\magnification = \magstep 1
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\def \B {{\cal B}}
\def \H {{\cal H}}
\def \K {{\cal K}}
\def \M {{\cal M}}
\def \N {{\cal N}}
\def \T {{\cal T}}
\def \U {{\cal U}}

\centerline{\bf A Note on the Similarity Theorem for Nest Algebras}
\medbreak

\centerline{\bf Kenneth~R.~Davidson}
\bigbreak

\centerline{Not for publication.} 
\bigbreak
 
In {\bf [4]}, nests were classified up to similarity simply in terms of
order type and dimension.  However the proof given there and exposited
in {\bf [6]} appears to rely on first classifying the compact
perturbations of the nest algebras.  This, in turn, required reliance
on some difficult results on derivations of C*-algebras and in
particular on the theorem of Johnson and Parrott {\bf [7]}.  However a
careful examination of the proofs shows that this circuitous route can
be avoided.  This note is a study guide to allow the reader to wend his
or her way through those necessary parts of the proof and avoid the
unneeded results on quasitriangular operators if they choose.

Historically, Andersen {\bf [1]} showed that any two continuous nests
were approximately unitarily equivalent; and used this to show that
their quasi-triangular operator algebras were unitarily equivalent. 
Larson {\bf [8]} used this result to show that any two continuous nests
were similar (although he was not able to control the induced order
isomorphism).  Then the author {\bf [4]} classified all nests up to
similarity, but took a route that passed first through a classification
of quasitriangular operators.  This mindset persisted in the treatment
in our text {\bf [6]}.

A {\it nest} $\N$ is a complete chain of subspaces of a Hilbert space
containing $\{0\}$ and $\H$.  For this note, all Hilbert spaces will be
separable.  The nest algebra $\T(\N)$ consists of all operators leaving
each element of $\N$ invariant---the set of operators with this
prescribed {\it upper triangular form}.  We let $P_N$ denote the
orthogonal projection of $\H$ onto $N$.  Then we obtain
$$
  \T(\N) = \{ T\in\B(\H) :
              P_N^\perp T P_N =0 \quad{\rm for\; all}\quad N\in\N \} .
$$

Two nests $\M$ and $\N$ are {\it order isomorphic} if
there is an order preserving bijection $\theta$ from $\M$ onto $\N$.
Such a map is said to {\it preserve dimension} if 
$$
  {\rm dim}\, \theta(M_2)/\theta(M_1) = {\rm dim}\, M_2/M_1 
  \quad{\rm for\; every }\quad M_1<M_2 \in \M .
$$
The nests $\M$ and $\N$ are {\it similar} if there is an invertible
operator $S$ such that $S\M=\N$.  It is easy to see that
$S\T(\M)S^{-1}=\T(S\M)$.  In this case, the map $\theta_S(M)=SM$ is an
order isomorphism of $\M$ onto $\N$ which preserves dimension.  The
Similarity Theorem establishes the converse.

First we introduce one more relation between two order isomorphic
nests.  Say that an order isomorphism $\theta$ from $\M$ onto $\N$ is
implemented by an {\it approximate unitary equivalence} provided
that there is a sequence of unitary operators $U_n\in\U$ such that
$$
  \lim_{n\to\infty} \sup_{M\in\M} \| P_{\theta(M)} - U_nP_MU_n^* \| = 0.
$$
Note that the approximation of $\N$ by $\M$ is required to be uniform
over the whole nest.  We also say that $\theta$ is implemented by an
{\it approximate unitary equivalence modulo $\K$} provided that in
addition, the function 
$$f(M) = P_{\theta(M)} - U_nP_MU_n^*$$
is a continuous compact operator valued function on $\M$ with the order
topology.

We can now state the main result:

\medbreak\noindent{\bf The Similarity Theorem.  }
{\it
Let $\theta$ be an order isomorphism from a nest $\M$ onto a nest $\N$
on separable Hilbert spaces.  Then the following are equivalent:
\item{$(1)$} $\theta$ preserves dimension.
\item{$(2)$} $\theta$ is implemented by an approximate unitary
equivalence.
\item{$(2')$} $\theta$ is implemented by an approximate unitary
equivalence modulo $\K$.
\item{$(3)$} $\theta=\theta_{S_n}$ where $S_n$ is a sequence of
invertible operators such that
$\displaystyle\lim_{n\to\infty}{\rm dist}\,(S_n,\U)=0$.
\item{$(3')$} $\theta=\theta_{S_n}$ where $S_n$ is a sequence of
invertible operators such that
 $S_n=U_n+K_n$, where $U_n\in\U$,
$K_n\in\K$ and $\displaystyle\lim_{n\to\infty}\|K_n\|=0$.
\item{$(4)$} $\theta=\theta_{S}$ for an invertible operator $S$.
}
\medbreak

Certain of these implications are obvious.  Indeed, $(3')$ implies both
$(2')$ and $(3)$; while $(2')$ implies $(2)$, and $(3)$ implies both
$(2)$ and $(4)$.  Finally $(4)$ readily implies $(1)$.

The implications $(2)$ implies $(3)$ and $(2')$ implies $(3')$ are the
content of Lemma 12.17 and Theorem 12.18 of {\bf [6]}.  It is here that
we save a lot of work by sidestepping the issues of compact perturbations.
These results rely on the Arveson distance formula {\bf [2]} (c.f. Theorem
9.5 of {\bf [6]}) and the fact that this formula persists within the compact
operators (Theorem 12.1 of {\bf [6]}).  The main idea is to show that if
$\M$ and $\N$ are close, say within $\varepsilon$, and a unitary operator
$U$ moves $\M$ closer to $\N$, say within $\delta$, then $U$ is always
within $2\varepsilon$ of lying in $\T(\M)$.  The distance formula is
perfectly suited for this.  The element $A\in\T(\M)$ with
$\|U-A\|<2\varepsilon$ will be invertible (if $2\varepsilon<1$) and so 
$UA^{-1}$ is close to the identity and moves $\M$ in the same way.  An
iteration of this idea provides an infinite product that converges to
the desired invertible operator implementing the similarity.

So the remainder of the proof deals with the difficult step
establishing that $(1)$ implies $(2')$.  This is the content of chapter
13 of {\bf [6]}, and the proof remains the same. 
Results 13.1--13.9 do the case of continuous nests, a
result due to Andersen {\bf [1]}.  This result has two other proofs, one
due to Arveson {\bf [3]} and one due to the author {\bf [5]}.  It is
this last argument which is presented in {\bf [6]} as it is the easiest.
It is closely based on the ideas in {\bf [3]}, which in turn builds on
the ideas of Voiculescu's generalized Weyl--von Neumann Theorem for
C*-algebras {\bf [9]}. 

The results 13.15--13.20 of {\bf [6]} extend this to arbitrary nests. 
The idea of {\bf [4]} was to perform similarities on the nests $\M$ and
$\N$ so that they each support a continuous part of the nest on every
uncountable interval.  This is accomplished, again by an infinite product
of iterations, using an idea of Larson {\bf [8]} to show that there are
invertible operators acting on any uncountable nest that are not
implemented by unitaries.  Applying this to uncountable atomic nests
necessarily introduces some continuous part.  This is one of the
mysteries of the similarity theorem because no explicit example of this
phenomenon is known.  Once this is done, the atoms of $\M$ are mapped
to the corresponding atoms of $\N$ while the continuous parts are
intertwined using Andersen's theorem resulting in an approximate
unitary equivalence.

\bigbreak
\centerline{\bf References}
\medbreak
  
\item {\bf [1]} N.T. Andersen, {\it Compact perturbations of
reflexive algebras}, J. Func.\ Anal.\ {\bf 38} (1980), 366--400.

\item {\bf [2]} W.B. Arveson, {\it Interpolation problems in nest
algebras}, J. Func.\ Anal.\ {\bf 20} (1975), 208--233.

\item {\bf [3]} W.B. Arveson, {\it Perturbation theory for groups and
lattices}, J. Func.\ Anal.\ {\bf 53} (1983), 22--73.

\item {\bf [4]} K.R. Davidson,  {\it Similarity and compact
perturbations of nest  algebras}, J. reine angew. Math. 348 (1984),
286--294.

\item {\bf [5]} K.R. Davidson, {\it Approximate unitary equivalences 
of continuous  nests},  Proc.\ Amer.\ Math.\ 	Soc.\  {\bf 97}  (1986),
655--660.

\item {\bf [6]} K.R. Davidson,  {\it  Nest Algebras},  Pitman
Research Notes in Mathematics Series, vol.\ {\bf 191}, Longman
Scientific and Technical Pub.\  Co., London,  New York,  1988.

\item {\bf [7]} Johnson, B.E. and Parrott, S.K., {\it Operators
commuting with a von  Neumann algebra modulo the set of compact
operators}, J. Func. Anal. 11 (1972), 39--61.

\item {\bf [8]} D.R. Larson, {\it Nest algebras and similarity
transformations}, Ann.\ Math.\ {\bf 121} (1985), 409-427.
    
\item {\bf [9]} Voiculescu, D.V., {\it A non-commutative
Weyl--von~Neumann Theorem}, Rev.\ Roum.\ Pures Appl.\ {\bf 21} (1976),
97--113.

\bigbreak\noindent
{\sl Pure Math.\ Dept., U. Waterloo,
Waterloo, ON\quad  N2L 3G1, CANADA }

\noindent
{\sl krdavidson@math.uwaterloo.ca}

\end
