11institutetext:  1 Harvard University, USA
2 Boston College, USA
3 Carnegie Mellon University, USA
4 University of Montreal, Canada
5 EPFL, Switzerland
11email<EMAIL_ADDRESS><EMAIL_ADDRESS>
# FreSeg: Frenet-Frame-based Part Segmentation for 3D Curvilinear Structures

Shixuan Gu1 Jason Ken Adhinarta2 Mikhail Bessmeltsev4 Jiancheng Yang5 Jessica
Zhang3 Daniel Berger1 Jeff W. Lichtman1 Hanspeter Pfister1 Donglai Wei2

###### Abstract

Part segmentation is a crucial task for 3D curvilinear structures like neuron
dendrites and blood vessels, enabling the analysis of dendritic spines and
aneurysms with scientific and clinical significance. However, their diversely
winded morphology poses a generalization challenge to existing deep learning
methods, which leads to labor-intensive manual correction. In this work, we
propose FreSeg, a framework of part segmentation tasks for 3D curvilinear
structures. With a Frenet-Frame-based point cloud transformation, it enables
the models to learn more generalizable features and have significant
performance improvements on tasks involving elongated and curvy geometries. We
evaluate FreSeg on 2 datasets: 1) DenSpineEM, an in-house dataset for
dendritic spine segmentation, and 2) IntrA, a public 3D dataset for
intracranial aneurysm segmentation. Further, we will release the DenSpineEM
dataset, which includes roughly 6,000 spines from 69 dendrites from 3 public
electron microscopy (EM) datasets, to foster the development of effective
dendritic spine instance extraction methods and, consequently, large-scale
connectivity analysis to better understand mammalian brains.

###### Keywords:

point cloud reconstruction 3D segmentation dendritic spine medical image
dataset electron microscopy

##  1 Introduction

Deep learning-enabled 3D biomedical imaging serves both as a catalyst for
scientific breakthroughs (e.g., connectomics [5], protein prediction [11]) and
as a crucial tool in medical care (e.g., rib analysis [19], aneurysm detection
[20]). While the ability of semantic segmentation algorithms to facilitate
image analysis and measurement in diverse domains is well-established (e.g.,
nn-UNet [9]), it remains a challenging task to devise solutions for scenarios
with elongated and curvy structures. In clinical research, such generalized
cylinders have great biological significance and are common in vasculature
systems among different organs and cells (e.g., brains, lungs, retinas). For
instance, in neuroscience, the morphology and density of neurites contain
critical information about the connectivity strength among neurons [16].
Likewise, in the field of medicine, understanding the shapes and locations of
intracranial aneurysms is essential to clinical diagnosis and surgical
planning [8]. However, manual inspection and measurement of such structures
are labor-intensive for professionals, as generalized cylinder geometries are
typically intricate and messy in 3D volumes. Several works have built datasets
of such intricate structures to enable the evaluation of deep learning-based
methods on such modalities [18, 17, 20, 10].

However, learning robust feature representations on geometries in 3D Euclidean
space (e.g., point clouds, volumes, meshes [22, 23]) is difficult for deep
learning models, since the coordinate systems are sensitive to the symmetries
of 3D space, such as rotations and translations [6]. This problem is
exacerbated in generalized cylinder geometries which may exhibit symmetries
under deformation to its centerlines. Unfortunately, most previous works
overlook the geometric complexity inherent in the data representations, and
merely devise task-specific methods which lack the capacity to generalize to
different domains; for instance, lung vasculature reconstruction tasks model
the targets as topological graphs, neglecting the surface geometry which is
crucial in other domains such as aneurysm detection. While the self-
configuring voxel-based method such as nnU-Net [9] achieved great segmentation
results in various biomedical image domains, it failed to embed intricate
global geometry or handle dense neurite volumes in high-resolution electron
microscopy images (EM images) [18, 17]. A few works handle the intricacies of
3D geometry by increasing the expressive capacity of the network with
equivariant layers and a high degree of weight sharing [4, 6], but the
explicit geometry representation is often ignored, which would lead to
degradation on extreme scenarios.

Figure 1: Part segmentation for 3D curvilinear structures. a) Dendritic spine
(blue) segmentation. b) Intracranial aneurysm (blue) segmentation.

In this study, we propose FreSeg, a skeleton-based cylindrical transformation
that reconstructs the twisted, elongated, and curvy structure in a more
generalizable representation to facilitate deep learning-based tasks.
Specifically, FreSeg 1) models the dense 3D volume into sparse point clouds
(locally isotropic) to preserve global geometry while reducing computational
expense, and 2) reconstructs the point clouds in a cylindrical coordinate
system based on the Frenet-Serret Frame (i.e., moving observational frame of
reference) along the main skeleton of the target object. With the desirable
properties inherent in the transformation (i.e., bijectivity,
differentiability, equivariance), the transformed structures, denoted as
generalized cylinders, can substantially increase the generalization capacity
of the deep learning models. We evaluate the method on two datasets: 1) IntrA,
a public 3D dataset for intracranial aneurysm segmentation (reconstructed from
MRI images), and 2) DenSpineEM, an in-house EM dataset for dendritic spine
segmentation, which includes 6,000 spine instances from 69 thoroughly
segmented dendrites from three public EM datasets. We conduct comparative
experiments of few-shot learning by training the model with and without the
transformation under different ratios of the training set to evaluate its
capacity for generalization. And by ablation study of influence-time
augmentation, we further showed that the rotation-invariant property of the
generalized cylinders could generalize the model by decreasing the need for
data augmentation. By result analysis, we showed that FreSeg can enable models
to learn generalizable features with fewer training samples and has the
potential to foster cross-domain learning topics (e.g., transfer learning,
synthetic data generation, and few-shot learning) with massive vasculature
systems which require huge annotation effort.

Contributions. 1) FreSeg, an equivariant transformation that reconstructs
elongated and curvy structures into generalized cylinders, which substantially
increases the generalization capacity of deep learning methods. 2) FreSeg has
a huge potential to foster cross-domain learning applications in massive
vasculature systems. 3) The first large public benchmark for 3D dendritic
spine segmentation, which enables downstream applications and method
comparison.

Figure 2: The Pipeline of FreSeg. The pipeline is divided into 3 steps: 1)
Skeleton-based Point Sampling: the dense medical volume is converted to a
point cloud for skeletonization [7], and it is further sampled into fragments
along the main trunk skeletons. 2) Frenet-Frame-based Transformation: the
fragments are reconstructed in a cylindrical coordinate system via a Frenet-
Frame-based transformation and propagated into the backbone model for
inference. 3) Point Cloud Network Prediction: the final volume prediction is
obtained by remapping and combining all predictions with majority voting.

##  2 Methodology

In this study, we proposed FreSeg, a framework that 1) samples the curvilinear
structure into point cloud fragments based on its skeleton, and 2) utilizes
the Frenet-Serret-Frame of the skeleton to reconstruct fragments in a more
generalizable representation to facilitate deep learning-based tasks, as
depicted in Fig. 2. To evaluate the method, we applied the framework to tasks
in different domains as shown in Fig. 1, i.e., dendritic spine segmentation
and intracranial aneurysm segmentation, by modifying the setting of skeleton-
based sampling.

###  2.1 Skeleton-based Point Sampling

Point-based representation. Intuitively, FreSeg reconstructs curvilinear
structures in a cylindrical coordinate system to reshape them into tubular
structures, as depicted in Fig. 3a. Hence, it models 3D medical imaging as
point clouds instead of volume, which also reduces computational expense while
preserving the global geometry. Specifically, for DenSpineEM dataset with EM
volume, the conversion is simply voxel indexing, while for IntrA dataset with
reconstructed artery surfaces, we voxelize the surface [2] and convert it to
point clouds.

Skeleton-based sampling. The extreme foreground-background imbalance and data
density are common challenges of part segmentation tasks in biomedical
domains. To alleviate it, we design a skeleton-based sampling for curvilinear
structures which is task-specific. We first skeletonize the point clouds [7]
and choose a path as the sampling trajectory. Specifically, for the DenSpineEM
dataset where neurites have one trunk connected with multiple spines, we
select the path containing most vertices with a degree greater than 2. While
for the IntrA dataset where the cerebral artery consists of messy vessels, we
traverse the skeleton of the entire artery, prune short branches, and obtain
multiple paths to cover the filtered skeleton. And then the point clouds can
be easily sampled into fragments, as depicted in Fig. 2.

###  2.2 Frenet-Frame-based Transformation

Preliminaries. In three-dimensional Euclidean space $\mathbb{R}^{3}$, the
Frenet–Serret frame (TNB frame) of a differentiable curve at a point is a
triplet of three mutually orthogonal unit vectors (tangent, normal, and
binormal). Specifically, let $\mathbf{r}(t)$ be a curve in Euclidean space,
representing the position vector of the particle as a function of time, and
$\mathbf{s}(t)$ represent the arc length which the particle has moved along
the curve in time t [14]. The TNB frame is defined by:

Figure 3: Calculation of Frenet-Frame-based Transformation. a) For a given
point $P_{i}$, we find its closest vertex $S_{i}$ on the curve. b) The orange
parts, $(\rho,\phi,g)$, represent the calculations based on the Cartesian
coordinates and are used as the cylindrical coordinates of the target point
$P_{i}$, respectively. $P_{i}^{{}^{\prime}}$ is the projection of $P_{i}$ on
the NB plane. c) Point reconstruction in a cylindrical coordinate system.

$\mathbf{T}:=\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}s},~{}\mathbf{N}:=\frac{\mathrm{d}\mathbf{T}}{\mathrm{d}s}/\left\|\frac{\mathrm{d}\mathbf{T}}{\mathrm{d}s}\right\|,~{}\mathbf{B}:=\mathbf{T}\times\mathbf{N}.$
(1)

Implementation. Given a point cloud $P=\\{(x_{i},y_{i},z_{i})|i=1,...,n\\}$
and its sampling trajectory $S=\\{(x_{i},y_{i},z_{i})|i=1,...,m\\}$, denoted
as the main skeleton, we first calculate the TNB frame of the skeleton
according to the definition 1, denoted as
$TNB_{S}=\\{(\textit{{t}}_{i},\textit{{n}}_{i},\textit{{b}}_{i})|i=1,...,m\\}$,
where $\textit{{t}}_{i}$, $\textit{{n}}_{i}$, and $\textit{{b}}_{i}$ are the
unit tangent, normal, and binormal vectors of the skeleton $S$ at the point
$S_{i}$ (Fig. 3b). Based on the TNB frame, the point cloud $P$ is
reconstructed in a cylindrical coordinate system, denoted as $C$, where
$C=\\{(\rho_{i},\phi_{i},g_{i})|i=1,...,n\\}$. Specifically, given the point
$P_{i}$, we first find its closet vertex on the skeleton, denoted as $S_{j}$.
Assuming $S$ is an ordered set, we have:

$\displaystyle\rho_{i}=\operatorname{dis}_{\text{Euclidean
}}\left(P_{i},S_{j}\right),~{}~{}~{}~{}g_{i}=\sum_{n=0}^{j-1}\operatorname{dis}_{\text{Euclidean
}}\left(S_{n},S_{n}+1\right)$
$\displaystyle\phi_{i}=\left\\{\begin{array}[]{rl}\arccos\left(\frac{\textit{{v}}_{i}\cdot\textit{{n}}_{j}}{|\textit{{v}}_{i}||\textit{{n}}_{j}|}\right),&\text{
if }\textit{{v}}_{i}\cdot\textit{{b}}_{j}\geq 0\\\
2\pi-\arccos\left(\frac{\textit{{v}}_{i}\cdot\textit{{n}}_{j}}{|\textit{{v}}_{i}||\textit{{n}}_{j}|}\right),&\text{
if }\textit{{v}}_{i}\cdot\textit{{b}}_{j}<0\end{array}\right.$ (4)

where $\textit{{v}}_{i}$ presents the projection of the vector
$\overrightarrow{S_{j}P_{i}}$ (denoted as $\textit{{u}}_{i}$) on the NB plane,
which could be calculated by $\textit{{v}}_{i}=A_{j}A_{j}^{T}\textit{{u}}_{i}$
($A_{j}=[\textit{{t}}_{j},\textit{{b}}_{j}]$ is a column orthogonal matrix).
Fig. 3c gives a visual interpretation of the transformation from Cartesian
coordinates to cylindrical coordinates.

Properties. The elegant properties inherent in the transformation (e.g.,
bijectivity, differentiability, equivariance) enable the model to achieve
significant performance improvement and learn more generalizable features. The
proof can be found in the supplementary material. (1)Bijectivity: with
skeleton vertices, the cylindrical representation could be transformed back to
the original representation, hence our method does not suffer any information
loss. (2)Rotation invariance: the representation is consistent regardless of
the rotation (and translation) augmentations, hence allowing the model to
learn generalizable features.

###  2.3 Point Cloud Network Prediction

Specifically, for the point cloud fragments with a large scale that is
computationally prohibitive, we divide it into equally-sized batches of 30K
points. For the batch with insufficient points, we ’ceil’ it up by randomly
sampling points from other batches, and applying majority voting for repeated
points. Finally, the point predictions of all batches are concatenated to
obtain the voxel prediction, as depicted in Fig. 2.

##  3 DenSpineEM Dataset

The morphology and density of dendritic spines contain critical information
about the connectivity strength among neurons. To foster new research to
address the spine segmentation challenge, we curated a large-scale 3D
dendritic spine segmentation benchmark, DenSpineEM, as depicted in Fig. 3.
DenSpineEM contains 6,000 spine instances from around 69 thoroughly segmented
dendrites, which consists of 3 3D EM image stacks acquired from mouse (50
${\mu}m)^{3}$, rat (30 ${\mu}m)^{3}$, and human (30 ${\mu}m)^{3}$ cortical
regions, whose neuron segmentation is available from previous publications
[12, 17]. The development of DenSpineEM took 3 neuroscientists about 30 days
in total. Tab. 1 shows the data distribution of the DenSpineEM dataset.

Table 1: Overview of DenSpineEM Dataset. We build upon 3 EM volumes with instance segmentation and annotate spine segmentation for 90 dendrites. Name | Tissue | Size ($\mu m^{3}$) | #Dendrites | #Spines
---|---|---|---|---
DenSpine-M50 | Mouse Somatosensory Cortex [12] | 50x50x30 | 50 | 3,886
DenSpine-M | Mouse Visual Cortex [17] | 30x30x30 | 10 | 315
DenSpine-H | Human Frontal Lobe [17] | 30x30x30 | 9 | 319

Dataset Construction. We first curate DenSpine-M50 from [12] as our main
dataset due its existing segmented dendrites (100+) and spines (4,000+) which
are analyzed in [13]. However, the spine segmentation is not thorough on most
dendrites, making it hard to train models for practical use due to the unknown
false negative errors. Thus, we pick 50 largest dendrites from the existing
annotation and manually proofread all spine instance segmentation. In the end,
we obtain 3,886 spine instances. We further divide the dendrites into
train/val/test split with 30/5/15 dendrites by a rough 60/10/30 ratio for
learning-based methods, leading to 2239/512/1135 spine instances respectively.
To evaluate the generalization performance of the model trained on
DenSpine-M50 across regions and species, we build two small datasets from
AxonEM data [17]: DenSpine-M from mouse visual cortex, and DenSpine-H from
human frontal lobe.

Annotation Protocol. For a high-quality ground truth annotation, we segment
spines manually with the VAST software [3] to avoid introducing bias from
automatic methods. To detect errors, we use the Neuroglancer software [1] to
generate and visualize 3D meshes of the segmentation of dendrites and spines.
Four neuroscience experts were recruited to proofread and double confirm the
annotation results for spine instance segmentation.

To evaluate FreSeg, we apply it to both the DenSpineEM and IntrA datasets for
part segmentation task. For generalization test, we conducted a comparison
study by decreasing training data and applying inference-time augmentation.

###  3.1 Experiment Setup

DenSpineEM. We trained the model on DenSpine-M50 which contains sufficient
samples to be divided into train/val/test, and evaluated the methods on the
test set. Since DenSpine-M and DenSpine-H contains much fewer dendrites, we
use them as additional test sets to evaluate the generalization performance.
For evaluation, we report the dice similarity coefficient (DSC) of the spines.

IntrA. We followed the two-step baseline method [21] to evaluate the methods
with five-fold cross-validation, and for a fair comparison with the baseline,
we converted point clouds segmentation back to surface models. For
generalization evaluation, we compared the performance of models trained with
$100\%/75\%/50\%/25\%$ training data scale as well as applying inference-time
augmentation (spine-rotation), as shown in Fig. 4. For metrics, we use the DSC
of the aneurysm parts for comparison as used in the baseline. The voxelization
code, voxelized data, and metrics tables of all 5 folds are in the supplement.

###  3.2 Benchmark Results

Table 2: Quantitative results. The FreSeg is evaluated on two datasets. Dataset | DenSpineEM | IntrA* [21]
---|---|---
M50-test | M | H
SO-Net [21] | -/- | -/- | -/- | 71.79
PointNet++ [15] | 52.23 | 29.30 | 20.10 | 31.43
FreSeg w/o Trans (Ours) | 60.53 | 31.10 | 19.79 | 70.05
FreSeg (Ours) | 66.58 | 42.88 | 37.84 | 77.08

DenSpineEM. We report the metrics for the 3 DenSpineEM datasets with the
models trained on the training set of the DenSpineEM-M50 dataset. As reported
in Tab. 2, we showed that FreSeg achieves at $14\%/13\%/17\%$ higher Dice
value than the PointNet++ baseline on the 3 datasets, respectively.

IntrA. As reported in Tab. 2, FreSeg brings a more distinct performance boost
on relatively small datasets like IntrA, where it achieves $5\%$ higher Dice
value compared to PointNet++, and beats the baseline method by $3\%$.
Qualitative results are provided in the supplementary material.

###  3.3 Ablation Studies

To further evaluate its capacity of generalization, we conducted an ablation
study by decreasing percentage of training data used from $100\%$ to $25\%$,
the method with FreSeg has a much lower performance degradation ($-8.5\%$)
than the one without FreNet ($-20\%$) over this range. We also attempt to
verify the rotation invariance of FreSeg by applying inference-time rotation
augmentations. As reported in Fig. 4b, FreSet maintains relatively stable
performance levels ($10\%$ drop) while the baseline method suffers a $35\%$
performance drop.

Figure 4: Results on IntrA. To evaluate the generalization capabilities of
FreSeg, we report the performance on intracranial aneurysm segmentation vs. a)
percentage of training data used and b) inference-time rotation augmentation.
The complete 5-fold validation metrics table can be found in the supplement.

##  4 Conclusion

In this study, we proposed FreSeg, a framework of part segmentation tasks for
3D curvilinear structures. It reconstructs elongated and curvy geometry to
facilitate deep learning applications in (bio)medical scenarios. With the
elegant properties inherent in the transformation (e.g., bijectivity,
differentiability, equivariance), it enables the model to learn more
generalizable features across different domains, which has the potential to
foster cross-domain deep learning topics. For instance, FreSeg makes it
possible for transfer learning across massive different vasculature datasets
to alleviate annotation efforts. In addition, it has the potential to generate
synthetic data and facilitate few-shot learning.

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# Quasitriangular Operator Algebras

Massoud Amini, Mehdi Moradi, Ismaeil Mousavi  Department of Pure Mathematics
Faculty of Mathematical Sciences
Tarbiat Modares University
Tehran
Iran<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract.

We give characterizations of quasitriangular operator algebras along the line
of Voiculescu’s characterization of quasidiagonal $C^{*}$-algebras.

###### Key words and phrases:

operator algebras, quasidiagonal, quasitriangular, block triangular,
triangular AF, residual FD, essentially embedded

###### 2010 Mathematics Subject Classification:

Primary: 46K50; Secondary: 46L07

##  1\. Introduction

Quasitriangular operators on a Hilbert space were introduced and studied in
the 60’s by Paul Halmos in [31] (and later in [32]). As Halmos also mentions,
the notion was already used (and indeed played a central role) in the proof of
the Aronszjn-Smith theorem [3] on the existence of invariant subspaces for
compact operators and its extensions (like that of Arveson and Feldman [9]).
These operators naturally generalize the notion of triangular matrices.

Triangular operator algebras were studied by John Ringrose [48], under the
name of nest algebras, and discussed in the Ph.D. thesis of his student,
Christopher Lance [40]. The terminology was used first in a paper by Kadison
and Singer [37]. There have been various characterizations of triangular
subalgebras of specific classes of $C^{*}$-algebras, including that of Power
for triangular subalgebras of AF-algebras [47].

The natural generalization of these to quasitriangular algebras were
introduced by William Arveson [6] and studied by him and coauthors in [29].
All these studies are in the spirit of nest algebras. The first instance of a
theory independent of the notion of nest algebras was presented by Dan
Voiculescu in [56]. Here quasidiagonal $C^{*}$-algebras are defined in terms
of representations, something which is anticipated by the original ideas of
Halmos (compare this with the approach of Bruce Wagner [58] based on nest
algebras). Further characterizations are given by Marius Dadarlat [18].

The main objective of this paper is to characterize quasitriangular (non
selfadjoint) operator algebras in terms of their representations. Note that in
the selfadjoint case (that is, for $C^{*}$-algebras) the notions of
quasidiagonality and quasitriangularity coincide, and so such a
characterization is a non selfadjoint version of the results of Voiculescu and
Dadarlat. The first characterization relies on a result of Arveson on the
extension of completely positive (c.p.) maps from operator systems to the
enveloping $C^{*}$-algebra [4] and the second is proved directly. We also
investigate basic properties of quasitriangular operator algebras (like stable
finiteness and existence of trace) and provide examples and non-examples. Our
approach is based on finite dimensional approximation and we do not allude to
the traditional approach based on nest algebras.

This paper could be regarded only as the starting point in comparing and
contrasting quasitriangularity (of the non selfadjoint operator algebras)
versus quasidiagonality (of selfadjoint ones). Although the two phenomena
(which coincide in the selfadjoint case) show many similarities, the
quasitriangularity has a pathological behavior which does not appear in
quasidiagonality: while a diagonal (i.e., commutative) algebra is always
quasidiagonal, the analogous thing does not happen in the non selfadjoint
realm, namely, a triangular algebra may fail to be quasitriangular. Indeed the
Cuntz $C^{*}$-algebra $\mathcal{O}_{n}$ has a wealth of maximal triangular
subalgebras which are not finite and so could not be quasitriangular. This
phenomenon is due to the break of symmetry in the non selfadjoint case.
Indeed, one has two options for a one way version of quasidiagonality (that
is, upper or lower quasitriangularity) and choosing each of these is in the
price of loosing the other. This might seem a bit annoying at the first
glance, but we trust that it would be a source of diversity for
quasitriangular algebras which is absent in the symmetric quasidiagonal case.
Though this feature may prove to be useful and so calls for an independent
research, here we could only sketch it and yet have no clear idea how one may
use this potential in applications with an intrinsic lack of symmetry.

##  2\. Definitions and Results

A Banach algebra $T$ is called triangular approximate finite dimensional (TAF)
if it is a norm-closed subalgebra of a unital AF $C^{*}$-algebra $A$ such that
$D:=T\cap T^{*}$ is a masa in $A$ which satisfies the Strătilă-Voiculescu
condition [53]: there is a nested sequence $(A_{k})$ of finite dimensional
unital subalgebras of $A$ with a dense union such that $D_{k}:=D\cap A_{k}$ is
a masa in $A_{k}$ [45]. If $T$ is TAF then $T$ is the direct limit of the
nested family $(T\cap A_{k})$ [45, Corollary 2.3], and conversely, for an
increasing sequence of triangular algebras $T_{k}\subseteq A_{k}$ with
diagonal $D_{k}$, the direct limit $T$ of $T_{k}$’s is TAF [45, Theorem 2.6].

######  Definition 2.1.

An operator algebra $A$ is called TAF-embeddable if it is embedded completely
isometrically into a TAF algebra.

An operator algebra $A$ is called residually finite dimensional (RFD) if there
is a family (possibly uncountable) of c.c. homomorphisms
$\rho_{i}:A\to\mathbb{M}_{k(i)}(\mathbb{C})$ which are asymptotically
isometric, that is, $\|a\|=\lim_{i}\|\rho_{i}(a)\|$, for each $a\in A$ (see
[13, Chapter 7], for the same notion for $C^{*}$-algebras, where $\rho_{i}$’s
are assumed to be $*$-homomorphisms). Next we need to define a triangular
version of the above notion. By a triangular subalgebra of $\mathbb{M}_{k}$ we
mean a subalgebra of the form
$\mathbb{T}_{k_{1}}\oplus\cdots\oplus\mathbb{T}_{k_{n}}$, where
$\mathbb{T}_{k_{i}}$ is the algebra of all upper triangular matrices in
$\mathbb{M}_{k_{i}}$.

######  Definition 2.2.

An operator algebra $A$ is called TRFD if it is RFD and the range of each
homomorphism $\rho_{i}:A\to\mathbb{M}_{k(i)}(\mathbb{C})$ in the above
definition is a triangular subalgebra of $\mathbb{M}_{k_{i}}$.

Two representations $\pi:A\to\mathbb{B}(H)$ and $\sigma:A\to\mathbb{B}(K)$ are
approximately unitarily equivalent if there is a sequence $u_{n}:H\to K$ of
unitary operators such that $\|\sigma(a)-u_{n}\pi(a)u_{n}^{*}\|\to 0,$ for
each $a\in A$. Note that when both $\pi$ and $\sigma$ have ranges inside
triangular operator algebras (in the sense of Kadison and Singer [37], that
is, the operator algebras $A$ of $\mathbb{B}(H)$ with $A\cap A^{*}=D$, where
$D$ is the set of diagonal operators), then in each step,
$u_{n}\pi(A)u_{n}^{*}$ is again a triangular operator algebra (since
triangularity is preserved under unitary equivalence). This is along the
general idea of Kadison and Singer to have non selfadjoint operator algebras
whose elements could be simultaneously triangularized (c.f. [42, page 245],
[37, Theorem 3.2.1]).

Our first main result asserts that a separable TRFD operator algebras is TAF-
embeddable if any pair of finite dimensional representations are approximately
unitarily equivalent after adding an enough large multiple of one of them to
both, providing a non selfadjoint analog of a result due to Dadarlat [19].

######  Theorem 2.3.

Let $A$ be a separable TRFD operator algebra such that for each pair of finite
dimensional representations $\sigma_{i}:A\to\mathbb{M}_{k}(\mathbb{C})$,
$i=1,2$, with triangular ranges, there is a positive integer $N\geq 1$ such
that $(N+1)\sigma_{1}$ is approximately unitarily equivalent to
$\sigma_{2}\oplus N\sigma_{1}$. Then $A$ is TAF-embeddable.

######  Definition 2.4.

An operator algebra $A$ is called quasitriangular (QT) if there is a net of
completely contractive maps $\phi_{i}:A\to\mathbb{M}_{k_{i}}(\mathbb{C})$
which are asymptotically isometric and asymptotically multiplicative, that is,
$\lim_{i}\|\phi_{i}(a)\|=\|a\|$ and
$\lim_{i}\|\phi_{i}(ab)-\phi_{i}(a)\phi_{i}(b)\|=0$, for each $a,b\in A$. A
collection of bounded operators $S\subseteq\mathbb{B}(H)$ is called
quasitriangular (QT) if given $\varepsilon>0$ and finite subsets
$\mathfrak{F}\subseteq S$ and $F\subseteq H$, there is a finite rank
projection $P\in\mathbb{B}(H)$ with $\|PT-PTP\|<\varepsilon$ and
$\|Pv-v\|<\varepsilon$, for each $T\in\mathfrak{F}$ and $v\in F$. Finally, a
representation $\pi:A\to\mathbb{B}(H)$ of an operator algebra $A$ is called
quasitriangular (QT) if $\pi(A)\subseteq\mathbb{B}(H)$ is quasitriangular.

A representation of an operator algebra $A$ is a completely contractive (c.c.)
homomorphism $\pi:A\to\mathbb{B}(H)$, where $H$ is a Hilbert space. A faithful
representation is assumed moreover to be completely isometric (c.i.). A
representation $\pi:A\to\mathbb{B}(H)$ is called essential if
$\pi(A)\cap\mathbb{K}(H)=\\{0\\}$.

The second main result of this paper gives a characterization of separable
unital quasitriangular operator algebras in terms of their faithful
representations. This is an extension of a result of Voiculescu which asserts
that a separable C*-algebra is QD iff it forms a QD family of operators in
some concrete Hilbert space. Since the “third isomorphism theorem” fails in
its most general form for operator algebras [2], the QT version of this result
for operator algebras satisfying an extra condition (here called essentially
embedded, requiring that the quotient map onto the Calkin algebra induces a
complete isometry on the given operator algebra; see the next section for
details and examples).

######  Theorem 2.5.

For a separable unital operator algebra $A$, consider the following
assertions:

$(i)$ $A$ is QT,

$(ii)$ $A$ has a faithful QT representation on a separable Hilbert space,

$(iii)$ Every faithful unital essential representation of $A$ on a separable
Hilbert space is QT.

Then $(iii)\Rightarrow(ii)\Rightarrow(i)$. If moreover $A$ is essentially
embedded, then we also have $(i)\Rightarrow(iii)$.

The above theorem could be used to construct QT and non-QT operator algebras.
It is shown by Halmos [31] that the unilateral shift $S$ on a separable
Hilbert space is not QT (Douglas and Pearcy later constructed an operator $T$
on a separable Hilbert space so that neither $T$ nor $T^{*}$ is QT, answering
a question of Halmos in negative [27]). Now the operator algebra generated by
$S$ (which is the non selfadjoint Toeplitz algebra, here denoted by
$\mathfrak{T}$) would not be QT by the above theorem. This has some
consequences. For instance, if a separable unital operator algebra $A$ is QT,
then $A$ could not contain a non surjective isometry (since otherwise it
should contain a copy of non-QT algebra $\mathfrak{T}$), and neither does
$A\otimes\mathbb{M}_{n}(\mathbb{C})$, for each $n$. In other words, $A$ is
stably finite (which simply means $\mathbb{M}_{n}(A)$ is finite–i.e., does not
contain non unitary isometries–for each $n\geq 1$). We may get the same result
for non-unital algebras by passing to the minimal unitization (which remains
separable, if the original algebra is so). Hence we have the following result.

######  Proposition 2.6.

An operator algebra is QT if and only if all of its finitely generated closed
subalgebras are QT.

It immediately follows from the above proposition and Arveson Extension
Theorem that the separability condition could be removed from Theorem 2.5.
This allows one to repeat the above argument for the lack of non surjective
isometries in the non separable case, leading to the next result.

######  Corollary 2.7.

A QT operator algebra $A$ is stably finite.

The above corollary could be employed to construct more examples of non-QT
operator algebras. It is known that in the Cuntz $C^{*}$-algebra
$\mathcal{O}_{n}$ we have an abundance of analytic subalgebras [36]. In
particular, it contains a canonical UHF subalgebra (such that each strongly
maximal triangular subalgebra of the UHF subalgebra has an extension to a
strongly maximal triangular subalgebra of $\mathcal{O}_{n}$). This shows that
there are lots of triangular subalgebras in $\mathcal{O}_{n}$. One concrete
example of such a subalgebra is the Volterra algebra (already studied by Power
[46]), which is maximal triangular, but not strongly maximal triangular [36,
pages 29 & 32]. However, this operator algebra is not QT, since it is
generated by the Cuntz partial isometries it contains (and so is not finite).

On the other hand, if the weight sequence of a bilateral weighted shift on a
separable Hilbert space has 0 as a limit point (in both directions) then it is
quasidiagonal [52, Theorem 1]. By almost the same proof as in [52], one could
check that if the weight sequence of a unilateral weighted forward shift has 0
as a limit point then it is quasitriangular (take a subsequence of weights
going to 0 and project on the the subspace generated by the basis elements
with the same indices as the subsequence). This way we could construct a
separable (singly generated) QT operator algebra (which is not QD if we choose
the weights not to go to zero when indices are negated).

Our next result asserts that that every QT operator algebra can be locally
approximated by a residually finite dimensional (RFD) operator algebra. The
proof is an adaptation of the Halmos original proof that every quasitriangular
operator can be written as a block triangular operator plus a compact (see
also, [34]). We call an operator algebra $A$ block triangular if there exists
an increasing sequence of finite rank projections $P_{n}$ going SOT to 1 such
that $P_{n}a=P_{n}aP_{n}$, for all $a\in A$ and $n\geq 1$. As for
$C^{*}$-algebras, it is easy to see that an operator algebra is RFD if and
only if it has a faithful representation with block triangular image (c.f.
[14], [15] for alternative approaches). Note that an operator algebra might be
RFD while its enveloping C*-algebra is not so [14, Example 3.4] and [55,
Example 1].

If $\varepsilon>0$ and $F,A\subseteq\mathbb{B}(H)$ are sets of operators then
$F$ is $\varepsilon$-contained in $A$ if for each $x\in F$ there exists $y\in
A$ with $\|x-y\|<\varepsilon$. When this is the case we write
$F\subseteq_{\varepsilon}B$.

######  Theorem 2.8.

A separable operator algebra $A\subseteq\mathbb{B}(H)$ is QT (as a set of
operators) iff for every finite set $F\subseteq A$ and every $\varepsilon>0$,
there exists a block triangular algebra
$B\subseteq\mathbb{\mathbb{missing}}B(H)$ such that
$F\subseteq_{\varepsilon}B$ and $A+\mathbb{K}(H)=B+\mathbb{K}(H)$. If moreover
$A$ is essentially embedded and $A\cap\mathbb{K}(H)=0$, then it is a quotient
of $B$.

A state $\tau$ on a unital operator algebra $A$ is a linear functional
satisfying $\tau(1)=\|\tau\|=1$. It is called a tracial state if moreover,
$\tau(ab)=\tau(ba)$, for each $a,b\in A$. We have the following extension of a
classical result of Voiculescu [57, 2.4].

######  Proposition 2.9.

Every unital QT operator algebra has a tracial state.

##  3\. Proofs

In this section we gives proofs of the results of the previous sections.
Because the notion of positivity is not available for non selfadjoint operator
algebras, in some cases we have to deviate from the line of the proof of the
analogous result for the selfadjoint case, or allude to the basic idea of
Arveson to get c.p. extensions of c.c. maps [4].

In the first lemma we deal with direct limit of operator algebras. The
connecting maps are homomorphism, and unlike $*$-homomorphisms these are not
guaranteed to be contractive (even continuous). Therefore, we only work with
c.c. homomorphisms.

######  Lemma 3.1.

Let $A$ be a separable unital operator algebra and $H$ be a separable Hilbert
space. If there is an increasing sequence of finite dimensional triangular
subalgebras $B_{n}\subseteq\mathbb{B}(H)$, constant non-negative integer
$k\geq 1$, injective c.c. homomorphisms $\pi_{n}:B_{n}\to B_{n+k}$, and u.c.c.
maps $\sigma_{n}:A\to B_{n}$ such that

$(i)$ the family $\\{\sigma_{n}\\}$ is asymptotically completely isometric and
asymptotically multiplicative,

$(ii)$
$\sum_{n=1}^{\infty}\|\pi_{n}\circ\sigma_{n}(a)-\sigma_{n+k}(a)\|<\infty,$ for
some subset $G\subseteq A$ with norm dense linear span and each $a\in G$,
where the norm in the summand is calculated in $\mathbb{B}(H)$,

then $A$ is TAF-embeddable.

###### Proof.

By a rearrangement of indices we may assume that $k=1$. Identify $B_{n}$ with
a subalgebra of the direct limit $B$ of the system $\\{(B_{n},\pi_{n})\\}$ and
regard each $\sigma_{n}$ as a map into $B$. Under this identification, one
could rewrite $(ii)$ as
$\sum_{n=1}^{\infty}\|\sigma_{n}(a)-\sigma_{n+1}(a)\|<\infty,$ which means
that $\\{\sigma_{n}(a)\\}$ is Cauchy in $B$, for each $a\in G$. This shows
that $\sigma:span(G)\to B;$ $\sigma(x):=\lim_{n}\sigma_{n}(x)$ is well
defined. By the first assumption in $(i)$, the linear map $\sigma$ is
completely isometric, and so has a complete isometric extension to
$\bar{\sigma}:A\to B$, which is also a homomorphism, by the second assumption
in $(i)$. Finally, observe that $B$ is TAF by [45, Theorem 2.6]. ∎

The proof of the first main result of this paper is a careful adaptation of an
argument of M. Dadarlat [19] to the triangular setting.

###### Proof of Theorem 2.3.

Choose a asymptotically isometric sequence (by separability) of finite
dimensional c.c. representations (by TRFD assumption)
$\rho_{i}:A\to\mathbb{M}_{k(i)}(\mathbb{C})$ such that

$A_{i}:={\rm
Im}(\rho_{i})=\mathbb{T}_{k_{1}(i)}\oplus\cdots\oplus\mathbb{T}_{k_{\ell}(i)}\subseteq\mathbb{M}_{k(i)}(\mathbb{C}),$

with $k_{1}(i)+\cdots k_{\ell}(i)=k(i)$, where $\ell$ depends on $i$. Put
$B_{1}=A_{1}$ and $\sigma_{1}=\rho_{1}:A\to
B_{1}\subseteq\mathbb{M}_{k(1)}(\mathbb{C})$. For finite dimensional
representations

$\sigma_{1}\otimes 1_{k(2)}=k(2)\sigma_{1}:A\to
B_{1}\otimes\mathbb{T}_{k(2)}\subseteq\mathbb{M}_{k(1)k(2)}(\mathbb{C}),$

and

$1_{B_{1}}\otimes\rho_{2}:A\to B_{1}\otimes
A_{2}\subseteq\mathbb{M}_{k(1)k(2)}(\mathbb{C}),$

choose positive integer $N_{1}\geq 1$ such that

$k(2)(N_{1}+1)\sigma_{1}:A\to
B_{1}\otimes\mathbb{T}_{k(2)(N_{1}+1)}\subseteq\mathbb{M}_{n(1)}(\mathbb{C}),$

and

$(1_{B_{1}}\otimes\rho_{2})\oplus k(2)N_{1}\sigma_{1}:A\to(B_{1}\otimes
A_{2})\oplus(B_{1}\otimes\mathbb{T}_{k(2)N_{1}})\subseteq\mathbb{M}_{n(1)}(\mathbb{C}),$

are approximately unitarily equivalent, where $n(1):=k(1)k(2)(N_{1}+1)$.

Choose a countable dense subset $\\{a_{i}\\}$ in $A$ and for $a_{1}$, choose a
unitary $u_{1}\in\mathbb{M}_{n(1)}(\mathbb{C})$ with

$\|u_{1}k(2)(N_{1}+1)\sigma_{1}(a_{1})u_{1}^{*}-(1_{B_{1}}\otimes\rho_{2}(a_{1}))\oplus
k(2)N_{1}\sigma_{1}(a_{1})\|<\frac{1}{2}.$

Put $C_{1}:=B_{1}$,
$B_{2}:=u_{1}(C_{1}\otimes\mathbb{T}_{k(2)N_{1}})u_{1}^{*}\subseteq\mathbb{M}_{n(1)}(\mathbb{C})$,
and define $\pi_{1}:C_{1}\to B_{2}$ by $\pi_{1}(c):=u_{1}(c\otimes
1_{k(2)(N_{1}+1)})u_{1}^{*}$. Put $C_{2}:=(B_{1}\otimes
A_{2})\oplus(C_{1}\otimes\mathbb{T}_{k(2)N_{1}})\subseteq\mathbb{M}_{n(1)}(\mathbb{C})$
and define $\sigma_{2}:A\to C_{2}$ by
$\sigma_{2}(a):=(1_{B_{1}}\otimes\rho_{2}(a))\oplus k(2)N_{1}\sigma_{1}(a)$,
then we can rewrite the last inequality as
$\|\pi_{1}\sigma_{1}(a_{1})-\sigma_{2}(a_{1})\|<\frac{1}{2}.$ Next, for finite
dimensional representations

$\sigma_{2}\otimes 1_{k(3)}=k(3)\sigma_{2}:A\to
C_{2}\otimes\mathbb{T}_{k(3)}\subseteq\mathbb{M}_{k(1)k(2)k(3)(N_{1}+1)}(\mathbb{C}),$

and

$1_{B_{2}}\otimes\rho_{3}:A\to B_{2}\otimes
A_{3}\subseteq\mathbb{M}_{k(1)k(2)k(3)(N_{1}+1)}(\mathbb{C}),$

choose positive integer $N_{2}\geq 1$ such that

$k(3)(N_{2}+1)\sigma_{2}:A\to
C_{2}\otimes\mathbb{T}_{k(3)(N_{2}+1)}\subseteq\mathbb{M}_{n(2)}(\mathbb{C}),$

and

$(1_{B_{2}}\otimes\rho_{3})\oplus k(3)N_{2}\sigma_{2}:A\to(B_{2}\otimes
A_{3})\oplus(C_{2}\otimes\mathbb{T}_{k(3)N_{2}})\subseteq\mathbb{M}_{n(2)}(\mathbb{C}),$

are approximately unitarily equivalent, where
$n(2):=k(1)k(2)k(3)(N_{1}+1)(N_{2}+1)$. Choose a unitary
$u_{2}\in\mathbb{M}_{n(2)}(\mathbb{C})$ with

$\|u_{2}k(3)(N_{2}+1)\sigma_{2}(a_{j})u_{2}^{*}-(1_{B_{2}}\otimes\rho_{3}(a_{j}))\oplus
k(3)N_{2}\sigma_{2}(a_{j})\|<\frac{1}{4},$

for $j=1,2$. Put
$B_{3}:=u_{2}(C_{2}\otimes\mathbb{T}_{k(3)N_{2}})u_{2}^{*}\subseteq\mathbb{M}_{n(2)}(\mathbb{C})$,
and define $\pi_{2}:C_{2}\to B_{3}$ by $\pi_{2}(c):=u_{2}(c\otimes
1_{k(3)(N_{2}+1)})u_{2}^{*}$. Put $C_{3}:=(B_{2}\otimes
A_{3})\oplus(C_{2}\otimes\mathbb{T}_{k(3)N_{2}})\subseteq\mathbb{M}_{n(2)}(\mathbb{C})$
and define $\sigma_{3}:A\to C_{3}$ by
$\sigma_{3}(a):=(1_{B_{2}}\otimes\rho_{3}(a))\oplus k(3)N_{2}\sigma_{2}(a)$,
then we can rewrite the last inequality as
$\|\pi_{2}\sigma_{2}(a_{j})-\sigma_{3}(a_{j})\|<\frac{1}{4},$ for $j=1,2$.

Since all the algebras involved are unital, we may identify both $B_{n}$ and
$C_{n}$ with a subalgebra of $C_{n+1}$. Continuing this way, we get finite
dimensional triangular algebras $C_{n}$, injective homomorphisms
$\pi_{n}:C_{n}\to C_{n+2}$ and u.c.c. maps $\sigma_{n}:A\to C_{n}$, satisfying
the conditions of the previous lemma (with $k=2$). Therefore, $A$ is TAF-
embeddable. ∎

######  Remark 3.2.

$(i)$ If the condition of approximate unitary equivalence in Proposition 3.19
is replaced with genuine unitary equivalence, the statement becomes trivial
(at least for the case of C*-algebras, c.f., [13, Exercice 8.1.1]).

$(ii)$ Let $\mathfrak{F}\subseteq A$ be a finite subset and $\varepsilon>0$. A
finite dimensional representation $\sigma:A\to\mathbb{M}_{k}(\mathbb{C})$ is
$(\mathfrak{F},\varepsilon)$-admissible if there is a faithful essential
representation $\pi:A\to\mathbb{B}(H)$ on a separable Hilbert space $H$, and a
unitary $u:\oplus_{1}^{\infty}\ell^{2}_{k}\to H$, such that
$\|u\pi(a)u^{*}-\sigma_{\infty}(a)\|<\varepsilon,\ (a\in\mathfrak{F})$,
writing $\pi\approx_{(\mathfrak{F},\varepsilon)}\sigma_{\infty}$. Let us
observe that in the proof of Proposition 3.14, in each step we need to know
that the finite dimensional representations involved are homotopic and the one
added with large multiplicity is $(\mathfrak{F},\varepsilon)$-admissible, for
a suitable choice of $(\mathfrak{F},\varepsilon)$. Indeed, for two homotopic
representations $\sigma_{1}$ and $\sigma_{2}$ with $\sigma_{1}$ being
$(\mathfrak{F},\varepsilon)$-admissible, there in $N\geq 1$ with
$(N+1)\sigma_{1}\approx_{(\mathfrak{F},3\varepsilon)}\sigma_{2}\oplus\sigma_{1}$
(this is proved verbatim to [13, Theorem 8.1.8]).

######  Remark 3.3.

$(i)$ We warn the reader that the last condition on the range of c.c.
homomorphisms $\rho_{i}$ is a rather strong condition, and an RFD
$C^{*}$-algebra is not necessarily TRFD as an operator algebra (unless it is
commutative). Also, even if $A$ is TRFD and all representations $\rho_{i}$ are
boundary representations, then the enveloping $C^{*}$-algebra $C^{*}_{e}(A)$
may fail to be is RFD (though each $\rho_{i}$ uniquely extends to a finite
dimensional representation $\tilde{\rho}_{i}$ of $C^{*}_{e}(A)$, here is no
way to guarantee that $\tilde{\rho}_{i}$’s are asymptotically isometric).

$(ii)$ As an example of a RFD operator algebra, let $\\{P_{n}\\}$ be an
increasing sequence of finite rank projections in $\mathbb{B}(H)$, SOT-
converging to the identiti. Consider,

$A:=\\{(T_{n})\in\prod_{n\geq 1}P_{n}\mathbb{B}(H)P_{n}:\ {\rm there\ exists}\
T\in\mathbb{B}(H)\ {\rm with}\ T_{n}\xrightarrow[]{\text{SOT}}T\\}.$

Then $A$ is norm closed: Given $\\{(T_{n}^{k})\\}_{k}\subseteq A$, if
$(T_{n}^{k})\to(T_{n})$ in $\prod_{n\geq 1}P_{n}\mathbb{B}(H)P_{n}$, as
$k\to\infty$, choose $T^{k}\in\mathbb{B}(H)$ with
$T_{n}^{k}\xrightarrow[]{\text{SOT}}T^{k}$, as $n\to\infty$, then since
$\|T_{n}^{k}-T_{n}\|\to 0$, uniformly on $n$, as $k\to\infty$, a standard
triangle inequality argument shows that $\\{T^{k}\\}$ is SOT-Cauchy, and so
there is $T\in\mathbb{B}(H)$ with $T^{k}\xrightarrow[]{\text{SOT}}T$. Again
using the above uniform convergence in norm, another triangle inequality
argument shows that $T_{n}\xrightarrow[]{\text{SOT}}T$, that is, $(T_{n})\in
A$. Since SOT is jointly sequentially continuous [43, Page 136], $A$ is an
operator algebra (but obviously not a C*-algebra). Finally, $A$ is RFD, since
for $k(i):={\rm rank}(P_{i})$, the sequence of c.c. homomorphisms,

$\rho_{i}:A\to\mathbb{M}_{k(i)}(\mathbb{C});\ \ (T_{n})\mapsto T_{i},\ \
(i\geq 1),$

is clearly asymptotically isometric.

$(iii)$ It is not hard to modify the above example such that $A$ is also TRFD:
Let $\mathcal{T}_{n}\subseteq\mathbb{M}_{k(n)}(\mathbb{C})$ be a (maximal)
triangular subalgebra. Let,

$A:=\\{(T_{n})\in\prod_{n\geq 1}\mathcal{T}_{n}:\ {\rm there\ exists}\
T\in\mathbb{B}(\bigoplus_{n}\ell^{2}_{k(n)})\ {\rm with}\
T_{n}\xrightarrow[]{\text{SOT}}T\\}.$

Then $A$ is a TRFD operator algebra.

Let $A$ be a unital operator algebra and $\pi:A\to\mathbb{B}(H)$ be a unital
completely contractive map. Let
$q:\mathbb{B}(H)\to\mathbb{B}(H)/\mathbb{K}(H)$ be the quotient map onto the
Calkin algebra. We say that $\pi$ is a (faithful) representation modulo the
compacts if $q\circ\pi:A\to\mathbb{B}(H)/\mathbb{K}(H)$ is a (completely
isometric) completely contractive homomorphism.

######  Definition 3.4.

An operator algebra $A\subseteq\mathbb{B}(H)$ is called essentially closed if
its image $q(A)$ in the Calkin algebra is norm closed. This is equivalent to
requiring that $A+\mathbb{K}(H)$ to be closed in $\mathbb{B}(H)$.

######  Example 3.5.

$(i)$ $C^{*}$-algebras are automatically essentially closed, since the range
of $*$-homomorphisms on $C^{*}$-algebras are closed. More generally, if $B$ is
a $C^{*}$-algebra, $A$ is a $C^{*}$-subalgebra and $J$ is a closed ideal, then
$A+J$ is closed in $B$ [24, 1.8.4].

$(ii)$ It was shown by Arveson the discrete triangular algebra is essentially
closed: every set $\mathcal{P}$ of projections in $\mathbb{B}(H)$ determines a
weakly closed unital algebra alg$\mathcal{P}$, consisting of all operators
$T\in\mathbb{B}(H)$ satisfying $(1-P)TP=0$, for every $P\in\mathcal{P}$. Given
an increasing sequence $\\{P_{n}\\}$ of finite rank projections such that
$P_{n}\uparrow 1$ in SOT, $\mathcal{T}:=$alg$\\{P_{n}\\}$ is essentially
closed [6, Proposition 2.1]. Indeed, $\mathcal{T}+\mathbb{K}(H)=\mathcal{QT}$,
consisting of all quasitriangular operators w.r.t. $\\{P_{n}\\}$ [6, Corollary
of Theorem 2.2].

$(ii)$ The result of Arveson is extended by Loebl and Muhly to all nest
algebras [41].

$(iii)$ The algebra of subnormal operators and Toeplitz algebra are also known
to be essentially closed [22].

$(iv)$ There is a commutative weakly closed operator algebra $A$ such that
$A+\mathbb{K}(H)$ is not closed [22, Proposition 2]: let $\\{e_{n}\\}$ be a
basis for $\ell^{2}(\mathbb{Z})$. Let $U$ be the bilateral shift $P$ be the
rank one projection onto $\mathbb{C}e_{0}$. Let $T=U+P$ and let $A$ be the
weakly closed algebra generated by $T$, then $A\cap\mathbb{K}(H)=0$ and
$\|T^{n}\|\geq\sqrt{n}$ while $\|q(T^{n})\|=\|q(U^{n})\|=1$, and so
$A+\mathbb{K}(H)$ is not closed by Remark 3.6$(i)$.

######  Remark 3.6.

$(i)$ An operator algebra $A\subseteq\mathbb{B}(H)$ is essentially closed if
and only if $A/(A\cap\mathbb{K}(H))$ and
$q(A)=(A+\mathbb{K}(H))/\mathbb{K}(H)$ are isomorphic as Banach algebras [22,
Proposition 1]. When $A\cap\mathbb{K}(H)=0$, this is equivalent to the
quotient map $q$ being bounded below on $A$ [22, Corollary 1].

$(ii)$ The assumption than an operator algebra $A\subseteq\mathbb{B}(H)$ is
essentially closed doesn’t mean that the canonical map from
$A/(A\cap\mathbb{K}(H))$ to $(A+\mathbb{K}(H))/\mathbb{K}(H)$ is an isometry
(where as the conserve is clearly true, since complete subspaces of Banach
spaces are closed).

$(iii)$ As a special case of Example 3.5$(i)$, if $I$ and $J$ are closed two-
sided ideals of a $C^{*}$-algebra, then $I+J$. Combes and Perdrizet extended
this observation by showing that the sum of a closed left ideal and a closed
right ideal of a $C^{*}$-algebra is always closed [15, Proposition 6.2]. This
was rediscovered by Rudin [49, Example 4.6] (see also, [26, Corollary 32.2]),
and again by Kirchberg [39, Lemma 4.9(iv)], and Wassermann [61, Lemma 8.1]. In
fact, more is true: if $L$ is a closed left ideal and $R$ is a closed right
ideal in a Banach algebra $B$ such that $L$ has a bounded right approximate
identity, or $R$ has a bounded left approximate identity, then $L+R$ is
closed. If moreover, the approximate identity is bounded by 1, then the
canonical map from $L/(L\cap R)$ to $(L+R)/R$ is an isometry [25, Proposition
2.4].

$(iv)$ If $B$ is an approximately unital operator algebra (i.e., an operator
algebra with a contractive approximate identity) and $I$ and $J$ are ideals in
$B$ such that $I$ has a contractive approximate identity (cai), then $I/(I\cap
J)=(I+J)/J$, completely isometrically isomorphically. In particular, $I+J$ is
closed in $B$ [2, Theorem 2.3] (the closedness of $I+J$ also follows under the
weaker condition that $I$ is only a hereditary subalgebra with cai, where
hereditary here means that $IBI\subseteq I$ [2, Corollary 4.3]). A closed
inspection of the proof shows that a complete isometric version of part $(ii)$
above also holds here: if $B$ is an approximately unital operator algebra and
$I$ and $J$ are left and right ideals in $B$, respectively, such that $I$ has
a contractive right approximate identity, then $I/(I\cap J)=(I+J)/J$,
completely isometrically isomorphically. In particular, for $B=\mathbb{B}(H)$
and $J=\mathbb{K}(H)$, if an operator algebra $A\subseteq\mathbb{B}(H)$ is a
left ideal in $\mathbb{B}(H)$ with a contractive right approximate identity,
then the canonical map from $A/(A\cap\mathbb{K}(H))$ to
$(A+\mathbb{K}(H))/\mathbb{K}(H)$ is not only an isometry, but indeed a
complete isometry. In particular, this holds when $A$ is unital and a left
ideal (indeed following the proof, one can see that it is enough to have
$\mathbb{K}(H)A\subseteq A$), however such a condition might be too strong
(for instance when we further assume that $A\cap\mathbb{K}(H)=0$, then
$\mathbb{K}(H)A\subseteq A$ could not hold unless $A=0$).

$(v)$ It is not true that if the restriction of the quotient map
$q:\mathbb{B}(H)\to\mathbb{B}(H)/\mathbb{K}(H)$ to an operator algebra
$A\subseteq\mathbb{B}(H)$ is a complete isometry, then the same holds for the
C*-algebra $C^{*}(A)$ generated by $A$: let $A$ be the norm-closed algebra
generated by the unilateral shift $S$ in $\mathbb{B}(\ell^{2}(\mathbb{Z}))$.
This is complete isometrically isomorphic to the disc algebra, and the image
in the Calkin algebra is the same, while
$C^{*}(S)\supseteq\mathbb{K}(\ell^{2}(\mathbb{Z}))$.

Remark 3.6$(iv)$ motivates the following stronger version of Definition 3.4.

######  Definition 3.7.

An operator algebra $A\subseteq\mathbb{B}(H)$ is called essentially embedded
if the canonical map from $A/(A\cap\mathbb{K}(H))$ to
$(A+\mathbb{K}(H))/\mathbb{K}(H)$ is a complete isometry.

######  Example 3.8.

$(i)$ All $C^{*}$-algebras are automatically essentially embedded, since
injective $*$-homomorphisms on $C^{*}$-algebras are automatically complete
isometric.

$(ii)$ The discrete triangular algebra $\mathcal{T}$ considered by Arveson is
essentially embedded [6, Proposition 2.1]: for $T\in\mathbb{B}(H)$,
$\|q(T)\|=\lim_{n}\|(1-P_{n})T(1-P_{n})\|$ by [5, Lemma 1, page 292]. Indeed,
the limit on the RHS is clearly 0 when $T$ is of rank 1, and so when $T$ is of
finite rank, and so it also happens when $T$ is compact, as the set of
operators for which RHS=0 is closed under norm. Next
$\limsup_{n}\|(1-P_{n})T(1-P_{n})\|=\limsup_{n}\|(1-P_{n})(T+K)(1-P_{i})\|\leq\|T+K\|$,
for each $K\in\mathbb{K}(H)$ and so
$\limsup_{i}\|(1-P_{n})T(1-P_{n})\|\leq\|q(T)\|$, where as,
$\|q(T)\|\leq\liminf_{n}\|(1-P_{n})T(1-P_{n})\|$, as each
$(1-P_{n})T(1-P_{n})$ is a finite rank perturbation of $T$. Now if
$T\in\mathcal{T}$ then $(1-P_{n})T(1-P_{n})=T+K_{n}$, where
$K_{n}=-P_{n}T-TP_{n}+P_{n}TP_{i}$ is a finite rank operator in $\mathcal{T}$
(this is a crucial part of the proof, as in general one cannot guarantee that
$K_{n}$ is in the given operator algebra). This implies that
$\|q(T)\|\geq\|T+(A\cap\mathbb{K}(H))\|$, while the inequality in other
direction always hold. A similar argument for amplifications shows that the
quotient map $q$ is indeed a complete isometry.

$(iii)$ If $S$ is a weak∗-closed subspace of $\mathbb{B}(H)$ such that
$S\cap\mathbb{K}(H)$ is weak∗-dense in $S$, then $S$ is essentially embedded
[20, Corollary 11.7].

$(iv)$ Any non-commutative irreducible algebra $A$ of almost normal operators
containing the identity fails to be essentially embedded (since the identity
representation on $C^{*}(A)$ is a boundary representation [5, Theorem 2.2.2],
see Remark 3.14$(iii)$).

The next lemma related the notion of essential embedding to representations
modulo the compacts.

######  Lemma 3.9.

Let $A$ be an operator algebra and the complete contraction
$\pi:A\to\mathbb{B}(H)$ be a faithful representation modulo the compacts, then
the operator algebra generated by $\pi(A)$ is essentially embedded in
$\mathbb{B}(H)$.

###### Proof.

Let $q$ be the quotient map onto the Calkin algebra on $H$. Since $\pi$ is
faithful modulo the compacts, for each $a\in A$,

$\|q(\pi(a))\|=\|a\|\geq\|\pi(a)\|,$

and as the reverse inequality is automatic, $q$ is isometric on $\pi(A)$.
Since $q$ is a continuous homomorphism, it follows that $q$ is also isometric
on the operator algebra generated by $\pi(A)$. A similar argument on
amplifications shows that $q$ is indeed a complete isometry on this operator
algebra. ∎

######  Remark 3.10.

$(i)$ Note that the converse of the above lemma is not true: if the operator
algebra generated by the range of a complete contraction
$\pi:A\to\mathbb{B}(H)$ is essentially embedded, $\pi$ may fail to be faithful
modulo the compacts (indeed, if $\pi$ is not an isometry, then so is
$q\circ\pi$).

$(ii)$ One could use the above lemma to construct examples of operator
algebras which are essentially embedded and QT.

$(iii)$ More examples could be constructed using nest algebras as follows: Let
$\mathcal{P}:=\\{P_{n}H\\}$ be the nest consisting of the ranges of finite
rank projection SOT-increasing to the identity. Let $\mathcal{T}(\mathcal{P})$
and $\mathcal{QT}(\mathcal{P})$ be the set of operators $T$ with
$(1-P_{n})TP_{n}=0$ and $(1-P_{n})TP_{n}\to 0$, respectively. It follows from
[20, Theorem 12.2] that
$\mathcal{QT}(\mathcal{P})=\mathcal{T}(\mathcal{P})+\mathbb{K}(H)$ (we warn
the reader that $\mathcal{T}(\mathcal{N})+\mathbb{K}(H)$ need not consist of
quasitriangular operators, for a general nest $\mathcal{N}$; c.f., [20,
Chapter 21]). By [20, Theorem 12.1], $\mathcal{QT}(\mathcal{P})$ is norm
closed, and the quotient map:

$\sigma:\mathcal{T}(\mathcal{P})/(\mathcal{T}(\mathcal{P})\cap\mathbb{K}(H))\to(\mathcal{T}(\mathcal{P})+\mathbb{K}(H))/\mathbb{K}(H)$

is isometric (Indeed, $\mathcal{T}(\mathcal{P})$ is weak∗-closed and by the
Erdös Density Theorem [20, Theorem 3.11],
$\mathcal{T}(\mathcal{P})\cap\mathbb{K}(H)$ is weak∗ dense, and so an M-ideal
in $\mathcal{T}(\mathcal{P})$ [20, Theorem 11.5, Corollary 11.6] and so the
quotient map is isometric by [20, Theorem 11.3]). Repeating this argument for
the inflation $\sigma^{(n)}$ for the Hilbert space $H\otimes\ell^{2}_{n}$, we
get that $\sigma$ is indeed completely isometric). Thus
$\mathcal{T}(\mathcal{P})$ is essentially embedded. Since
$\mathcal{QT}(\mathcal{P})+\mathbb{K}(H)=\mathcal{T}(\mathcal{P})+\mathbb{K}(H)$
and
$\mathcal{QT}(\mathcal{P})\cap\mathbb{K}(H)=\mathcal{T}(\mathcal{P})\cap\mathbb{K}(H)$,
it follows that $\mathcal{QT}(\mathcal{P})$ is also essentially embedded. Now,
$\mathcal{QT}(\mathcal{P})$ is clearly QT, but it is not triangular or block
triangular, as it contains all compact operators.

######  Lemma 3.11.

Let $A\subseteq\mathbb{B}(H)$ be an essentially embedded operator algebra with
$A\cap\mathbb{K}(H)=0$. Then each c.c. representation $\pi:A\to\mathbb{B}(K)$
extends to c.c. representation $\bar{\pi}:A+\mathbb{K}(H)\to\mathbb{B}(K)$
which is zero on $\mathbb{K}(H)$.

###### Proof.

Since $A$ is essentially embedded and $A\cap\mathbb{K}(H)=0$, there is a
surjective c.i. homomorphism $\sigma:(A+\mathbb{K}(H))/\mathbb{K}(H)\to A$.
The composition $\bar{\pi}:=\pi\circ\sigma\circ
q:A+\mathbb{K}(H)\to\mathbb{B}(K)$ has all the required properties. ∎

The next result is an slight extension of Voiculescu-Weyl-von Neumann theorem
to separable operator algebras.

######  Lemma 3.12.

Let $A\subseteq\mathbb{B}(H)$ be a separable unital essentially embedded
operator algebra and $\pi:A\to\mathbb{B}(K)$ be a u.c.c. map with
$\pi(A\cap\mathbb{K}(H))=0$. Then there are isometries $V_{n}:K\to H$ with
$\pi(a)-V_{n}^{*}aV_{n}\in\mathbb{K}(K)$, for each $a\in A$ and $n\geq 1$,
such that ad${}_{V_{n}}\to\pi$, point-norm on $A$.

###### Proof.

By Lemma 3.11, we get a c.c. representation
$\bar{\pi}:A+\mathbb{K}(H)\to\mathbb{B}(K)$ which vanishes on $\mathbb{K}(H)$.
By [4, 1.2.3, 1.2.8], $\bar{\pi}$ extends to a completely positive map
$\tilde{\pi}:\mathbb{B}(H)\to\mathbb{B}(K)$, still vanishing on
$\mathbb{K}(H)$. Thus without loss of generality we may assume that $A$
contains $\mathbb{K}(H)$ and $\pi$ vanishes on $\mathbb{K}(H)$. Next, the
$C^{*}$-algebra $B$ generated by the operator system $\overline{A+A^{*}}$ in
$\mathbb{B}(H)$ is a separable $C^{*}$-algebra containing $\mathbb{K}(H)$
satisfying $\tilde{\pi}(B\cap\mathbb{K}(H))=\tilde{\pi}(\mathbb{K}(H))=0$. By
the Voiculescu-Weyl-von Neumann theorem (c.f. [21, II.5.3]), there are
isometries $V_{n}:K\to H$ with
$\tilde{\pi}(b)-V_{n}^{*}bV_{n}\in\mathbb{K}(K)$, for each $b\in B$ and $n\geq
1$ such that ad${}_{V_{n}}\to\tilde{\pi}$, point-norm on $B$. The same
isometries do the job for $A$ as well. ∎

######  Lemma 3.13.

Let $A\subseteq\mathbb{B}(H)$ be a separable operator algebra and let
$(\pi,H_{\pi})$ and $(\sigma,H_{\sigma})$ be representations of $A$ such that
there are partial isometries $V_{n}:H_{\pi}\to H_{\sigma}$ with image
projection 1 such that $\|\sigma(a)V_{n}-V_{n}\pi(a)\|\to 0$, for $a\in A$.
Then if $\sigma(A)$ is a QT set of operators, so is $\pi(A)$.

###### Proof.

By definition, for finite subsets $F_{n}\subseteq H_{\sigma}$ and
$\mathfrak{F}_{n}\subseteq A$, there are finite rank projections $P_{n}\in
B(H_{\sigma})$ such that

$\|P_{n}\sigma(a)-P_{n}\sigma(a)P_{n}\|\approx 0,\ \|P_{n}\xi\|\approx\|\xi\|\
\ \ (a\in\mathfrak{F}_{n},\xi\in F_{n}).$

Given $G_{n}\subseteq H_{\pi}$, let $F_{n}:=V_{n}(G_{n})$ and
$Q_{n}:=V_{n}^{*}P_{n}V_{n}$. Then each $Q_{n}\in B(H_{\pi})$ is a projection,
since $V_{n}V_{n}^{*}=1$, and it is of finite rank, since $P_{n}$ is so. Also,

$\displaystyle\|Q_{n}\pi(a)-Q_{n}\pi(a)Q_{n}\|$
$\displaystyle=\|V_{n}^{*}P_{n}V_{n}\pi(a)-V_{n}^{*}P_{n}V_{n}\pi(a)V_{n}^{*}P_{n}V_{n}\|$
$\displaystyle=\|V_{n}^{*}P_{n}\sigma(a)V_{n}-V_{n}^{*}P_{n}\sigma(a)P_{n}V_{n}\|$
$\displaystyle\leq\|P_{n}\sigma(a)-P_{n}\sigma(a)P_{n}\|\approx 0,$

as required. ∎

Now we are ready to prove the analog of Voiculescu theorem for QT algebras.
The original proof (c.f. [13, 7.2.5]) is based on Voiculescu constants

$\eta_{\pi}(a):=2{\rm
max}\big{(}\|\pi(a^{*}a)-\pi(a^{*})\pi(a)\|^{\frac{1}{2}},\|\pi(aa^{*})-\pi(a)\pi(a^{*})\|^{\frac{1}{2}}\big{)},$

defined for faithful representations $\pi$ modulo the compacts for the case of
$C^{*}$-algebras (c.f. [13, 1.7.6]). This argument is obviously not usable for
operator algebras, and instead we employ the above lemma (which gives an
alternative way, also suggested by the work of Voiculescu). Also note that a
proof by reduction to the $C^{*}$-algebra generated by the given operator
algebra does not seem to be possible, as it is not the case that if an
operator algebra is QT as a family of operators, the it generated a
$C^{*}$-algebra which is a QD family of operators (this is because the notion
of QT families defined by Halmos is an upper quasitriangular family which
switches to a lower quasitriangular family, when taking adjoint).

###### Proof of Theorem 2.5.

$(i)\Rightarrow(iii).$ Since $A$ is separable, we may choose a sequence of
completely contractive maps $\phi_{n}:A\to\mathbb{M}_{k_{n}}(\mathbb{C})$
which are asymptotically isometric and asymptotically multiplicative, that is,

$\|\phi_{n}(a)\|\approx\|a\|,\ \
\|\phi_{n}(ab)-\phi_{n}(a)\phi_{n}(a)\|\approx 0,$

for $a\in F_{n}$, where $1\in F_{1}\subseteq F_{2}\subseteq\cdots$ is an
increasing sequence of finite subsets of $A$ with dense union. Take any
faithful unital essential representation $\pi:A\to\mathbb{B}(H)$ in a
separable Hilbert space $H$. Consider the faithful representation modulo the
compacts
$\Phi:=\oplus\phi_{n}:A\to\prod_{n=1}^{\infty}\mathbb{M}_{k_{n}}(\mathbb{C})$,
which is a u.c.c. map. Compose $\Phi$ with the quotient map $q$ from
$B(\bigoplus_{n=1}^{\infty}\ell^{2}_{k(n)})$ onto its Calkin algebra and embed
the latter in $\mathbb{B}(K)$, for some Hilbert space $K$, to get a u.c.c. map
$\Pi:=q\circ\Phi:A\to\mathbb{B}(K)$. Since $\pi$ is essential, we may assume
that $A\subseteq\mathbb{B}(H)$ with $\pi(A\cap\mathbb{K}(H))=0$. By Lemma 3.12
(switching the role of $V_{n}$ and $V_{n}^{*}$), there are partial isometries
$V_{n}:H\to K$ with range projection 1 such that
$\Pi(a)-V_{n}\pi(a)V_{n}^{*}\in\mathbb{K}(K)$, for each $a\in A$ and $n\geq
1$, and

$\|\Pi(a)-V_{n}\pi(a)V_{n}^{*}\|\approx 0,\ \ (a\in A).$

By Lemma 3.13, it is enough to show that $\Pi(A)$ is a QT set of operators,
which is in turn follows if we show that $\Phi(A)$ is a QT set of operators
(since $q$ is norm decreasing). Put
$X:=\bigoplus_{n=1}^{\infty}\ell^{2}_{k(n)}$ and let $X^{(k)}$ and
$X^{(\infty)}$ be the direct sum of $k$ and countably many copies of $X$ with
itself, respectively. Consider representations $\Phi^{(k)}$ and
$\Phi^{(\infty)}$ of $A$ on $X^{(k)}$ and $X^{(\infty)}$. Let $P_{n}$ be the
projection in $X$ onto $\ell^{2}_{k(n)}$, and not that the relations of
asymptotic isometry and asymptotic multiplicativity could be rewritten in
terms of projections $P_{n}$ as

$\|P_{n}\Phi(a)P_{n}\|\approx\|a\|,\ \
\|P_{n}\Phi(a)-P_{n}\Phi(a)P_{n}\|\approx 0,$

for $a\in F_{n}$. Let $X_{k}:=\bigoplus_{n=k}^{\infty}\ell^{2}_{k(n)}$and let
$Q_{k}$ be the orthogonal projection onto $X_{k}$. Then

$[Q_{k},\Phi(a)]=\bigoplus_{n\geq k}[P_{n},\phi_{n}(a)]\in\mathbb{K}(X),$

for $a\in\cup_{n}F_{n}$, and so by density for any $a\in A$. Now since ${\rm
ad}_{Q_{n}}\circ\Phi:A\to B(X)$ is a faithful representation modulo the
compacts and $A$ is essentially embedded, then applying Lemma 3.12 again,
there are partial isometries $U_{n}:X^{(k)}\to X$ with $\Im(V_{n})\subseteq
X_{n}$ such that $\|U_{n}\Phi^{(k)}(a)-\Phi(a)U_{n}\|\approx 0$, for $a\in A$.
To see that $\Phi(A)$ is a QT set of operators, without loss of generality we
may consider the finite subsets $\Phi(F_{n})\subseteq\Phi(A)$ and $F\subseteq
X^{(k)}$ and first choose $N_{n}\geq 1$ such that for the orthogonal
projection $\tilde{Q}_{n}$ onto $\bigoplus_{i=n}^{N-n}\ell^{2}_{k(i)}$,

$\|\tilde{Q}_{n}\Phi(a)-\tilde{Q}_{n}\Phi(a)\tilde{Q}_{n}\|\approx 0,\ \
\|(1-\tilde{Q}_{n})U_{n}\xi\|\approx 0\ \ (a\in F_{n},\xi\in F).$

Next, let $W_{k}:X\to X^{(\infty)}$ be the isometry mapping $X$ onto its
$(k+1)$-copy in $X^{(\infty)}$ and put $\tilde{W}_{n}:=U_{n}^{*}\oplus
W_{n}(1-U_{n}U_{n}^{*})$, and observe that
$\|\tilde{W}_{n}\Phi(a)-\Phi^{(\infty)}(a)\tilde{W}_{n}\|\approx 0$, for $a\in
A$. Finally, put $\tilde{P}_{n}:=\tilde{W}_{n}\tilde{Q}_{n}\tilde{W}_{n}^{*}$,
and observe that

$\|\tilde{P}_{n}\Phi(a)-\tilde{P}_{n}\Phi(a)\tilde{P}_{n}\|\leq\|\tilde{Q}_{n}\Phi(a)-\tilde{Q}_{n}\Phi(a)\tilde{Q}_{n}\|\approx
0,$

and

$\|\tilde{P}_{n}\xi\|\leq\|\tilde{Q}_{n}U_{n}\xi\|\approx\|\xi\|,$

for $a\in F_{n}$ and $\xi\in F$, as required.

$(iii)\Rightarrow(ii).$ This is immediate.

$(ii)\Rightarrow(i).$ If $\pi:A\to\mathbb{B}(H)$ is a faithful QT
representation on a separable Hilbert space and $P_{n}$’s are finite rank
projections with $P_{n}\uparrow 1$ in SOT, and $\|P_{n}a-P_{n}aP_{n}\|\to 0$,
for each $a\in A$, as $n\rightarrow\infty$. If $k(n)$ is the rank of $P_{n}$,
we identify $P_{n}\mathbb{B}(H)P_{n}$ with $\mathbb{M}_{k(n)}(\mathbb{C})$ and
regard $\phi_{n}(a):=P_{n}aP_{n}$ as a map from $A$ into
$\mathbb{M}_{k(n)}(\mathbb{C})$. Then

$\displaystyle\|\phi_{n}(ab)-\phi_{n}(a)\phi_{n}(b)\|$
$\displaystyle=\|P_{n}abP_{n}-P_{n}aP_{n}bP_{n}\|$
$\displaystyle\leq\|P_{n}ab-P_{n}aP_{n}b\|$
$\displaystyle\leq\|P_{n}a-P_{n}aP_{n}\|\|b\|\approx 0,$

for each $a,b\in A$. Also,

$\|\phi_{n}(a)-a\|=\|P_{n}aP_{n}-a\|\approx\|P_{n}aP_{n}-P_{n}a\|\approx 0,$

for each $a\in A$. ∎

######  Remark 3.14.

$(i)$ A boundary representation of a unital operator algebra $A$ consists of a
completely isometric homomorphism $\phi:A\to C$, where $C$ is a C*-algebra and
$C^{*}(\phi(A))=C$, together with a representation $\pi:C\to\mathbb{B}(H)$
such that the only completely positive map on $C$ agreeing with $\pi$ on
$\phi(A)$ is $\pi$ itself. (In the original definition by Arveson, boundary
representations are also assumed to be irreducible, but this condition was
dropped later). It immediately follows that in Theorem 2.5, in part $(iii)$
“essential representation” could be replaced by “essential boundary
representation”. Furthermore, since every separable operator algebras is known
to posses a faithful boundary representation [7], [39] the same replacement
could be done in part $(ii)$. This observation is for those who prefer to work
with homomorphisms with extension to $*$-homomorphisms on the $C^{*}$-algebra
generated by the image of $A$. The boundary representations appear in the
study of the noncommutative Choquet boundary (the peak points of the Shilov
boundary) of operator systems [7]. Arveson showed that unital operator
algebras have enough boundary representations to generate the enveloping
$C^{*}$-algebras. This is extended to operator systems by Ken Davidson and
Matt Kennedy [23], and reconfirmed by a result of Dritschel and McCullough who
showed (dropping the irreducibility condition of Arveson) that any family of
representations of an operator algebra or an operator space (in Agler’s sense)
has boundary representations [28], and thereby gave a direct proof of
Arveson’s result (compare to the results in [33]).

$(ii)$ One might suspect that since boundary representations extend to
representations of an enveloping $C^{*}$-algebra, an indirect proof of the
above result is possible for such representations by extending and using
Voiculescu theorem for the enveloping $C^{*}$-algebra, but this is not
possible, since the the enveloping $C^{*}$-algebra of a QT operator algebra is
not necessarily QD. Also, the extension of a faithful (resp., essential)
boundary representation need not be faithful (resp., essential).

$(iii)$ The boundary theorem of Arveson relates boundary representations to
essentially embedded (non closed) operator systems as follows: if $S$ is an
irreducible set of operators on a Hilbert space $H$ containing the identity
and the enveloping $C^{*}$-algebra $C_{e}^{*}(S)$ contains $\mathbb{K}(H)$.
Then the identity representation of $C_{e}^{*}(S)$ is a boundary
representation on $S$ if and only if, the quotient map
$q:\mathbb{B}(H)\to\mathbb{B}(H)/\mathbb{K}(H)$ is not completely isometric on
the linear span of $S\cup S^{*}$ [5, Theorem 2.1.1].

To prove Proposition 2.6 we need a bit of preparation. The proof of the next
lemma is an easy adaptation of that of [12, Lemma 15.3], which is given here
just for the sake of completeness.

######  Lemma 3.15.

Let $A$ be a separable operator algebra with a faithful representation with QT
range (as a set of operators), then $A$ has such a representation on a
separable Hilbert space.

###### Proof.

We shall prove a bit more by showing that for each faithful representation
$\pi:A\to\mathbb{B}(H)$ with QT range, $H$ has a $\pi$-invariant separable
subspace $K$ such that the corresponding sub-representation $\pi^{K}$ has a QT
range. We do this by constructing an increasing sequence $(K_{n})$ of
separable $\pi$-invariant subspaces of $H$ and finite rank projections $Q_{n}$
with range inside $K_{n+1}$ such that $\|Q_{n}\pi(a)-Q_{n}\pi(a)Q_{n}\|\to 0$,
as $n\to\infty$ and $Q_{n}\to 1$ (SOT) on the union of $K_{n}$’s.

By separability of $A$, we may choose a dense sequence $(a_{i})$ in the unit
ball of $A$ and unit vectors $\xi_{i,n}$ with
$\|\pi(a_{i})\xi_{i,n}\|>\|a_{i}\|-2^{-n}$, for each $i,n$. Let $H_{1}$ be the
span closure of the set of all vectors $\xi_{i,n}$ and $K_{1}$ be the closure
of $\pi(A)H_{1}$. This is a separable $\pi$-invariant subspace of $H$ and the
sub-representation $\pi^{K_{1}}$, being isometric on the sequence $(a_{i})$,
is faithful. Let $(e_{1,k})$ be an ONB for $K_{1}$, then by assumption there
is a finite rank projection $Q_{1}$ in $\mathbb{B}(H)$ with
$\|Q_{1}\pi(a_{1})-Q_{1}\pi(a_{1})Q_{1}\|<2^{-1}$ and
$\|Q_{1}(e_{1,1})-e_{1,1}\|$ as small as we wish. Indeed by an argument
similar to that of [12, Proposition 3.4], we may also arrange for equality
$Q_{1}(e_{1,1})=e_{1,1}$ to hold. Let’s do so and take $H_{2}$ the be the span
closure of Im$(Q_{1})\cup K_{1}$ and $K_{2}$ be the closure of $\pi(A)H_{2}$.
Let $(e_{2,n})$ be an ONB for $K_{2}$ and choose a finite rank projection
$Q_{2}$ in $\mathbb{B}(H)$ with
$\|Q_{2}\pi(a_{i})-Q_{2}\pi(a_{i})Q_{2}\|<2^{-2}$, for $i=1,2$, such that
$Q_{2}$ fixes a finite ONB of Im$(Q_{1})$ (which means that $Q_{2}\geq
Q_{1}$). Proceeding inductively, we get an increasing sequence of separable
$\pi$-invariant subspaces $K_{n}$ with ONB $(e_{n,k})$ and increasing sequence
of finite rank projections $Q_{n}$ with range inside $K_{n+1}$ satisfying
$Q_{n}(e_{i,j})=e_{i,j}$, for $1\leq i,j\leq n$, and
$\|Q_{n}\pi(a_{i})-Q_{n}\pi(a_{i})Q_{n}\|<2^{-n}$, for $1\leq i\leq n$. Now
the closure $K$ of the union of $K_{n}$’s is a separable $\pi$-invariant
subspace of $H$ and the range of the corresponding (faithful) sub-
representation $\pi^{K}$ is QT. ∎

Now let $A$ be a separable operator algebra and $\pi:A\to\mathbb{B}(H)$ be a
faithful representation. Then by the above lemma and an argument as in the
proof of [12, Lemma 15.4] there exists a separable $\pi$-invariant subspace
$K$ of $H$ such that $\pi^{K}$ is faithful with $\pi^{K}(a)$ finite rank if
and only if $\pi(a)$ is so (with equal ranks). This could be used to make
another useful observation in the spirit of the previous lemma: for each
faithful representation $\pi:A\to\mathbb{B}(H)$, if $L$ is the span closure of
$\pi(A)H$, then $\pi$ has a QT range (as a set of operators) iff its non-
degenerate sub-representation $\pi^{L}$ is so (it is wise to handle the unital
case first, c.f. [12, Lemma 15.5]).

Next take any faithful essential representation $\pi:A\to\mathbb{B}(H)$. Since
quasitriangularity of a set of operators is defined via approximation on
finite sets, the range of $\pi$ is a QT set of operators if $\pi(B)$ is QT for
each separable operator subalgebra $B$ of $A$. Moreover, when $H$ is non
separable, for each finite subset $F\subseteq H$, by the first observation in
the last paragraph, there is a separable $\pi$-invariant subspace $K$ of $H$
containing $F$ such that the corresponding sub–representation $\pi^{K}$ is
faithful and essential. This plus Theorem 2.5 proves the next lemma, which
shows that part of Theorem 2.5 also holds for the non separable case. This is
recorded in the next lemma.

######  Lemma 3.16.

Let $A$ be a QT operator algebra then the range of each faithful essential
representation of $A$ is QT set of operators.

###### Proof of Proposition 2.6.

The necessity is obvious. To prove the sufficiency, let
$\pi:A\to\mathbb{B}(H)$ be a faithful essential representation. Then for each
finitely generated subalgebra $B$ of $A$, the restriction of $\pi$ to $B$ is a
faithful essential representation and so by Lemma 3.16, the image of $B$ under
$\pi$ is a QT set of operators. Since this holds for each finitely generated
subalgebra, it immediately follows from the definition that $\pi(A)$ is also
QT. ∎

In the last stage of the above proof, note that although $B$ is separable, we
still need to appeal to the above lemma as $H$ is not assumed to be separable.

The next proof is an adaptation of an argument by Arveson [8, Theorem 2].

###### Proof of Theorem 2.8.

We only need to show the necessity, since $B+\mathbb{K}(H)$ is a
quasitriangular set of operators. We may assume that $F$ is contained in the
unit ball of $A$. Take an increasing sequence $F\subseteq F_{1}\subseteq
F_{2}\subseteq F_{3}\subseteq\dots$ of finite sets with dense union in the
unit ball of $A$. Again by Theorem 2.5, $A$ is a quasitriangular set of
operators and there is an increasing sequence of finite rank projections
$P_{n}$ converging to 1 (SOT) such that $\|P_{n}a-P_{n}aP_{n}\|\to 0$ for each
$a\in A$. Passing to a subsequence, we let
$\|P_{n}a-P_{n}aP_{n}\|<\varepsilon/2n,$ for $a\in F_{n}$. Put $P_{0}=0$ and
let $E_{n}=P_{n}-P_{n-1}$, then $\sum_{n}E_{n}=1$ in SOT and
$a\mapsto\delta(a):=\sum_{n}E_{n}aE_{n}$ is a complete contraction on $A$
satisfying $a-\delta(a)=\sum_{n}(aE_{n}-E_{n}aE_{n})$ in SOT. Let $B$ be the
operator algebra generated by the image of $\delta$ in $\mathbb{B}(H)$, which
is clearly a block triangular algebra. When $a\in F_{n}$ for some $n\geq 1$,
the last sum also converges in the norm and $a-\delta(a)\in\mathbb{K}(H)$. By
density of the union of $F_{n}$’s and norm continuity of $\delta$,
$a-\delta(a)\in\mathbb{K}(H)$ for each $a\in A$. This in particular shows that
$A$ is essentially closed iff $B$ is so. Since $a-\delta(a)\in\mathbb{K}(H)$,
$\delta$ induces a well-defined linear map

$\tilde{\delta}:(A+\mathbb{K}(H))/\mathbb{K}(H)\to(B+\mathbb{K}(H))/\mathbb{K}(H);\
\ a+\mathbb{K}(H)\mapsto\delta(a)+\mathbb{K}(H),$

which is contractive, since

$\|a+k\|=\|\delta(a)+(k+a-\delta(a))\|\geq\|\delta(a)+\mathbb{K}(H)\|,$

for each $k\in\mathbb{K}(H)$, thus taking infimum over $k$,
$\|a+\mathbb{K}(H)\|\geq\|\delta(a)+\mathbb{K}(H)\|$. Indeed, $\tilde{\delta}$
is an isometry, as the above argument could be symmetrically repeated the
other way around. Now, by the same argument applied to amplifications of
$\delta$, $\tilde{\delta}$ is also a complete isometry. Also, $\delta$ is a
homomorphism modulo compacts, since

$\displaystyle\delta(ab)-\delta(a)\delta(b)$
$\displaystyle=(\delta(ab)-ab)-(\delta(a)-a)\delta(b)-\delta(a)(\delta(b)-b)$
$\displaystyle+(\delta(a)-a)(\delta(b)-b)\in\mathbb{K}(H),$

for each $a,b\in A$, therefore, $\tilde{\delta}$ is a homomorphism.

Now if $A$ is essentially embedded and meets compacts trivially, the
restriction of the quotient map
$q:\mathbb{B}(H)\to\mathbb{B}(H)/\mathbb{K}(H)$ to $A$ is a complete isometry.
On the other hand, since $B$ is block-triangular, it is essentially embedded
by Example 3.7$(ii)$, therefore, the canonical map
$q_{B}:B/(B\cap\mathbb{K}(H))\to(B+\mathbb{K}(H))/\mathbb{K}(H)$ is a complete
isometry, and so is the composition map
$q_{B}^{-1}\circ\tilde{\delta}\circ(q|_{A}):A\to B/(B\cap\mathbb{K}(H))$. ∎

Two operator algebras $A$ and $B$ are homotopic if there are homomorphisms
$\phi:A\to B$ and $\psi:B\to A$ such that $\phi\circ\psi$ is homotopic to the
identity on $B$ and $\psi\circ\phi$ is homotopic to the identity on $A$. It is
desirable to show that quasitriangularity passes from one operator algebra to
the other when they are homotopic, but at this point it is not clear if the
argument in [56, Theorem 5] passes to non selfadjoint operator algebras. In
particular, it is not known to us if contractible operator algebras–i.e.,
those homotopic to the zero algebra–are quasitriangular.

Fortunately, this could be taken care of for cones. The notions of the cone
and suspension could be defined for an operator algebra $A$ by
$CA:=C_{0}(0,1]\otimes A$ and since $C_{0}(0,1]$ is a nuclear $C^{*}$-algebra,
the min and max tensor products are the same here [44, Proposition 2.9]. The
same holds for the suspension $SA:=C_{0}(0,1)\otimes A$. If an operator
algebra $A$ is sitting in a C*-algebra $D$, then $CA$ sits in the QD
C*-algebra $CD$, and so $CA$ is QT.

######  Remark 3.17.

Indeed more is true by a result of Salinas [52]. In order to describe the
result of Salinas we need some notations: let $D$ be a unital separable
C*-algebra and let $Ext(D)$ be the set of equivalence classes of unital
$*$-monomorphisms of $D$ into the Calkin algebra $Q$, and $Ext_{-1}(D)$ be the
group of invertible in $Ext(D)$. An extension $\tau$ is said to be QT w.r.t. a
closed subalgebra $A$ if $\pi^{-1}(\tau(A))$ is QT, where $\pi$ is the
quotient map onto the Calkin algebra. Salinas showed that if $B$ sits in a
unital separable C*-algebra $A$ and $\gamma:[0,1]\to Ext_{-1}(D)$ is
continuous so that $\gamma(0)$ is QT then so is $\gamma(1)$ [52, Theorem
5.10].

######  Remark 3.18.

$(i)$ If the embedding $A\subseteq\mathbb{B}(H)$ is not essential, then
$\tilde{A}:=A/(A\cap\mathbb{K}(H))$ has a natural essentail embedding:
consider the chain of identification and inclusions

$\tilde{A}=(A+\mathbb{K}(H))/\mathbb{K}(H)\subseteq\mathbb{B}(H)/\mathbb{K}(H)\subseteq\mathbb{B}(K),$

for a Hilbert space $K$. It is known that the Calkin algebra
$\mathbb{B}(H)/\mathbb{K}(H)$ is antiliminal [25, 4.7.22(b)], so it has no
non-zero liminal closed ideal [25, page 99], in particular,
$(\mathbb{B}(H)/\mathbb{K}(H))\cap\mathbb{K}(K)=0$, thus
$\tilde{A}\cap\mathbb{K}(K)=0$.

$(ii)$ Every operator algebra has a faithful essential representation: take
any faithful representation $\sigma:A\to\mathbb{B}(H)$ and let
$\pi=\sigma^{(\infty)}:A\to\mathbb{B}(H^{(\infty)})$ be direct sum of
countably many copies of $\sigma$, then $\pi$ is faithful and essential.

######  Remark 3.19.

$(i)$ Theorem 2.8 provides a non selfadjoint analog of a result of Goodearl
and Menal [30] as follows: Every unital, separable operator algebra
$A\subseteq\mathbb{B}(H)$ whose cone $CA:=C_{0}(0,1]\otimes A$ is essentially
embedded in $\mathbb{B}(L^{2}(0,1]\otimes H)$, is a quotient of a block
triangular operator algebra. Indeed, let us consider a C*-algebra $D$
containing $A$ and observe that $CA$ is contained in $CD$, which is
contractible and so QD. Hence, $CA$ is QT, and so is $({\rm
id}\otimes\pi)(CA)$, for any faithful essential representation
$\pi:A\to\mathbb{B}(H)$ (which exists by Remark 3.18$(ii)$). Since $\pi$ is
faithful, ${\rm id}\otimes\pi$ is a complete isometry, thus $({\rm
id}\otimes\pi)(CA)$ is essentially embedded. By Theorem 2.8, we get an
essentially embedded RFD operator algebra
$B\subseteq\mathbb{B}(L^{2}(\Omega)\otimes H)$ with

$({\rm id}\otimes\pi)(CA)+\mathbb{K}(K)=B+\mathbb{K}(K),$

for $\Omega:=(0,1]$ and $K:=L^{2}(\Omega)\otimes H$. Identifying $A$ with
$\pi(A)$, we may identify $C\pi(A)=({\rm id}\otimes\pi)(CA)$ with $CA$.
Passing to the Calkin algebra and using the second equality and the fact that
both $B$ and $CA$ are essentially embedded, we observe that

$CA/(CA\cap\mathbb{K}(K))\simeq B/(B\cap\mathbb{K}(K)),$

complete isometrically. Finally,

$A\simeq
CA/SA\simeq\frac{CA/(CA\cap\mathbb{K}(K))}{(SA+CA\cap\mathbb{K}(K))/(CA\cap\mathbb{K}(K))},$

where the last (complete) isometry follows from the isomorphism theorems for
operator algebras [1, Theorems 3.1.4, 3.1.5] and Remark 3.6$(iv)$ applied to
$I:=SA$ and $J:=CA\cap\mathbb{K}(K)$ (note that $I$ has a cai, since $A$ is
unital). Now the completely isometric homomorphism from
$CA/CA\cap\mathbb{K}(K)$ onto $B/B\cap\mathbb{K}(K)$ maps
$(SA+CA\cap\mathbb{K}(K))/CA\cap\mathbb{K}(K)$ to a closed ideal
$L/B\cap\mathbb{K}(K)$, for a closed ideal $L$ of $B$ containing
$B\cap\mathbb{K}(K)$, and we have completely isometric isomorphisms

$\frac{CA/CA\cap\mathbb{K}(K)}{(SA+CA\cap\mathbb{K}(K))/CA\cap\mathbb{K}(K)}\simeq\frac{B/B\cap\mathbb{K}(K)}{L/B\cap\mathbb{K}(K)}\simeq
B/L,$

by the above mentioned isomorphism theorems. Summing up, $A$ is (isometrically
isomorphic to) a quotient of the block triangular operator algebra $B$.

$(ii)$ Note that the condition of $A$ or $CA$ being essentially embedded is
automatic in the selfadjoint case. The assumption of the cone being
essentially embedded (which is automatic for C*-algebras) seems to be crucial
in the non selfadjoint case (at least in the above proof). However, at this
point we don’t know examples of non selfadjoint operator algebras with an
essentially embedded cone (except trivial examples, such as finite dimensional
triangular matrix algebras).

$(iii)$ The above cited result of Goodearl and Menal fails when $A$ is not
separable (even if $H$ is separable). Indeed Larry Brown has used Zorn lemma
to construct maximal quasidiagonal subsets of $\mathbb{B}(H)$ and Nate Brown
have used this to give counterexamples even in the selfadjoint case (c.f. [12,
Remark 3.7]).

$(iv)$ Finally, the unitality assumption could be relaxed to approximate
unitality, i.e., the existence of a net consisting of real positive elements
in $A$ which is a contractive approximate identity in any C*-algebra which is
generated by $A$ as a closed Jordan subalgebra [11].

$(v)$ By the isomorphism $A\simeq CA/SA$, every operator algebra is a quotient
of a QT operator algebra and the quotient could be explicitly given. Also, for
any operator algebra $A\subseteq\mathbb{B}(H)$ with $H$ separable, since
$\mathbb{B}(H)$ is a quotient of an RFD C*-algebra [13, Excercise 7.1.4],
every separable operator algebra is a quotient of an RFD operator algebra. We
have just proved a stronger result for separable, essentially embedded
operator algebras.

###### Proof of Proposition 2.9.

By the proof of Theorem 2.5, we can choose a sequence of unital complete
contractions $\varphi_{k}:A\to M_{n(k)}(\mathbb{C})$ which form an approximate
isometric sequence of approximate homomorphisms. Take a WOT-cluster point
$\tau$ of $tr_{n(k)}\circ\varphi_{k}$ and observe that $\tau$ is a state. The
tracial property follows from the fact that the sequence $(\varphi_{k})$ is
approximately multiplicative.∎

Note that even nice unital QD $C^{*}$-algebras (like the unitization of the
compact operators) fail to have a faithful tracial state. On the other hand,
the situation could be saved in the simple case. An operator algebra is simple
if it has no proper closed ideal. A state $\tau$ on an operator algebra $A$ is
faithful if $x=0$ for each $x\in A_{+}:=A\cap C^{*}(A)_{+}$ with $\tau(x)=0$,
where $C^{*}(A)$ is the $C^{*}$-algebra of generated by $A$ in a faithful
representation. Now every simple unital operator algebra which is contained in
a QD C*-algebra has a faithful trace: First note that since QD passes to
subalgebras [13, Proposition 7.1.10], the last assumption is the same as
requiring that $C^{*}(A)$ is QD. Choose a tracial state $\tau$ on $C^{*}(A)$
by [13, Proposition 7.1.16]. We use simplicity to show that the restriction of
$\tau$ to $A$ is faithful. Let

$I_{\tau}:=\\{x\in C^{*}(A):\tau(xx^{*})=0\\}$

be the ideal kernel of $\tau$. Take $a\in A_{+}:=A\cap C^{*}(A)_{+}$ with
$\tau(a)=0$ and choose $y\in C^{*}(A)$ with $a=yy^{*}$. Then
$\tau(yy^{*})=\tau(a)=0$ and so $y\in I_{\tau}$. But this is an ideal, thus
$a=yy^{*}\in I_{\tau}$. If $a$ is not zero, then we have a nonzero closed
ideal $A\cap I_{\tilde{\tau}}$ of $A$, which by simplicity has to be equal to
$A$, which means that $A\subseteq I_{\tau}$. But then $A^{*}:=\\{x^{*}:x\in
A\\}\subseteq I_{\tau}$, leading to the equality $I_{\tau}=C^{*}(A)$, which is
absurd, since $A$ is unital and $\tau(1)=1$.

######  Remark 3.20.

$(i)$ The assumption that $A$ is contained in a QD C*-algebra is stronger than
assuming that $A$ is itself QT (see Remark 3.14$(ii)$).

$(ii)$ It doesn’t seem plausible to prove the existence of a faithful trace on
a simple unital QT operator algebra, using a modification of the above
argument: if we start with a tracial state $\tau$ on $A$ (which exists by
roposition 2.9 when $A$ is QT), it is not possible in general to extend it to
a trace on $C^{*}(A)$.

Indeed, let $A$ be the norm-closed algebra generated by the unilateral shift
in $\mathbb{B}(\ell^{2}(\mathbb{N}\cup\\{0\\}))$. This is completely
isometrically isomorphic the disc algebra, which is in turn the (universal)
operator algebra of a contraction. Since the disc algebra is abelian and its
character space is the closed unit disc, all its characters are tracial
states.

On the other hand, if we have a tracial state $\tau$ on the Toeplitz algebra
(the $C^{*}$-algebra generated by the unilateral shift) then it should
annihilate the compacts: Let $p_{n}$ be the orthogonal projection on
$\ell^{2}$ projecting to the $n$-th coordinate. Then,
$\tau(VV^{*})=\tau(V^{*}V)=1$, and so $\tau(p_{0})=\tau(I-VV^{*})=0$. Also,
$\tau(V^{2}(V^{2})^{*})=1$, and so
$\tau(p_{0}+p_{1})=\tau(I-V^{2}(V^{2})^{*})=0$, that is, $\tau(p_{1})=0$. By
an inductive argument, we get $\tau(p_{n})=0$, for each $n$. Now consider the
$(n,n)$-compression state $\tau_{n}$ on the Toeplitz algebra, that is,
$\tau_{n}(x)=p_{n}xp_{n}$. This state does not annihilate the compacts, and by
the above argument, its restriction to $A$ gives zero Fourier scalars (note
that, every element in $A$ admits a representation as an analytic function on
the open unit disc), and so is tracial.

## acknowledgement

The authors would like to thank David Blecher, Ken Davidson and Evgenios
Kakariadis for suggesting (non) examples and references. The authors also
thank the anonymous referee for careful reading of the paper and suggestions
to improve the paper.

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