|= \|s\|=1\) while \(s\in \mathcal{J}(M)\text{,}\) hence \(\projone{}-\projtwo{}\in \mathcal{J}(M)\text{,}\) and \((\projone{},\projtwo{})\) is a Fredholm pair with respect to \(M\text{.}\)
Proposition 2.7 (ii) holds without any changes if \(M\) is a factor, but does not hold for global algebras. Consider for instance \(M:=\B(\h_1)\oplus \B(\h_2)\) and \(\tau:=\tr\oplus \tr\text{.}\) Let \(\projtwo{}, \projone{}\) be rank one projections in \(\B(\h_1)\) and \(\B(\h_2)\) respectively. Then \(\projtwo{}\) and \(\projone{}\) are not equivalent (with respect to \(M\)) and hence a fortiori they are not unitarily equivalent. On the other hand \(\projone{}-\projtwo{} \in \mathcal{J}(M)=\mathcal K(\h_1)\oplus \mathcal K(\h_2)\text{,}\) hence \(\projone{},\projtwo{}\) is a Fredholm pair and furthermore \([\projone{}:\projtwo{}] = \tr(\projone{})-\tr(\projtwo{})=0\text{.}\)