# Dialogue Relation Extraction with Document-Level
Heterogeneous Graph Attention Networks

Hui Chen, Pengfei Hong, Wei Han, Navonil Majumder, Soujanya Poria
DeCLaRe Lab, Singapore University of Technology and Design, Singapore
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###### Abstract

Dialogue relation extraction (DRE) aims to detect the relation between pairs
of entities mentioned in a multi-party dialogue. It plays an essential role in
constructing knowledge graphs from conversational data increasingly abundant
on the internet and facilitating intelligent dialogue system development. The
prior methods of DRE do not meaningfully leverage speaker information—they
just prepend the utterances with the respective speaker names. Thus, they fail
to model the crucial inter-speaker relations that may provide additional
context to relevant argument entities through pronouns and triggers. We
present a graph attention network-based method for DRE where a graph that
contains meaningfully connected speaker, entity, type, and utterance nodes is
constructed. This graph is fed to a graph attention network for context
propagation among relevant nodes, which effectively captures the dialogue
context. We empirically show that this graph-based approach quite effectively
captures the relations between different argument pairs in a dialogue as it
outperforms the state-of-the-art approaches by a significant margin on the
benchmark dataset DialogRE. Our code is released at:
https://github.com/declare-lab/dialog-HGAT.

##  1 Introduction

The relation extraction (RE) task aims to identify relations between pairs of
entities that exist in a document. It plays a pivotal role in understanding
unstructured text and constructing knowledge bases Peng et al. (2017); Quirk
and Poon (2017). Although the task of document-level relation extraction has
been studied extensively in the past, the task of relation extraction from
dialogues has yet to receive extensive study.

Figure 1: An example adapted from DialogRE dataset. Words with red and blue
background represent subject and object entities. Words with yellow background
represent triggers that facilitate the relation inference. Solid and dash
lines stand for intra- and inter-utterance relations.

Most previous works in this field focus on the professional and formal
literature like biomedical documents Li et al. (2016); Wu et al. (2019) and
Wikipedia articles Elsahar et al. (2018); Yao et al. (2019); Mesquita et al.
(2019). These kinds of datasets are well-formatted and logically coherent with
clear referential semantics. Hence, for most NLP tasks, analyzing a few
continuous sentences is enough to grasp pivotal information. However, in
dialogue relation extraction, conversational text is sampled from daily chat,
which is more casual in nature. Hence its logic is simpler but more entangled,
and the referential ambiguity always occurs to an external reader. Compared
with formal literature, it has lower information density Wang and Liu (2011)
and thus is more difficult for models to understand. Moreover, compared with
other document-level RE datasets such as DocRED, dialogue text has much more
cross-sentence relations Yu et al. (2020).

Fig. 1 presents an example of the target dialogue, taken from DialogRE Yu et
al. (2020) dataset. In order to infer the relation between Speaker1 and Emma,
we may need to find some triggers to recognize the characteristics of Emma.
Triggers are shreds of evidence that can support the inference. As we can see,
the following utterances are talking about Emma, and the keyword baby daughter
mentioned by Speaker1 is a trigger, which provides evidence that Emma is
Speaker1’s daughter.

Prior works show that triggers of arguments facilitate the document-level
relation inference. Thus, the DocRED dataset Yao et al. (2019) provides
several supporting evidence for each argument pair. Some efforts utilize the
dependency paths of arguments to find possible triggers. For example, LSR
model Nan et al. (2020) constructs meta dependency paths of each argument pair
and aggregates all the word representations located in these paths to their
model to enhance the model’s reasoning ability. Sahu et al. (2019) uses
syntactic parsing and coreference resolution to find intra- and inter-related
words of each argument. Christopoulou et al. (2019) proposes an edge-oriented
graph to synthesize argument-related information. These models are graph-based
and have proven powerful in encoding long-distance information. However, for
dialogue relation extraction, interlocutors exist in every utterance of the
dialogue, and they are often considered as an argument. Although these
previous approaches have utilized entity features of arguments, most of them
employ meta dependency paths to find the related words, which neglect
necessary information related to speakers, since the speaker references have
very little dependency features in each utterance. In this work, we formulate
the dialogue relation extraction task as a classification problem, where we
design a graph attention network to model semantic, syntactic, and speaker
information. Compared with other graph-based models in the relation extraction
task, our model is lightweight, without any costly matrix operation, and it
can generalize to completely unseen graphs.

In this paper, we propose a simple yet effective attention-based heterogeneous
graph neural network to tackle the dialogue relation extraction task in an
inductive manner. We use multi-type features to create the graph and employ
graph attention mechanism to propagate contextual information. Different from
most of the previous works, our proposed model is customized for the relation
extraction task in dialogue background, as we have specially modeled speaker
information and designed a mechanism to propagate messages among different
sentences for better inter-sentence representation learning.

The remainder of this paper is organized as follows: Section 2 briefly
discusses relevant works of heterogeneous graph neural networks; Section 3
elaborates on our proposed framework; Section 4 introduces the used dataset
and baseline models; Section 5 lays out the experiment results and analysis;
Section 6 concludes the paper.

##  2 Related Work

Graph-based models have raised widespread attention from NLP researchers, as
it is demonstrated as a powerful mathematical tool to represent complicated
syntactic and semantic relations among structured language data. Early work
applies classic graph processing algorithms onto language graphs. Pang and Lee
(2004) constructed a text graph and adopt the minimum-cut method to cluster
the nodes for sentiment analysis. Agirre and Soroa (2009) leveraged PageRank
algorithm on personalized subgraphs of a wordnet to disambiguate polysemous
words according to connected context words.

Recently, graph neural networks (GNN) Kipf and Welling (2017) becomes popular
in relation extraction tasks. For example, Peng et al. (2017) tried to build a
computation graph from syntactic parsing trees and employed graph LSTM to
obtain better word embeddings for multi-ary relation extraction. Zhang et al.
(2018) designed a pruning algorithm for syntactic graphs and add a graph
convolution layer on top of the sequential LSTM encoder in the learning
process. The combination with typical attention-based language models such as
transformer Vaswani et al. (2017) is also studied. Cai and Lam (2020) and Yao
et al. (2020) used transformer-based graph convolutional networks to
explicitly encode relations among distant syntactic nodes, to address the
long-distance propagation issue.

Based on GNN, heterogeneous graph neural networks are proposed and have been
applied in many NLP tasks, like text classification Linmei et al. (2019), text
summarization Wang et al. (2020), user profiling Chen et al. (2019), and event
categorization Peng et al. (2019). The prior work proves that heterogeneous
graph neural network is a powerful tool in NLP. For the relation extraction
task, Christopoulou et al. (2019) constructed an edge-oriented heterogeneous
graph that contains sentence, mention, and entity information. However,
syntactic information is neglected in their model. Differently, homogeneous
nodes in our graph are all independent, and we take syntactic features to
initialize sentence information as well as edges features.

##  3 Method

###  3.1 Task Definition

Given a dialogue containing $N$ utterances
$\mathcal{D}=\\{u_{1},u_{2},...,u_{N}\\}$ and a couple of argument pairs
$\mathcal{A}=\\{(x_{1},y_{1}),(x_{2},y_{2}),\dots\\}$, where subject $x_{i}$
and object $y_{i}$ are entities mentioned in the dialogue, the goal is to
identify the relation between argument pairs $(x_{i},y_{i})$. For document-
level relation extraction task, there are many cross-sentence relations which
are supported by various sentences.

###  3.2 Model Overview

In this work, we introduce an attention-based graph network to tackle the
problem where each conversation is represented as a heterogeneous graph. We
first utilize an utterance encoder, which is composed of two Bidirectional
long short-term memory networks to encode conversational information. These
utterance encodings, along with word embeddings, speaker embeddings, argument
embeddings, and type embedding, are logically connected to form a
heterogeneous graph, which will be discussed in detail later in this section.
Further, this graph is fed through five graph attention layers Veličković et
al. (2018) that aggregate information from the neighboring nodes. Lastly, we
concatenate the learned argument embeddings and feed them to a classifier. An
overview of the proposed model is shown in Fig. 2.

Figure 2: An overview of the proposed model.

###  3.3 Data Preprocessing

In the data preprocessing period, we use spaCy111https://spacy.io to tokenize
utterances, and at the same time, we obtain part-of-speech (POS) tags as well
as named entity types of each token.

###  3.4 Utterance Encoder

Given a dialogue $\mathcal{D}=\\{u_{1},u_{2},...,u_{N}\\}$, we use GloVe
Pennington et al. (2014) to initialize the word embeddings and then feed them
to a contextual Bidirectional Long Short-Term Memory network (BiLSTM) to
obtain contextualized representations. The operation of BiLSTM can be defined
as:

$\displaystyle\overleftarrow{h_{j}^{i}}$
$\displaystyle=LSTM_{l}(\overleftarrow{h_{j+1}^{i}},e_{j}^{i})$ (1)
$\displaystyle\overrightarrow{h_{j}^{i}}$
$\displaystyle=LSTM_{r}(\overrightarrow{h_{j-1}^{i}},e_{j}^{i})$ (2)
$\displaystyle h_{j}^{i}$
$\displaystyle=[\overleftarrow{h_{j}^{i}};\overrightarrow{h_{j}^{i}}]$ (3)

where $\overleftarrow{h_{j}^{i}}$ and $\overrightarrow{h_{j}^{i}}$ denote the
hidden representations in the $j$-th layer of utterance $u_{i}$ from two
directions, $h_{j}^{i}$ is the contextual representation which is the
concatenation of $\overleftarrow{h_{j}^{i}}$ and $\overrightarrow{h_{j}^{i}}$,
and $e_{j}^{i}$ stands for the embedding of the $j$-th token in utterance
$u_{i}$. Unlike the previous approaches Christopoulou et al. (2019); Nan et
al. (2020) that only adopt semantic contextual features in utterance encoding,
we add syntactic features such as POS tags and named entity types to the
contextual representations. The embedding of each token in the utterance can
be described as:

$\displaystyle e=[e_{w};e_{p};e_{t}]$ (4)

where we concatenate word embedding $e_{w}$ initialized by GloVe Pennington et
al. (2014), syntactic POS embedding $e_{p}$, and type embedding $e_{t}$ to
form the token embedding $e$.

Moreover, we believe conversation-level contextual features play an important
role in understanding a conversation. To encode non-local contextual
information between each utterance, we apply max pool operation to the hidden
states of each utterance-level BiLSTM (local LSTM), and then feed the sequence
$c=\\{c_{1},c_{2},...,c_{N}\\}$ to a conversation-level BiLSTM (global LSTM).
The operation of global LSTM is the same as Eqs. 1, 2 and 3.

###  3.5 Graph Construction

####  3.5.1 Node Construction

In our model, we design a heterogeneous graph network containing five types of
nodes: utterance nodes, type nodes, word nodes, speaker nodes, and argument
nodes. Each type of node is used to encode a type of information in the
dialogue. In the task, only word nodes, speaker nodes and argument nodes are
probable to attend the final classification process. In other words, only
these types of nodes are possible arguments. For simplicity, we name them as
basic nodes in our illustration.

##### Utterance and Type Nodes

Utterance nodes are initialized by the utterance embeddings which we obtain
from the utterance encoder. They are connected with the basic nodes which
constitute the utterance. Type nodes represent the entity types of words in an
utterance, where a variety of named and numeric entities, such as PERSON or
LOCATION, are included. Since one mention may have different types in one
conversation, type nodes can facilitate information integration. For example,
‘Frank’ can be a string if it represents an alternative name, and at the same
time, it can be a person if it refers to a speaker in the conversation. Type
nodes are connected with the basic nodes having the type attribute in the
conversation. Each type node is initialized with a specific type of embedding.
We believe that type of information has a positive influence on the relation
inference process.

##### Basic Nodes

Word nodes represent the vocabulary of a conversation. Each word node is
connected with the utterance, which contains the word and it is also connected
with all the possible types that the word may have in the conversation. We
initialize the states of word nodes with GloVe Pennington et al. (2014).

Speaker nodes represent each unique speaker in the conversation. Each speaker
node is connected with the utterances uttered by the speaker himself/herself.
This type of node is initialized with some specific embeddings and can gather
information from different speakers.

Argument nodes are two special nodes that are used to encode relative
positional information of argument pairs. There are two argument nodes in each
graph in total. One stands for the subject argument and the other represents
the object argument. Similarly, argument nodes are also encoded by specific
embeddings.

####  3.5.2 Edge Construction

The proposed graph is undirected but the propagation has directions. There are
five types of edges: utterance-word, utterance-argument, utterance-speaker,
type-word, and type-argument edges. Each edge has its own type. These edges
are randomly initialized except the utterance-word edge.

For the edge between utterance and word nodes, we adopt POS tags to initialize
the edge features. This type of edge aggregates not only global semantic
features of the conversation but also local syntactic features to the word
nodes.

####  3.5.3 Graph Attention Mechanism

We use graph attention mechanism Veličković et al. (2018) to aggregate
neighboring information to the target node. Suppose we have a node $i$ and
some neighborhood nodes $j$, the graph attention mechanism can be described
as:

$\displaystyle\mathcal{F}(h_{i},h_{j})$
$\displaystyle=\operatorname*{\text{LeakyReLU}}(\textbf{a}^{T}(\textbf{W}_{i}h_{i};\textbf{W}_{j}h_{j};\textbf{E}_{ij})$
(5) $\displaystyle\alpha_{ij}$
$\displaystyle=\operatorname*{\text{softmax}}(\mathcal{F}(h_{i},h_{j}))$ (6)
$\displaystyle=\frac{\exp(\mathcal{F}(h_{i},h_{j}))}{\sum_{k}\exp(\mathcal{F}(h_{i},h_{k}))}$
(7) $\displaystyle h_{i}^{\prime}$
$\displaystyle=||_{k=1}^{K}\sigma(\sum_{j}\alpha_{ij}^{k}\textbf{W}_{q}^{k}h_{j})$
(8)

where ${h}_{i}$ and ${h}_{j}$ are representations of node $i$ and nodes $j$,
$\textbf{W}_{i}$, $\textbf{W}_{j}$, $\textbf{W}_{q}$ and $\textbf{a}^{T}$ are
trainable weight matrices, $\textbf{E}_{ij}$ is the edge weight matrix that is
mapped to the multi-dimensional embedding space, $\alpha_{ij}$ is the
attention weight between $i$ and $j$, $\sigma$ is an activation function, and
$||$ is concatenation operation.

####  3.5.4 Message Propagation

As shown in Fig. 2, there are five layers in our proposed graph module, where
each layer represents an aggregation. There are four types of layers that we
mark in the figure. LayerA and LayerD contain the message propagation between
utterance nodes and basic nodes, and similarly, LayerB and LayerC are the
message propagation between basic nodes and type nodes. We would call the
whole message propagation path meta path. Different meta path strategies may
lead to different performance.

Our meta path in this work can be described as follows: First, we use
utterance nodes to update word nodes, speaker nodes, and argument nodes;
secondly, the updated word nodes and argument nodes pass messages to type
nodes; then type nodes conversely update the word nodes and argument nodes;
next we use word nodes, speaker nodes, and argument nodes to update utterance
nodes; and lastly the updated utterance nodes update word nodes, speaker nodes
and argument nodes. The path can be denoted as
$V_{u}-V_{b}-V_{t}-V_{b}-V_{u}-V_{b}$, where $V_{u}$, $V_{b}$, and $V_{t}$
refer to utterance nodes, basic nodes, and type nodes.

Following Wang et al. (2020), we add a residual connection He et al. (2016) to
avoid gradient vanishing during updating:

$\displaystyle\hat{h_{i}}=\bar{h_{i}}+h_{i}^{\prime}$ (9)

where $\bar{h_{i}}$ is the output learned in the graph attention layer, and
$h_{i}^{\prime}$ is the original input of the graph attention layer.

In message passing, except for graph attention operation, there is also a two-
layer feed-forward network which can be denoted as:

$\displaystyle h_{i}^{new}=\operatorname*{\text{FFN}}(\hat{h_{i}})$ (10)

Suppose we have the initial embeddings of utterance nodes, basic nodes and
type nodes, denoted as embedding matrices
$\mathbf{H}_{u}=\\{\mathbf{H}_{u},\mathbf{H}_{b},\mathbf{H}_{t}\\}$, the
message propagating process can be written as:

$\displaystyle\mathbf{H}_{b}^{1}$
$\displaystyle=\operatorname*{\text{GAT}}(\mathbf{H}_{b}^{0},\mathbf{H}_{u}^{0})$
(11) $\displaystyle\mathbf{H}_{t}^{1}$
$\displaystyle=\operatorname*{\text{GAT}}(\mathbf{H}_{t}^{0},\mathbf{H}_{b}^{1})$
(12) $\displaystyle\mathbf{H}_{b}^{2}$
$\displaystyle=\operatorname*{\text{GAT}}(\mathbf{H}_{b}^{1},\mathbf{H}_{t}^{1})$
(13) $\displaystyle\mathbf{H}_{u}^{1}$
$\displaystyle=\operatorname*{\text{GAT}}(\mathbf{H}_{u}^{0},\mathbf{H}_{b}^{2})$
(14) $\displaystyle\mathbf{H}_{b}^{3}$
$\displaystyle=\operatorname*{\text{GAT}}(\mathbf{H}_{b}^{2},\mathbf{H}_{u}^{1})$
(15)

where the GAT operation is the same as Eqs. 5, 6, 7, 8, 9 and 10. The
superscripts represent the $n^{th}$ update of the matrix and 0 marks the
initial state.

###  3.6 Relation Classifier

After the message propagation in the heterogeneous graph, we obtain new
representations of all entities. We select the argument nodes $\tau_{x}$ and
$\tau_{y}$, as well as the corresponding word nodes $e_{x}$ and $e_{y}$ from
basic nodes, and concatenate them. Finally, they are fed to a linear
transformation and a sigmoid function to get the predictions:

$\displaystyle e_{x}^{\prime}$
$\displaystyle=[\operatorname*{\text{maxpool}}(\tau_{x});\operatorname*{\text{maxpool}}(e_{x})]$
(16) $\displaystyle e_{y}^{\prime}$
$\displaystyle=[\operatorname*{\text{maxpool}}(\tau_{y});\operatorname*{\text{maxpool}}(e_{y})]$
(17) $\displaystyle e^{\prime}$
$\displaystyle=[e_{x}^{\prime};e_{y}^{\prime}]$ (18) $\displaystyle
P(r|e_{x},e_{y})$ $\displaystyle=\sigma(\textbf{W}_{e}e^{\prime}+b_{e})_{r}$
(19)

where $P(r|e_{x},e_{y})$ is the probability of relation type $r$ given
argument pair $(e_{x},e_{y})$, $\textbf{W}_{e}$ and $b_{e}$ are linear
transformation weight and bias vector, $maxpool$ is max pooling operation, and
$\sigma$ is sigmoid function.

##  4 Experiments

###  4.1 Dataset Used

We evaluate the proposed framework on the DialogRE dataset Yu et al. (2020),
which contains 1,788 dialogues and 10,168 relational triples. The data
statistics are shown in Table 1. DialogRE is adapted from the complete
transcripts of Friends, a widely used corpus in dialogue research these years
Chen et al. (2017); Zhou and Choi (2018); Yang and Choi (2019); Poria et al.
(2019), and there are 36 possible relation types, most of which focus on
biographical attributes of person entities. Each dialogue contains several
relational triples $(x,y,r)$, and the task is to predict the relation $r$
between each argument pair $(x,y)$. In the experiments, the dataset is
partitioned into train, dev, and test set with a roughly 60/20/20 ratio.
Following the evaluation metrics of DialogRE, we report macro $F1$ scores of
the proposed model and all the baselines in both the standard and
conversational settings. In the following sections, we use $F1_{c}$ to
represent $F1$ scores in the conversational setting.

DialogRE | Train | Dev | Test
---|---|---|---
#Conversations | 1073 | 358 | 357
#Argument Pairs | 5963 | 1928 | 1858
Average dialogue length | 229.5 | 224.1 | 214.2
Average # of turns | 13.1 | 13.1 | 12.4
Average # of speakers | 3.3 | 3.2 | 3.3
Table 1: DialogRE dataset statistics.

###  4.2 Baseline models

####  4.2.1 Sequence-based Models

We select convolutional neural networks (CNN) Zeng et al. (2014), LSTM, and
BiLSTM Cai et al. (2016) as the sequence-based baselines. These models take
word embeddings, mention embeddings, and type embeddings as features.
Concretely, they use GloVe and spaCy to get word embeddings and label named-
entity types, and then take an average of all the embeddings of mention names
for each entity to get mention embeddings.

####  4.2.2 Graph-based Models

As our proposed model is graph-based, we also select two graph-based models
AGGCN Guo et al. (2019) and LSR Nan et al. (2020) as the baselines. AGGCN
directly feeds the full dependency tree of each sentence to a graph
convolutional network, which takes self-attention weights as soft edges. It
achieves state-of-the-art results in various relation extraction tasks. LSR
adopts an adaptation of Kirchhoff’s Matrix-Tree Theorem Tutte (1984); Koo et
al. (2007) to induce the latent dependency structure of each document and then
feeds the latent structure to a densely connected graph convolutional network
to inference the relations. These graph-based models both utilize dependency
information to construct the inference graph.

##  5 Result and Analysis

###  5.1 Comparison with Baselines

We present our main results on DialogRE dataset in Table 2. As shown in Table
2, our model surpasses the state-of-the-art method by 9.6%/7.5% $F1$ scores,
and 8.4%/5.7% $F1_{c}$ scores in both validation and test sets, which
demonstrates the effectiveness of the information propagation along task-
specific functional meta-paths in the heterogeneous graph. As a result, inter-
sentence communication usually passes through a long distance, which causes
information loss or degradation. However, this kind of information
transmission is critically important for dialog-style text, because logical
connections are not locally compact within adjacent sentences, instead, they
are spread over the whole conversations. Our proposed model constructs a
heterogeneous graph with shorter distances between logically closed but
syntactically faraway word pairs. Hence the long-distance issue is mitigated.

We also compare the model sizes as an efficiency indicator. Although creating
numerous nodes and edges inevitably brings overhead, the total number of
parameters is still moderate.

Model |  | Dev (%) | Test (%)
---|---|---|---
#params | $F1$ | $F1_{c}$ | $F1$ | $F1_{c}$
Majority Yu et al. (2020) | - | 38.9 | 38.7 | 35.8 | 35.8
CNN Yu et al. (2020) | - | 46.1 | 43.7 | 48.0 | 45.0
LSTM Yu et al. (2020) | - | 46.7 | 44.2 | 47.4 | 44.9
BiLSTM Yu et al. (2020) | 4.1M | 48.1 | 44.3 | 48.6 | 45.0
AGGCN Guo et al. (2019) | 3.7M | 46.6 | 40.5 | 46.2 | 39.5
LSR Nan et al. (2020) | 20.5M | 44.5 | - | 44.4 | -
This work | 4.0M | 57.7 | 52.7 | 56.1 | 50.7
Table 2: Main results on DialogRE dataset. Values in the #params column refer
to parameter sizes of the models. $F1$ and $F1_{c}$ are macro $F1$ scores
under standard setting and conversational setting, respectively. Word
embeddings of the models are captured by GloVe Pennington et al. (2014).

###  5.2 Ablation Study

To understand the impact of our model’s components, we perform ablation
studies using our proposed model on the DialogRE dataset. The ablation results
are shown in Table 3. First, we remove local LSTM and global LSTM. The
dropping accuracy proves that the contextual encoder plays an important role
in semantic feature extraction. Second, we remove the specific argument nodes
and have observed that $F1$ and ${F1_{c}}$ scores decrease to 55.0% and 50.2%
on test set. This proves that our design on argument nodes effectively
synthesizes argument features to the model. Further, we test the performance
of the syntactic features we inject by removing POS embedding, NER embedding,
and POS edge features. The scores record a decrease under all these experiment
settings. Notably, removing POS embedding leads to even about 2% drops in all
the evaluation metrics.

Model | Dev (%) | Test (%)
---|---|---
$F1$ | $F1_{c}$ | $F1$ | $F1_{c}$
Full model | 57.7 | 52.7 | 56.1 | 50.7
w/o Local BiLSTM | 54.9 | 50.0 | 55.3 | 50.3
w/o Global BiLSTM | 54.7 | 50.2 | 53.5 | 48.7
w/o Argument nodes | 56.0 | 51.3 | 55.0 | 50.2
w/o POS embedding | 54.6 | 50.9 | 53.0 | 48.5
w/o NER embedding | 56.8 | 51.5 | 54.2 | 49.2
w/o POS edge weights | 56.9 | 52.4 | 54.7 | 50.4
Table 3: Ablation results on DialogRE dataset.

###  5.3 Effect of the Meta Path

We test the performance of our message propagation strategy via changing meta-
path strategies. In our proposed model, there are five layers in the
heterogeneous graph. Those basic nodes, corresponding to different types of
words, speakers, and arguments, are updated totally three times, i.e., they
are first updated by utterance nodes, second updated by type nodes, and
ultimately updated by utterance again. To investigate the meta path’s effect,
we compare our proposed five-layer graph module with different strategies
where the numbers of layers are one, seven, and nine in Table 4. In Strategy1,
we only set up one LayerA, where the basic nodes are updated by the
initialized utterance nodes once. We observe that all the macro $F1$ scores
drop dramatically, showing the one-layer structure is not deep enough to grasp
complex dependencies. To make node features more informative, we would add
more layers. At this time, we may be curious about how many layers the module
should have to induct an optimal structure in this task. In Strategy2 and
Strategy3, we design a seven-layer module and a nine-layer module,
respectively. For Strategy2, the order of layers is A-B-C-D-A-D-A, where
A,B,C, and D are layer labels introduced in Fig. 2. Compared with our proposed
module, scores on validation set decrease about 1% and scores on test set
decrease 1.7% and 0.6 % with the standard-setting and the conversational
setting, respectively. However, the module with nine layers in Strategy3 shows
a larger gap between itself and the best performance, where the order of
layers is A-B-C-D-A-B-C-D-A. We think this is probably because the structure
is so complicated, which causes an over-smooth problem and prevents itself
from learning meaningful hidden representations.

Strategy | Dev (%) | Test (%)
---|---|---
$F1$ | $F1_{c}$ | $F1$ | $F1_{c}$
This work (L=5) | 57.7 | 52.7 | 56.1 | 50.7
Strategy1 (L=1) | 48.3 | 46.5 | 48.4 | 45.9
Strategy2 (L=7) | 56.8 | 51.6 | 54.4 | 50.1
Strategy3 (L=9) | 53.8 | 49.1 | 52.2 | 47.2
Table 4: Comparison with different meta-path strategies on DialogRE dataset.
‘L’ means the number of layers in the graph module.

###  5.4 Case Studies

In the dataset, 95% of argument pairs span in at least two consecutive
sentences instead of being restricted to the same sentence. Therefore, it is
crucial that the model can tackle long-distance learning issues. Compared with
the LSTM model, direct connections among different types of nodes in HGNN
reduce the length of information propagation paths between pairs of argument
nodes. Considering the following example in Fig. 3, subject a - ‘Mindy’ and
object b - ‘Speaker 1’ share the relationship ‘per:friends’, which is
indicated by the trigger ‘my best friend’ in the first utterance. The entity
information is relayed from ‘Mindy’ to ‘Speaker 1’ in the update process:
‘speaker 1’ node aggregates utterance level information from its neighbor
nodes containing a. the relation trigger ‘best friend’. b. in BiLSTM model,
the key information has to travel a long journey from the subject entity word
to the object one as there are too many words between them in the context.

Figure 3: An example to show the effective message propagation between
argument pairs

###  5.5 Error Analysis

Type information involves in the information propagation process and thus
affects the contents of output embeddings. The model is prone to make
incorrectly and biased predictions. If it fails to receive enough certainty
from other information sources and then can only rely on the entity types of
the two arguments. For example, given an argument pair of two human names,
both are named entity type ‘PERSON’. Sometimes the model inclines to deem the
relationship between the two arguments to be ‘per:alternate_name’ instead of
the correct answer ‘per:alumni’ or ‘per:roommate’. This is because among all
of these classes, ‘PERSON-PERSON’ is a preferable type pair. However, the
class ‘per:alternate_name’ (22.01%) presents more frequently than ‘per:alumni’
(1.83%) and ‘per:roommate’ (1.29%) in the dataset. When information aggregated
from all sources other than the argument pair is not evident for judgment,
entity bias misguides the model to the wrong classification results.

##  6 Conclusion

In this work, we present an attention-based heterogeneous graph network to
deal with the dialogue relation extraction task in an inductive manner. This
heterogeneous graph attention network has modeled multi-type features of the
conversation, such as utterance, word, speaker, argument, and entity type
information. On the benchmark DialogRE dataset, our proposed framework
outperforms the strongest baselines and the state-of-the-art approaches by a
significant margin, which proves the proposed framework can effectively
capture relations between different entities in the conversation. Future work
will focus on making use of latent relations between entities that exist in
dialogue history to develop intelligent conversational agents.

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##  Appendix A Settings and Hyperparameters

In our experiments, we tune the parameters of batch size, learning rate, and
BiLSTM hidden size by testing the performance on the validation set. Table 5
lists the major parameters used in our experiments.

Parameter | Value
---|---
Word embedding dimension | 300
NER embedding dimension | 30
POS embedding dimension | 30
Local BiLSTM hidden Size | 200
Local BiLSTM layers | 2
Global BiLSTM hidden Size | 128
Global BiLSTM layers | 2
# Multihead attention | 10
Learning rate | 0.0005
Batch size | 16
Edge embedding dimension | 50
Table 5: Parameter settings.

##  Appendix B Statistics of Relation Labels

Table 6 shows statistics of relation labels in DialogRE dataset. In the train
set and test set, there are 35 types of relations, while in the dev set, there
are 37 types. ‘gpe:birth_in_place’ and ‘per:place_of_birth’ only exist in the
dev set.

Relation Type | Quantity | Percentage (%)
---|---|---
train | dev | test | train | dev | test
per:alternate_names | 1319 | 410 | 409 | 22.12 | 21.26 | 22.01
unanswerable | 1308 | 404 | 388 | 21.94 | 20.95 | 20.88
per:girl/boyfriend | 502 | 170 | 136 | 8.42 | 8.82 | 7.32
per:positive_impression | 476 | 149 | 138 | 7.98 | 7.73 | 7.43
per:friends | 444 | 156 | 122 | 7.45 | 8.09 | 6.57
per:title | 250 | 86 | 78 | 4.19 | 4.46 | 4.20
per:spouse | 204 | 72 | 54 | 3.42 | 3.73 | 2.91
per:siblings | 196 | 64 | 58 | 3.29 | 3.32 | 3.12
per:children | 171 | 55 | 48 | 2.87 | 2.85 | 2.58
per:parents | 171 | 55 | 48 | 2.87 | 2.85 | 2.58
per:negative_impression | 156 | 46 | 56 | 2.62 | 2.39 | 3.01
per:roommate | 140 | 44 | 24 | 2.35 | 2.28 | 1.29
per:alumni | 110 | 38 | 34 | 1.84 | 1.97 | 1.83
per:other_family | 66 | 29 | 30 | 1.11 | 1.50 | 1.61
per:works | 58 | 12 | 19 | 0.97 | 0.62 | 1.02
per:age | 53 | 15 | 10 | 0.89 | 0.78 | 0.54
per:client | 52 | 18 | 18 | 0.87 | 0.93 | 0.97
per:place_of_residence | 49 | 12 | 23 | 0.82 | 0.62 | 1.24
gpe:residents_of_place | 49 | 12 | 23 | 0.82 | 0.62 | 1.24
per:boss | 49 | 13 | 12 | 0.82 | 0.67 | 0.65
per:subordinate | 49 | 13 | 12 | 0.82 | 0.67 | 0.65
per:visited_place | 48 | 20 | 25 | 0.80 | 1.04 | 1.35
gpe:visitors_of_place | 48 | 20 | 25 | 0.80 | 1.04 | 1.35
per:employee_or_member_of | 46 | 11 | 15 | 0.77 | 0.57 | 0.81
org:employees_or_members | 46 | 11 | 15 | 0.77 | 0.57 | 0.81
per:neighbor | 40 | 14 | 12 | 0.67 | 0.73 | 0.65
per:place_of_work | 37 | 9 | 25 | 0.62 | 0.47 | 1.35
per:pet | 30 | 10 | 8 | 0.50 | 0.52 | 0.43
per:acquaintance | 26 | 12 | 34 | 0.44 | 0.62 | 1.83
per:origin | 21 | 4 | 1 | 0.35 | 0.21 | 0.05
per:dates | 20 | 14 | 6 | 0.34 | 0.73 | 0.33
per:schools_attended | 5 | 2 | 1 | 0.08 | 0.10 | 0.05
org:students | 5 | 2 | 1 | 0.08 | 0.10 | 0.05
per:major | 2 | 1 | 3 | 0.03 | 0.05 | 0.16
per:date_of_birth | 1 | 2 | 3 | 0.02 | 0.10 | 0.16
gpe:birth_in_place | 0 | 1 | 0 | 0 | 0.05 | 0
per:place_of_birth | 0 | 1 | 0 | 0 | 0.05 | 0
Table 6: Statistics of relation labels in DialogRE dataset.


# Commutativity and Orthogonality of Similarity Orbits in Banach Algebras

Muhammad Hassen  University of Johannesburg, Johannesburg, South Africa
<EMAIL_ADDRESS>,  Rudi Brits  University of Johannesburg, Johannesburg,
South Africa<EMAIL_ADDRESS>and  Francois Schulz  University of
Johannesburg, Johannesburg, South Africa<EMAIL_ADDRESS>
###### Abstract.

For a semisimple unital Banach algebra $A$ over $\mathbb{C}$, and elements
$a,b\in A,$ we show that the similarity orbits, $\operatorname{orb}(a)$ and
$\operatorname{orb}(b),$ over the principal component of the invertible group
of $A$ commute precisely when there is at least one nonzero complex number not
belonging to the spectrum of any product $a^{\prime}b^{\prime}$—where
$(a^{\prime},b^{\prime})\in\operatorname{orb}(a)\times\operatorname{orb}(b)$.
In this case, the polynomially convex hull of the spectra of the
$a^{\prime}b^{\prime}$ is constant. When
$\operatorname{orb}(a)=\operatorname{orb}(b),$ then $a$ is central under the
aforementioned assumption—and the result then generalizes part of an old
theorem due to J. Zemánek. We show further that the two classical
characterizations of commutative Banach algebras via the spectral radius can
be algebraically localized in the sense of ‘local’ implies ‘global’.
Thereafter, in Section 3, we give a (somewhat weaker) localization of the
above situation involving spectral perturbation on small neighborhoods in a
similarity orbit. Finally, we apply the above results to algebraic elements
and idempotents in particular, so that orthogonality of similarity orbits of
two idempotents is equivalent to a pair of spectral radius properties. To
conclude with, a couple of localization theorems specific to idempotents and
algebraic elements are presented. Similar statements to all of the above hold
if $a^{\prime}b^{\prime}$ is replaced by $a^{\prime}+b^{\prime}$,
$a^{\prime}-b^{\prime}$, or $a^{\prime}+b^{\prime}-a^{\prime}b^{\prime}$.

###### Key words and phrases:

Banach algebra, spectrum, spectral radius, similarity orbit, idempotent,
algebraic element

###### 2020 Mathematics Subject Classification:

46H05, 46H15, 47A10

## 1 Introduction

We will assume throughout this paper that $(A,\left\lVert\cdot\right\rVert)$
is a semisimple complex and unital Banach algebra with unit denoted by
$\mathbf{1}$. Furthermore, the group of invertible elements in $A$ will be
denoted by $G(A)$, with the principal component containing $\mathbf{1}$
denoted by $G_{\mathbf{1}}(A)$ (or simply $G_{\mathbf{1}}$ if there is no
confusion). We shall use $\sigma_{A}$ to denote the spectrum

$\sigma_{A}(x)=\\{\lambda\in\mathbb{C}\colon\lambda\mathbf{1}-x\not\in
G(A)\\},$

and $\rho_{A}$ to denote the spectral radius

$\rho_{A}(x)=\sup\\{\lvert\lambda\rvert\colon\lambda\in\sigma_{A}(x)\\},$

of an element $x\in A$ (and obviously drop the subscript if the context is
clear as to which algebra is concerned). The nonzero spectrum of $x\in A$ will
be the set $\sigma^{\prime}(x)=\sigma(x)\setminus\\{0\\}$, and the
polynomially convex hull of the spectrum of $x$ will be denoted using
$\hat{\sigma}(x)$. Geometrically, $\hat{\sigma}(x)$ is the spectrum of $x$
together with all of its holes filled in—which ensures that
$\mathbb{C}\setminus\hat{\sigma}(x)$ is connected. Throughout this paper, we
will use $a\circ b$ to denote the element $a+b-ab$, for any given $a,b\in A$.
Notice then that $a\circ b=\mathbf{1}-(\mathbf{1}-a)(\mathbf{1}-b)$. The key
ideas in our study will utilise and be concerned with the similarity orbit of
an element $a\in A$, defined as:

$\operatorname{orb}(a)\coloneqq\left\\{waw^{-1}\colon w\in
G_{\mathbf{1}}\right\\}.$

Notice that, since $G_{\mathbf{1}}$ is a group, the collection of all
similarity orbits of elements of $A$ forms a partition of $A$. Due to the
nature of the elements of the similarity orbit, Jacobson’s Lemma is favourable
in the matter of reducing the work required to prove certain facts in our
upcoming results. As a reminder, Jacobson’s Lemma says that the following is
true for all $a,b\in A$:

$\sigma^{\prime}(ab)=\sigma^{\prime}(ba).$

Because $G(A)$ is a group, one can quickly show (without using the above
equality) that $\sigma(a)=\sigma(waw^{-1})$ for all $w\in G_{\mathbf{1}}(A)$.

If $E$ is the set of idempotents in a Banach algebra $A$ and $a\in E$, then J.
Zemánek showed, amongst a large number of other results, that the connected
component of $E$ containing $a$ is precisely the similarity orbit of $a$ ([5,
Theorem 3.3]). Likewise, in 2016, E. Makai together with J. Zemánek in a
collaborative effort had proven that if a polynomial $p$ whose roots all have
multiplicity 1 is fixed, then the connected components of the set
$E(p)=\\{x\in A\colon p(x)=0\\}$ are precisely the similarity orbits of each
of its members ([4, Theorem 3]). The connectedness of the similarity orbits in
$E(p)$ is deeply investigated in this latter paper, particularly connectedness
via polynomial and polygonal paths. Another important fact that will be used
in a few instances is that $a$ is central  if and only if
$\operatorname{orb}(a)=\\{a\\}$. Hence, if
$a^{\prime}\in\operatorname{orb}(a)$ is central, then $a=a^{\prime}$ since
$\operatorname{orb}(a)=\operatorname{orb}(a^{\prime})=\\{a^{\prime}\\}$.

The last major result in [5] provides ten statements which are equivalent to
an idempotent $a$ being central. In fact, these conditions are all intricately
related to the similarity orbit and the spectrum of $a$. To illustrate the
various ideas that will henceforth be discussed, it is worthwhile to list some
of the conditions given in [5, Theorem 4.3]. If $a\in E$, then the following
are equivalent:

  1. (i)

$a$ is central;

  2. (ii)

$aE\subseteq E$;

  3. (iii)

$a\operatorname{orb}(a)\subseteq\operatorname{orb}(a)$;

  4. (iv)

$\displaystyle\sup_{r\in\operatorname{orb}(a)}\rho(ar)<\infty$;

  5. (v)

$\displaystyle\sup_{r\in\operatorname{orb}(a)}\rho(r+a)<\infty$;

  6. (vi)

$\displaystyle\sup_{r\in\operatorname{orb}(a)}\rho(r-a)<\infty$;

  7. (vii)

$\displaystyle\sup_{r\in\operatorname{orb}(a)}\rho(a\circ r)<\infty$;

  8. (viii)

$a\circ E\subseteq E$;

  9. (ix)

$a\circ\operatorname{orb}(a)\subseteq\operatorname{orb}(a)$.

In the present paper, we not only show this list can be extended to members of
$E(p)$, but also a portion thereof is true for arbitrary Banach algebra
elements. Furthermore, we had sought to obtain new information concerning the
interactions between the elements belonging to different similarity orbits,
say $\operatorname{orb}(a)$ and $\operatorname{orb}(b)$, in $A$. This is in
stark contrast to the results in [5] where members of $\operatorname{orb}(a)$
and only their interactions with $a$ were studied. To clarify, compare the
next question with (iv) in the above list: Suppose there is some $\epsilon>0$
such that $\rho(ar)\leq\epsilon$ for all $r\in\operatorname{orb}(b)$; what is
the connection between $\operatorname{orb}(a)$ and $\operatorname{orb}(b)$? We
answer this question via Corollary 2.5, which says that, under the above
premise, the similarity orbits must commute (with the converse also being
true). Keep in mind that there need not be anything extra assumed for the
elements; they need not necessarily be idempotents or even algebraic.
Moreover, we also consider other algebraic variations, namely, $r+a$, $r-a$,
and $a\circ r$, as $r$ varies through $\operatorname{orb}(b)$—and the same
conclusions relating $\operatorname{orb}(a)$ and $\operatorname{orb}(b)$ are
deduced.

The improvements to the results due to Zemánek which we have in our current
paper are corollaries to a more general theorem—namely—Theorem 2.2. In order
to guarantee the commutation of the similarity orbits of $a$ and $b$, we need
only require that the union of all $\sigma^{\prime}(ar)$—as $r$ runs through
$\operatorname{orb}(b)$—misses at least one nonzero complex number. We obtain
the same statement for $r+a$ and $r-a$ and an ever-so-slightly weaker
statement for $a\circ r$. The details are thoroughly discussed in the theorem
together with its proof. This is the highlighting result of Section 2.
Rounding off this section is Theorem 2.6, wherein we obtain an improvement
upon two classical characterizations of commutative semisimple Banach algebras
related to subadditivity and submultiplicativity (respectively) of the
spectral radius. The precise details of this will be expounded upon thoroughly
prior to Theorem 2.6.

In Section 3, a weak localization of Theorem 2.2 is proven. Instead of $r$
varying throughout $\operatorname{orb}(b)$, we consider when $r$ varies in
some open subset $N$ of $\operatorname{orb}(b)$—but—we must add a few
additional conditions in the premise. The first being that
$\sigma(ar)\subseteq C$, for all $r\in N$ where $C$ is a fixed countable
subset of $\mathbb{C}$; the second assumption is that $ar$ has a finite
spectrum for some $r\in\operatorname{orb}(b)$. Then we can conclude that the
similarity orbits of $a$ and $b$ commute. These additional assumptions are
actually quite natural when one considers algebraic elements, as we do in the
next section—wherein we apply the theory resulting from the prior sections in
Section 4 to algebraic elements, and then to idempotents, so that new facts
concerning the similarity orbits of these types of elements are brought to
light.

In this final section, supposing that $a$ and $b$ belong to $E(p)$, we expose
some conditions under which $a^{\prime}b^{\prime}=\mu\mathbf{1}$, for all
$(a^{\prime},b^{\prime})\in\operatorname{orb}(a)\times\operatorname{orb}(b)$—and
$\mu\in\mathbb{C}$ is fixed. Then, appropriately, if $a,b\in E$, it turns out
that $\mu=0$, so that the similarity orbits of $a$ and $b$ are orthogonal. The
pair of concluding results in this section discusses some localization aspects
regarding idempotents and algebraic elements in a similar vein to the main
theorem of Section 3.

Much of the early work in this paper deals with representation theory for
Banach algebras. For a brief idea of a typical argument using this, one can
look at the proof of [3, Theorem 2.3] wherein a characterization of the
Jacobson radical, $\operatorname{rad}(A)$, is given. A Banach algebra $A$ is
said to be semisimple when $\operatorname{rad}(A)=\\{0\\}$. We shall use the
fact that $\operatorname{rad}(A)$ is precisely those members of $A$ which
belong to the kernels of all continuous irreducible representations $\pi$ of
$A$ throughout our work (see [1, Theorem 4.2.1]). Along with that, the next
result due to A. Sinclair will be crucial to our proofs:

######  Theorem 1.1 (Sinclair Density Theorem, [1, Corollary 4.2.6]).

Let $\pi$ be a continuous irreducible representation of a Banach algebra $A$
on a Banach space $X$. If $\left\\{\xi_{1},\ldots,\xi_{n}\right\\}$ and
$\left\\{\eta_{1},\ldots,\eta_{n}\right\\}$ are linearly independent subsets
of $X$, then there exists $y\in G_{\mathbf{1}}(A)$ such that

$\pi(y)\xi_{i}=\eta_{i}\text{ for each }i\in\\{1,\ldots,n\\}.$

We set forth a few additional conventions which will be the standard for the
rest of this paper. Let $L,M\subseteq\mathbb{C}$. To denote the number of
distinct elements in $L$, we will simply write $\\#L$, and we also define the
following sets:

$\displaystyle LM\coloneqq$ $\displaystyle\left\\{\lambda\mu\colon\lambda\in
L,\mu\in M\right\\},$ $\displaystyle L+M\coloneqq$
$\displaystyle\left\\{\lambda+\mu\colon\lambda\in L,\mu\in M\right\\},$
$\displaystyle\alpha L\coloneqq$
$\displaystyle\left\\{\alpha\lambda\colon\lambda\in L\right\\}\text{ for
}\alpha\in\mathbb{C}.$

For the sake of brevity, we will let $L^{2}$ denote the set $LM$ whenever
$L=M$. If $a,b\in A$ commute, then the following spectral containments
concerning their sum and product are true:

$\sigma(a+b)\subseteq\sigma(a)+\sigma(b),\quad\sigma(ab)\subseteq\sigma(a)\sigma(b).$

For any metric space $\mathcal{M}$, we will denote the open and closed balls
with centre $x\in\mathcal{M}$ and radius $\epsilon$ by
$B_{\mathcal{M}}(x,\epsilon)$ and $\overline{B}_{\mathcal{M}}(x,\epsilon)$,
respectively.

## 2 Commutativity of Similarity Orbits

The lemma below is the backbone of the proof of the main result—Theorem 2.2—in
that most of the nuances regarding the representation theory used lie herein.
The proof hinges on the fact that, if $X$ is a Banach space and
$T\in\mathcal{L}(X)$, the Banach algebra of bounded linear operators on $X$,
then $T$ is a scalar multiple of the identity operator whenever every nonzero
$\xi\in X$ is an eigenvector of $T$. Hence if $T$ is non-central we must be
able to find $\xi\in X$ such that $\\{T\xi,\xi\\}$ is linearly independent in
$X$.

######  Lemma 2.1.

Let $\pi$ be a continuous irreducible representation of a Banach algebra $A$
on a Banach space $X$. Suppose that $\pi(a)$ is non-central in
$\mathcal{L}(X)$ and $\alpha\in\mathbb{C}\setminus\\{0\\}$. Then there exist
$\eta\in X\setminus\\{0\\}$ and $\\{x,y,z\\}\subseteq G_{\mathbf{1}}(A)$ such
that $\\{\eta,\pi(a)\eta\\}$ is a linearly independent set and

$\displaystyle\pi(x)^{-1}\pi(a)\pi(x)\pi(a)\eta=$ $\displaystyle\alpha\eta,$
$\displaystyle\left(\pi(y)^{-1}\pi(a)\pi(y)+\pi(a)\right)\eta=$
$\displaystyle\alpha\eta,$
$\displaystyle\left(\pi(z)^{-1}\pi(a)\pi(z)-\pi(a)\right)\eta=$
$\displaystyle\alpha\eta.$

Moreover, there exist $\gamma\in X\setminus\\{0\\}$ and $w\in
G_{\mathbf{1}}(A)$ such that $\\{\gamma,\pi(a)\gamma\\}$ is linearly
independent and $\pi\left((w^{-1}aw)\circ a\right)\gamma=(1-\alpha)\gamma.$

###### Proof.

If $\\{\pi(a)\eta,\eta\\}$ is a linearly dependent set in $X$ for all $\eta\in
X$, then it can be shown that $\pi(a)$ is a scalar multiple of the identity
operator, implying that $\pi(a)$ is central—which is obviously a
contradiction. So let $\eta\neq 0$ be such that $\\{\pi(a)\eta,\eta\\}$ is a
linearly independent set. We can certainly find some $\lambda\in\mathbb{C}$
such that $\lambda^{2}=\alpha$, and then $\\{\pi(a)\eta,\lambda\eta\\}$ is
also linearly independent. By Sinclair’s Density Theorem we obtain some $x\in
G_{\mathbf{1}}(A)$ such that

$\displaystyle\pi(x)\pi(a)\eta=$ $\displaystyle\lambda\eta,$
$\displaystyle\pi(x)(\lambda\eta)=$ $\displaystyle\pi(a)\eta.$

Rewriting the second equality here gives $\lambda\eta=\pi(x)^{-1}\pi(a)\eta$.
Therefore,

$\pi\left(x^{-1}axa\right)\eta=\pi\left(x^{-1}a\right)(\lambda\eta)=\lambda^{2}\eta=\alpha\eta.$

One can easily show that $\\{\beta\pi(a)\eta-\beta\alpha\eta,\eta\\}$ is
linearly independent for all complex numbers $\beta\neq 0$. Then recall
$\\{\pi(a)\eta,\eta\\}$ is linearly independent, so that an application of
Sinclair’s Density Theorem yields some $y_{\beta}\in G_{\mathbf{1}}(A)$ such
that

$\displaystyle\pi\left(y_{\beta}\right)\pi(a)\eta=$
$\displaystyle\beta\pi(a)\eta-\beta\alpha\eta$
$\displaystyle\pi\left(y_{\beta}\right)\eta=$ $\displaystyle\eta.$

With the second equality above we have that

$\pi(a)\pi\left(y_{\beta}\right)\eta=\pi(a)\eta,$

and then the following is true:

$\displaystyle\pi\left(y_{\beta}\right)^{-1}\left[\pi\left(y_{\beta}\right)\pi(a)+\pi(a)\pi\left(y_{\beta}\right)\right]\eta=$
$\displaystyle\pi\left(y_{\beta}\right)^{-1}\left[(\beta+1)\pi(a)\eta-\beta\alpha\eta\right],$
$\displaystyle\pi\left(y_{\beta}\right)^{-1}\left[\pi(a)\pi\left(y_{\beta}\right)-\pi\left(y_{\beta}\right)\pi(a)\right]\eta=$
$\displaystyle\pi\left(y_{\beta}\right)^{-1}\left[(1-\beta)\pi(a)\eta+\beta\alpha\eta\right].$

Since $\pi\left(y_{\beta}\right)^{-1}\eta=\eta$, taking $y=y_{\beta}$ when
$\beta=-1$ gives us

$\left(\pi(y)^{-1}\pi(a)\pi(y)+\pi(a)\right)\eta=\alpha\eta.$

On the other hand if we set $z=y_{\beta}$ when $\beta=1$ we obtain

$\left(\pi(z)^{-1}\pi(a)\pi(z)-\pi(a)\right)\eta=\alpha\eta.$

Finally, if we take $a^{\prime}=\mathbf{1}-a$, then $\pi(a^{\prime})$ is non-
central, and hence, by the first part, there is $w\in G_{\mathbf{1}}(A)$ and
$\gamma\in X\setminus\\{0\\}$ such that $\\{\pi(a^{\prime})\gamma,\gamma\\}$
is linearly independent and

$\pi(w^{-1}a^{\prime}wa^{\prime})\gamma=\alpha\gamma.$

Hence it follows that $\\{\pi(a)\gamma,\gamma\\}$ is linearly independent and

$\displaystyle\pi((w^{-1}aw)\circ a)\gamma=$
$\displaystyle\pi(\mathbf{1}-(\mathbf{1}-w^{-1}aw)(\mathbf{1}-a))\gamma$
$\displaystyle=$
$\displaystyle\pi(\mathbf{1}-w^{-1}a^{\prime}wa^{\prime})\gamma=\gamma-\alpha\gamma=(1-\alpha)\gamma.$

∎

Since most of the technicalities were taken care of in the above lemma, the
proof of Theorem 2.2 is straightforward. The key point here is that the
eigenvalues of an operator $\pi(a)$ (for any continuous irreducible
representation $\pi$ of $A$) belong to $\sigma_{A}(a)$. Considering the
hypothesis of the lemma once more, we observe that we have near-total freedom
in our choice of the complex number $\alpha$ (the only restriction being that
it is nonzero, with 1 also being excluded for the form $a\circ r$). From this,
conditions (1)–(4) become obvious.

######  Theorem 2.2.

Let $A$ be a semisimple Banach algebra and suppose that $a,b\in A$. Then every
element of $\operatorname{orb}(a)$ commutes with every element of
$\operatorname{orb}(b)$ if and only if any one of the following holds:

(1)
$\displaystyle\bigcup_{r\in\operatorname{orb}(b)}\sigma^{\prime}(ar)\neq\mathbb{C}\setminus\\{0\\}$
(2)
$\displaystyle\bigcup_{r\in\operatorname{orb}(b)}\sigma^{\prime}(r+a)\neq\mathbb{C}\setminus\\{0\\}$
(3)
$\displaystyle\bigcup_{r\in\operatorname{orb}(b)}\sigma^{\prime}(r-a)\neq\mathbb{C}\setminus\\{0\\}$
(4)
$\displaystyle\bigcup_{r\in\operatorname{orb}(b)}\left(\sigma^{\prime}(a\circ
r)\setminus\\{1\\}\right)\neq\mathbb{C}\setminus\\{0,1\\}.$

###### Proof.

For the forward implication, we may use commutation of $r$ and $a$ to obtain

$\displaystyle\sigma(ar)$ $\displaystyle\subseteq\sigma(a)\sigma(r),$
$\displaystyle\quad\sigma(r+a)$ $\displaystyle\subseteq\sigma(a)+\sigma(r),$
$\displaystyle\sigma(r-a)$ $\displaystyle\subseteq\sigma(r)-\sigma(a),$
$\displaystyle\quad\sigma(a\circ r)$
$\displaystyle\subseteq\sigma(a)+\sigma(r)-\sigma(a)\sigma(r).$

Recall that $\sigma(r)=\sigma(b)$ for all $r\in\operatorname{orb}(b)$, and
hence our claim is true. With regards to the converse, we will show
commutation of $a$ and $b$ under the assumption of (1)-(4) individually. So
suppose that $ab\neq ba$, and let (1) hold. By semisimplicity there is a
continuous irreducible representation $\pi$ of $A$ on some Banach space $X$
such that $\pi(a)$ and $\pi(b)$ are not central. Let
$\alpha\in\mathbb{C}\setminus\\{0\\}$ such that $\alpha\not\in\sigma(ar)$ for
all $r\in\operatorname{orb}(b)$. Apply Lemma 2.1 to obtain some $\eta\neq 0$
and $y\in G_{\mathbf{1}}(A)$ such that $\\{\pi(a)\eta,\eta\\}$ is linearly
independent and

$\pi(y)^{-1}\pi(a)\pi(y)\pi(a)\eta=\alpha\eta.$

Moreover, fix $\gamma\neq 0$ such that $\left\\{\pi(b)\gamma,\gamma\right\\}$
is linearly independent, so that with Sinclair’s Density Theorem there must
exist $w\in G_{\mathbf{1}}(A)$ such that

$\displaystyle\pi(w)\pi(a)\eta=$ $\displaystyle\pi(b)\gamma,$
$\displaystyle\pi(w)\eta=$ $\displaystyle\gamma.$

Therefore, one has that

$\displaystyle\pi\left(y^{-1}ayw^{-1}bw\right)\eta=\pi\left(y^{-1}ay\right)\pi\left(w^{-1}b\right)\gamma=\pi\left(y^{-1}ay\right)\pi(a)\eta=\alpha\eta.$

Notice that this together with Jacobson’s Lemma forces
$\alpha\in\sigma\left(ayw^{-1}bwy^{-1}\right)$ which is a contradiction to our
choice of $\alpha$. Thus, $a$ and $b$ commute.
Now assume that (2) holds but $ab\neq ba$ and let
$\alpha\in\mathbb{C}\setminus\\{0\\}$ such that $\alpha\not\in\sigma(a+r)$ for
all $r\in\operatorname{orb}(b)$. Once more, there is a continuous irreducible
representation $\pi$ on $A$ for some Banach space $X$ such that $\pi(a)$ and
$\pi(b)$ are not central. With Lemma 2.1 we obtain some $\eta\neq 0$ and $y\in
G_{\mathbf{1}}(A)$ such that $\\{\pi(a)\eta,\eta\\}$ is linearly independent
and

$\left(\pi(y)^{-1}\pi(a)\pi(y)+\pi(a)\right)\eta=\alpha\eta.$

Also fix $\gamma\neq 0$ such that $\\{\pi(b)\gamma,\gamma\\}$ is a linearly
independent set. By Sinclair’s Density Theorem there exists $w\in
G_{\mathbf{1}}(A)$ such that

$\displaystyle\pi(w)\pi(a)\eta=$ $\displaystyle\pi(b)\gamma,$
$\displaystyle\pi(w)\eta=$ $\displaystyle\gamma.$

Hence it follows that

$\displaystyle\pi\left(y^{-1}ay+w^{-1}bw\right)\eta=$
$\displaystyle\pi\left(y^{-1}ay\right)\eta+\pi\left(w^{-1}bw\right)\eta=\pi\left(y^{-1}ay\right)\eta+\pi\left(w^{-1}b\right)\gamma$
$\displaystyle=$
$\displaystyle\pi\left(y^{-1}ay\right)\eta+\pi(a)\eta=\alpha\eta.$

Notice that this implies $\alpha\in\sigma\left(a+yw^{-1}bwy^{-1}\right)$ which
is a contradiction to our choice of $\alpha$. Thus, $a$ and $b$ commute.
The case for (3) is analogous to the one above. So assume that (4) holds but
$ab\neq ba$. Then $\pi(a)$ and $\pi(b)$ are non-central for a continuous
irreducible representation $\pi$ of $A$ on a Banach space $X$. Pick
$\alpha^{\prime}\not\in\mathbb{C}\setminus\\{0,1\\}$, so that
$1-\alpha^{\prime}\neq 0$. Therefore, Lemma 2.1 says that there exist
$\zeta\in X\setminus\\{0\\}$ and $z\in G_{\mathbf{1}}(A)$ such that
$\\{\pi(a)\zeta,\zeta\\}$ is linearly independent and

$\pi((z^{-1}az)\circ a)\zeta=\alpha^{\prime}\zeta.$

Moreover, let $\omega\in X\setminus\\{0\\}$ be chosen so that
$\\{\pi(b)\omega,\omega\\}$ is linearly independent in $X$; hence, Sinclair’s
Density Theorem gives rise to $u\in G_{\mathbf{1}}(A)$ such that

$\displaystyle\pi(u)\pi(a)\zeta=$ $\displaystyle\pi(b)\omega,$
$\displaystyle\pi(u)\zeta=$ $\displaystyle\omega.$

Now observe that we have the following:

$\displaystyle\pi((zaz^{-1})\circ(u^{-1}bu))\zeta=$
$\displaystyle\pi(zaz^{-1})\zeta+\pi(u^{-1}bu)\zeta-\pi(zaz^{-1}u^{-1}bu)\zeta$
$\displaystyle=$
$\displaystyle\pi(zaz^{-1})\zeta+\pi(a)\zeta-\pi(zaz^{-1})\pi(a)\zeta$
$\displaystyle=$ $\displaystyle\pi((zaz^{-1})\circ
a)\zeta=\alpha^{\prime}\zeta.$

Thus, Jacobson’s Lemma says that
$\alpha^{\prime}\in\sigma(a\circ(z^{-1}u^{-1}buz))$. But because
$\alpha^{\prime}$ was arbitrary in $\mathbb{C}\setminus\\{0,1\\}$, we now have
a contradiction to (4). So $a$ and $b$ do commute.
Finally, observe that, for all the above arguments, we may use Jacobson’s
Lemma to replace $a$ and $b$ by arbitrary members coming from
$\operatorname{orb}(a)$ and $\operatorname{orb}(b)$ respectively, so that the
proof is now complete. ∎

Under commutation of the similarity orbits, it follows from the commutation of
$a$ and $r$ (for each $r\in\operatorname{orb}(b)$) that $\sigma(ar)$,
$\sigma(r+a)$, $\sigma(r-a)$, and $\sigma(a\circ r)$ are all contained in
respective fixed compact subsets of $\mathbb{C}$ (specific to each of the four
forms). The precise details of this is actually used in the proof below. Now
by replacing $\sigma$ here by $\hat{\sigma}$ (the polynomially convex hull of
the spectrum), we get not just set containment—but equality to some fixed
compact subsets of the complex plane.

######  Corollary 2.3.

Let $A$ be a semisimple Banach algebra and suppose that $a,b\in A$. If any one
of the conditions (1)-(4) holds, then for each $r\in\operatorname{orb}(b)$ we
have that

$\displaystyle\hat{\sigma}(ar)$ $\displaystyle=\hat{\sigma}(ab),\quad$
$\displaystyle\hat{\sigma}(r+a)$ $\displaystyle=\hat{\sigma}(b+a),$
$\displaystyle\hat{\sigma}(r-a)$ $\displaystyle=\hat{\sigma}(b-a),\quad$
$\displaystyle\hat{\sigma}(a\circ r)$ $\displaystyle=\hat{\sigma}(a\circ b).$

###### Proof.

If any of (1)-(4) is true, then by Theorem 2.2 it follows that $a$ commutes
with any $r\in\operatorname{orb}(b)$. Suppose that $r=e^{x_{1}}\cdots
e^{x_{n}}be^{-x_{n}}\cdots e^{-x_{1}}$ is a fixed member of
$\operatorname{orb}(b)$ and define the entire function
$f\colon\mathbb{C}\to\operatorname{orb}(b)$ as follows:

$f(\lambda)\coloneqq e^{\lambda x_{1}}\cdots e^{\lambda x_{n}}be^{-\lambda
x_{n}}\cdots e^{-\lambda x_{1}}.$

Since $a$ commutes with $f(\lambda)$ and $\sigma(f(\lambda))=\sigma(b)$, for
all $\lambda\in\mathbb{C}$, we have the following:

$\displaystyle\sigma(f(\lambda)+a)$
$\displaystyle\subseteq\sigma(b)+\sigma(a),\quad$
$\displaystyle\sigma(f(\lambda)-a)$
$\displaystyle\subseteq\sigma(b)-\sigma(a),$ $\displaystyle\sigma(a\circ
f(\lambda))$
$\displaystyle\subseteq\sigma(a)+\sigma(b)-\sigma(a)\sigma(b)\quad$
$\displaystyle\sigma(af(\lambda))$ $\displaystyle\subseteq\sigma(a)\sigma(b).$

Hence Liouville’s Spectral Theorem ([1, Theorem 3.4.14]) implies that
$\hat{\sigma}(af(\lambda))$, $\hat{\sigma}(f(\lambda)+a)$,
$\hat{\sigma}(f(\lambda)-a)$, $\hat{\sigma}(a\circ f(\lambda))$ are all
constant, so we have that

$\displaystyle\hat{\sigma}(ar)$
$\displaystyle=\hat{\sigma}(af(1))=\hat{\sigma}(af(0))=\hat{\sigma}(ab),$
$\displaystyle\hat{\sigma}(r+a)$
$\displaystyle=\hat{\sigma}(f(1)+a)=\hat{\sigma}(f(0)+a)=\hat{\sigma}(b+a),$
$\displaystyle\hat{\sigma}(r-a)$
$\displaystyle=\hat{\sigma}(f(1)-a)=\hat{\sigma}(f(0)-a)=\hat{\sigma}(b-a),$
$\displaystyle\hat{\sigma}(a\circ r)$ $\displaystyle=\hat{\sigma}(a\circ
f(1))=\hat{\sigma}(a\circ f(0))=\hat{\sigma}(a\circ b).$

Since $r$ was arbitrarily chosen from $\operatorname{orb}(b)$, we now have the
result. ∎

Our next corollary considers the case where $a$ and $b$ have the same
similarity orbit. It turns out that $a$ commutes with every member of $A$ when
we have that $\operatorname{orb}(a)=\operatorname{orb}(b)$ and if one of
(1)-(4) holds.

######  Corollary 2.4.

Let $A$ be a semisimple Banach algebra and suppose that $a\in A$. Then $a$ is
central if and only if any one of the following holds:

(5)
$\displaystyle\bigcup_{r\in\operatorname{orb}(a)}\sigma^{\prime}(ar)\neq\mathbb{C}\setminus\\{0\\}$
(6)
$\displaystyle\bigcup_{r\in\operatorname{orb}(a)}\sigma^{\prime}(r+a)\neq\mathbb{C}\setminus\\{0\\}$
(7)
$\displaystyle\bigcup_{r\in\operatorname{orb}(a)}\sigma^{\prime}(r-a)\neq\mathbb{C}\setminus\\{0\\}$
(8)
$\displaystyle\bigcup_{r\in\operatorname{orb}(a)}\left(\sigma^{\prime}(a\circ
r)\setminus\\{1\\}\right)\neq\mathbb{C}\setminus\\{0,1\\}.$

###### Proof.

Clearly, when $a$ is central, we will have that
$\operatorname{orb}(a)=\\{a\\}$, so then (5)-(8) are all true. Conversely, if
any of the conditions (5)-(8) holds, then by Theorem 2.2 it follows that $a$
commutes with every member of $\operatorname{orb}(a)$. If, to the contrary,
$a$ is not central, then there is a continuous irreducible representation
$\pi$ on a Banach space $X$ such that $\pi(a)$ is non-central. Pick
$\alpha\in\mathbb{C}\setminus\left(\\{0\\}\cup\sigma(a)^{2}\right)$. By Lemma
2.1 there exist $w\in\operatorname{orb}(a)$ and $0\neq\eta\in X$ such that

$\pi(w^{-1}awa)\eta=\alpha\eta.$

Hence, $\alpha\in\sigma(w^{-1}awa)$. But because $a$ commutes with $w^{-1}aw$,
we may conclude that
$\sigma(w^{-1}awa)\subseteq\sigma(w^{-1}aw)\sigma(a)=\sigma(a)^{2}$, and
therefore $\alpha\in\sigma(a)^{2}$—which is clearly a contradiction.
Therefore, $a$ is central. ∎

The following corollary, which is immediate from Theorem 2.2 and Corollary
2.4, generalizes some of the statements given in [5, Theorem 4.3] to general
Banach algebra elements:

######  Corollary 2.5.

Let $A$ be a semisimple Banach algebra and suppose that $a,b\in A$. Then every
element of $\operatorname{orb}(a)$ commutes with every element of
$\operatorname{orb}(b)$ if and only if any one of the following holds:

  1. (i)

$\displaystyle\sup_{r\in\operatorname{orb}(b)}\rho(ar)<\infty$;

  2. (ii)

$\displaystyle\sup_{r\in\operatorname{orb}(b)}\rho(r+a)<\infty$;

  3. (iii)

$\displaystyle\sup_{r\in\operatorname{orb}(b)}\rho(r-a)<\infty$;

  4. (iv)

$\displaystyle\sup_{r\in\operatorname{orb}(b)}\rho(a\circ r)<\infty$.

In particular, if $\operatorname{orb}(a)=\operatorname{orb}(b)$, then $a$ is
central if and only if any one of the conditions (i)-(iv) holds.

The above corollary can be used to strengthen two well-known characterizations
of commutative semisimple Banach algebras. It is known (see for instance [1,
Corollary 5.2.3] or the main theorem in [6]) that a semisimple Banach algebra
$A$ is commutative if and only if any one of the following conditions hold:

  1. (i)

there exists $M>0$ such that $\rho(a+b)\leq M(\rho(a)+\rho(b))$ for all
$a,b\in A$;

  2. (ii)

there exists $N>0$ such that $\rho(ab)\leq N\rho(a)\rho(b)$ for all $a,b\in
A$.

Obviously, if $A$ is commutative, then we can take $M=N=1$. Observe that both
$M$ and $N$ are independent of all members $a,b\in A$, so the conditions are
required to hold globally on $A$. The following theorem localizes these
results to similarity orbits—which, as a reminder, partition $A$.

######  Theorem 2.6.

Let $A$ be a semisimple Banach algebra. Then the following properties are
equivalent:

  1. (i)

$A$ is commutative;

  2. (ii)

for each similarity orbit $\operatorname{orb}(a)$ of $A$, there exist
$M_{\operatorname{orb}(a)}>0$ and $a^{\prime}\in\operatorname{orb}(a)$ such
that $\rho(a^{\prime}+r)\leq
M_{\operatorname{orb}(a)}(\rho(a^{\prime})+\rho(r))$ for all
$r\in\operatorname{orb}(a)$;

  3. (iii)

for each similarity orbit $\operatorname{orb}(a)$ of $A$, there exist
$N_{\operatorname{orb}(a)}>0$ and $a^{\prime}\in\operatorname{orb}(a)$ such
that $\rho(a^{\prime}r)\leq N_{\operatorname{orb}(a)}\rho(a^{\prime})\rho(r)$
for all $r\in\operatorname{orb}(a)$.

###### Proof.

If $A$ is commutative then (ii) and (iii) are trivially true. So suppose that
(ii) holds. Then, for each $r\in\operatorname{orb}(a)$, it follows that

$\rho(a^{\prime}+r)\leq
M_{\operatorname{orb}(a)}(\rho(a^{\prime})+\rho(r))=2M_{\operatorname{orb}(a)}\rho(a),$

whence Corollary 2.5 implies that $a^{\prime}$ is central, and so,
$a^{\prime}=a$. Since $a\in A$ was arbitrary, it follows that $A$ is
commutative. If (iii) holds then the argument is analogous. ∎

## 3 Localization to Neighbourhoods in Similarity Orbits

If $K_{1}$ and $K_{2}$ are nonempty compact subsets of $\mathbb{C}$, then the
Hausdorff distance between these sets is defined as:

$\Delta(K_{1},K_{2})\coloneqq\max\left\\{\sup_{\lambda\in
K_{1}}\left(\inf_{\alpha\in
K_{2}}\lvert\lambda-\alpha\rvert\right),\sup_{\lambda\in
K_{2}}\left(\inf_{\alpha\in K_{1}}\lvert\lambda-\alpha\rvert\right)\right\\}.$

The main result of this section is Theorem 3.1 and, for the sake of brevity,
we state it for only one type of operation, namely, multiplication. That is,
the hypothesis and conclusion to the theorem stated below pertain only to the
spectra of elements of the form $ra$ as $r$ varies through
$\operatorname{orb}(b)$. The reader will no doubt find after perusing the
proof of the theorem that there is really no need to limit the result to
products of elements, but we may obtain identical results for each of the
remaining combinations of elements: $r+a$, $r-a$ and $r\circ a$. Refer to the
remarks at the end of this section to view the tweaks necessary to achieve
this.

######  Theorem 3.1.

Let $A$ be a semisimple Banach algebra. Suppose that $a,b\in A$ and $C$ is a
fixed countable subset of $\mathbb{C}$. If there exists a nonempty open subset
$N$ of $\operatorname{orb}(b)$ and a pair
$(a^{\prime},b^{\prime})\in\operatorname{orb}(a)\times\operatorname{orb}(b)$
such that $\sigma(ra)\subseteq C$ for all $r\in N$ and
$\\#\sigma(b^{\prime}a^{\prime})<\infty$, then every member of
$\operatorname{orb}(a)$ commutes with every member of $\operatorname{orb}(b)$.
Moreover, we have that $\sigma(ra)=\sigma(ba)$ for all
$r\in\operatorname{orb}(b)$.

The following two lemmas are a requisite for proving Theorem 3.1, and the
hypothesis thereof will be assumed in the premise of each of these lemmas.

######  Lemma 3.2.

$\sigma(ra)$ is countable for all $r\in\operatorname{orb}(b)$.

###### Proof.

Because the map $v\mapsto vbv^{-1}$ is continuous on $G_{\mathbf{1}}$, it
follows that there exists an open ball $B=B_{A}(v_{0},\epsilon)$ in
$G_{\mathbf{1}}$ such that $v\in B$ implies $vbv^{-1}\in N$ and hence
$\sigma(vbv^{-1}a)\subseteq C$ for all $v\in B$. Suppose that
$v_{0}=e^{x_{1}}\cdots e^{x_{n}}$. Now take $y=e^{y_{1}}\cdots e^{y_{m}}\in
G_{\mathbf{1}}$ to be arbitrary. By appropriately using $e^{0}=\mathbf{1}$, we
may assume without loss of generality that $m=n$. Next we define the entire
function $f\colon\mathbb{C}\to G_{\mathbf{1}}$ to be

$f(\lambda)\coloneqq e^{(1-\lambda)x_{1}+\lambda y_{1}}\cdots
e^{(1-\lambda)x_{n}+\lambda y_{n}}.$

Then if $g(\lambda)=f(\lambda)b\left[f(\lambda)\right]^{-1}a$, we have via [1,
Theorem 7.1.13] that $\lambda\mapsto\sigma(g(\lambda))$ is an analytic
multifunction on $\mathbb{C}$. Furthermore, the continuity of $f$ says that
there exists $\delta>0$ such that $\lvert\lambda\rvert<\delta$ implies
$f(\lambda)\in B$, whence $\sigma(g(\lambda))\subseteq C$ whenever
$\lvert\lambda\rvert<\delta$. But then the set of $\lambda\in\mathbb{C}$ such
that $\sigma(g(\lambda))$ is at most countable has nonzero capacity—in which
case, [1, Theorem 7.2.8] forces $\sigma(g(\lambda))$ to be at most countable
for all $\lambda\in\mathbb{C}$. In particular then,
$\sigma(g(1))=\sigma(yby^{-1}a)$ is at most countable. Because $y\in
G_{\mathbf{1}}$ was arbitrary, we now have the result. ∎

######  Lemma 3.3.

There exists an open ball $B$ in $G_{\mathbf{1}}$ such that
$\sigma(vbv^{-1}a)$ is constant for all $v\in B$.

###### Proof.

Denote by $\mathcal{K}$ the metric space consisting of all nonempty compact
subsets of $\mathbb{C}$ under the Hausdorff metric. Let $T\colon
G_{\mathbf{1}}\to\mathcal{K}$ be the map $v\mapsto\sigma(vbv^{-1}a)$. The open
ball $B$ will be the same one which we obtained in Lemma 3.2, that is,
$B=B_{A}(v_{0},\epsilon)$. By continuity of $T$ and the connectedness of $B$,
it follows that $T(B)$ is connected in $\mathcal{K}$. To complete the proof,
we need only show that $T(B)$ is a singleton. To this end, we will assume to
the contrary that there is some $v\in B$ such that $T(v)\neq T(v_{0})$. Now
there is an open ball $D_{0}=B_{T(B)}(T(v_{0}),\delta_{0})$ in $T(B)$ to which
$T(v)$ does not belong. Moreover, if there does not exist any $T(y)$ on the
boundary $\partial D_{0}$ of the open ball then that would mean that $T(B)$ is
the union of two nonempty disjoint open sets, that is,

$T(B)=D_{0}\cup(T(B)\setminus\overline{D_{0}}),$

which is obviously a contradiction. So let $T(y_{0})\in\partial D_{0}$.
Therefore, we can use the same reasoning to guarantee that for each real
number $\delta\in\left(0,\delta_{0}\right]$, there is some $y_{\delta}\in B$
such that $T(y_{\delta})\in\partial D_{\delta}$, where
$D_{\delta}=B_{T(B)}(T(v_{0}),\delta)$ here. Recall that by the hypothesis, we
have that $T(v)\subseteq C$ for all $v\in B$, where $C$ is a fixed countable
subset of $\mathbb{C}$. Because the spectrum is nonempty and compact, we have
that if $T(y_{\delta})\in\partial D_{\delta}$ then the distance of
$T(y_{\delta})$ to $T(v_{0})$ is assumed at some complex numbers belonging to
$C$. Therefore, to each $\delta\in(0,\delta_{0}]$ we can associated a pair
$(\alpha_{\delta},\beta_{\delta})\in T(y_{\delta})\times T(v_{0})$ such that
$\lvert\alpha_{\delta}-\beta_{\delta}\rvert=\delta$. Clearly, $C\times C$ is
countable whilst the interval $(0,\delta_{0}]$ is uncountable, and so the
preceding statement leads to the following pigeonhole argument: There must be
distinct real numbers, say $\delta$ and $\gamma$ in $(0,\delta_{0}]$ for which
the corresponding pairs
$(\alpha_{\delta},\beta_{\delta}),(\alpha_{\gamma},\beta_{\gamma})\in
T(y_{\delta})\times T(v_{0})$ are the same. That is,
$\alpha_{\delta}=\alpha_{\gamma}$ and $\beta_{\delta}=\beta_{\gamma}$ in the
above, in addition to the fact that

$\lvert\alpha_{\delta}-\beta_{\delta}\rvert=\delta,\quad\lvert\alpha_{\gamma}-\beta_{\gamma}\rvert=\gamma.$

Notice then that the contradiction $\delta=\gamma$ is reached. ∎

###### Proof of Theorem 3.1.

In Lemma 3.3 we established that $\sigma(vbv^{-1}a)$ is constant for all $v\in
B$, where $B=B_{A}(v_{0},\epsilon)$ is an open ball in $G_{\mathbf{1}}$. Let
us assume that, for each $v\in B$,

$\sigma(vbv^{-1}a)=\left\\{\alpha_{n}\right\\}_{n\in\mathbb{N}}=M,$

where $M\subseteq C$. Suppose that $y\in G_{\mathbf{1}}$ is arbitrary, and
define the functions $f$ and $g$ as in Lemma 3.2, with $v_{0}=e^{x_{1}}\cdots
e^{x_{n}}$ and $y=e^{y_{1}}\cdots e^{y_{n}}$. Then the map
$\lambda\mapsto\sigma(g(\lambda))$ is an analytic multifunction; with Lemma
3.2 we have that $\sigma(g(\lambda))$ is countable for all
$\lambda\in\mathbb{C}$ and by our preceding remarks, there exists $\delta>0$
such that $\lvert\lambda\rvert<\delta$ implies $\sigma(g(\lambda))=M$. Now for
each $n\in\mathbb{N}$ we define the set

$Z_{n}\coloneqq\left\\{\lambda\in\mathbb{C}\colon\alpha_{n}\in\sigma(g(\lambda))\right\\},$

and we observe that $B_{\mathbb{C}}(0,\delta)\subseteq Z_{n}$ for all
$n\in\mathbb{N}$—in which case, [1, Theorem 7.2.13] says that
$Z_{n}=\mathbb{C}$ for each $n\in\mathbb{N}$. In particular then,
$\alpha_{n}\in\sigma(g(1))=\sigma(yby^{-1}a)$ for all $n\in\mathbb{N}$. As $y$
was arbitrarily picked in $G_{\mathbf{1}}$ we can conclude that
$M\subseteq\sigma(yby^{-1}a)$ for all $y\in G_{\mathbf{1}}$.
By the hypothesis and Jacobson’s Lemma, it follows that there is some
$y_{0}\in G_{\mathbf{1}}$ such that $\sigma(y_{0}by_{0}^{-1}a)$ is finite, and
consequently, $M$ is finite. But recall we had that $\sigma(vbv^{-1}a)=M$ for
all $v\in B$. So we may fix an arbitrary $y\in G_{\mathbf{1}}$ as before and
define the corresponding analytic function $g$ according to Lemma 3.2 to
obtain existence of a $\delta>0$ (via continuity of $g$) such that
$\lvert\lambda\rvert<\delta$ implies that $\\#\sigma(g(\lambda))<\infty$.
Hence the Scarcity Theorem (refer to [1, Theorem 3.4.25]) now says that
$\\#\sigma(g(\lambda))\leq\\#M$ for all $\lambda\in\mathbb{C}$. But by the
first part, we have that $\sigma(g(\lambda))$ contains $M$ for all
$\lambda\in\mathbb{C}$ and, in particular, $M\subseteq\sigma(g(1))$, so it
follows that $\sigma(g(1))=M$. But $y$ was arbitrary in $G_{\mathbf{1}}$ so we
conclude that $\sigma(yby^{-1}a)=M$ for all $y\in G_{\mathbf{1}}$. Finally, a
simple application of Theorem 2.2 now gives the result. ∎

######  Remarks.

We conclude this section with the following observations:

  1. (i)

Notice that the condition in Theorem 3.1 assuming that there exists a pair
$(a^{\prime},b^{\prime})\in\operatorname{orb}(a)\times\operatorname{orb}(b)$
such that $\\#\sigma(b^{\prime}a^{\prime})<\infty$ implies the following
weaker statement:

$\\#\left[\bigcap_{r\in\operatorname{orb}(b)}\sigma(ra)\right]<\infty.$

Furthermore, the above statement is sufficient to deduce the last part in the
proof of Theorem 3.1.

  2. (ii)

If we go to the proof of Lemma 3.2 and replace $g(\lambda)$ by one of the
following entire functions:

$\lambda\mapsto f(\lambda)b\left[f(\lambda)\right]^{-1}+a,\quad\lambda\mapsto
f(\lambda)b\left[f(\lambda)\right]^{-1}-a,\quad\lambda\mapsto
f(\lambda)b\left[f(\lambda)\right]^{-1}\circ a,$

then we obtain Theorem 3.1 (in exactly the same manner) for the forms $r+a$,
$r-a$, $r\circ a$ respectively.

## 4 Applications to Algebraic Elements

It will be interesting to consider the results highlighted in the preceding
section for elements in $A$ with more properties—specifically—algebraic
elements. For this, it will be agreed upon that, for the remainder of this
section, $p$ will be a polynomial of the form

$p(\lambda)\coloneqq\prod_{i=1}^{n}\left(\lambda-\lambda_{i}\right),$

where $K=\left\\{\lambda_{1},\ldots,\lambda_{n}\right\\}$ is a fixed set of
distinct complex numbers. The set of elements of $A$ which are algebraic under
$p$ will be denoted by $E(p)$. If $K=\\{0,1\\}$, then the elements of $E(p)$
are idempotents. If $a\in E(p)$, then one can show that $\sigma(a)\subseteq
K$. Something quite useful is the next result which decomposes algebraic
elements into linear combinations of idempotents:

######  Theorem 4.1 ([4, Proposition 1]).

Let $A$ be a Banach algebra and suppose that $a\in A$. Then $a\in E(p)$ if and
only if there exists a set of idempotents
$\left\\{e_{1},\ldots,e_{n}\right\\}$ such that $e_{i}e_{j}=0$ for $i\neq j$,
$\sum_{i=1}^{n}e_{i}=\mathbf{1}$, and

$a=\sum_{i=1}^{n}\lambda_{i}e_{i}.$

In fact, these idempotents are the Riesz idempotents obtained via the
Holomorphic Functional Calculus. Hence if $a,b\in E(p)$ commute and their
decompositions are respectively given by:

$a=\sum_{i=1}^{n}\lambda_{i}e_{i},\quad
b=\sum_{i=1}^{n}\lambda_{i}\tilde{e}_{i},$

then $e_{i}\tilde{e}_{j}=\tilde{e}_{j}e_{i}$ for all $i,j\in\\{1,\ldots,n\\}$.
Of course, $E(p)$ is not necessarily connected; in fact if $a\in E(p)$, then
the connected component of $E(p)$ containing $a$ is $\operatorname{orb}(a)$.
This was established for idempotents in [5] and for algebraic elements in [4].
Before we directly apply the theory obtained in Section 2, we establish that
the product of two commuting algebraic elements is once again algebraic, via
the following proposition:

######  Proposition 4.2.

Let $A$ be a semisimple Banach algebra and suppose that $a,b\in E(p)$. If
every element of $\operatorname{orb}(a)$ commutes with every element of
$\operatorname{orb}(b)$ then
$\operatorname{orb}(a)\operatorname{orb}(b)=\operatorname{orb}(b)\operatorname{orb}(a)$
consists of algebraic elements.

###### Proof.

If we decompose $a^{\prime}\in\operatorname{orb}(a)$ and
$b^{\prime}\in\operatorname{orb}(b)$ according to Theorem 4.1 as

$a^{\prime}=\sum_{i=1}^{n}\lambda_{i}e_{i},\quad
b^{\prime}=\sum_{i=1}^{n}\lambda_{i}\tilde{e}_{i},$

then by direct calculation (and the aid of commutativity and orthogonality of
the Riesz idempotents) we find that

$a^{\prime}b^{\prime}=\sum_{i=1}^{n}\sum_{j=1}^{n}\lambda_{i}\lambda_{j}e_{i}\tilde{e}_{j},=\sum_{\alpha\in
K^{2}}\alpha\hat{e}_{\alpha},$

where $\hat{e}_{\alpha}\hat{e}_{\beta}=0$ for all $\alpha\neq\beta$ in
$K^{2}$, and $\sum_{\alpha\in K^{2}}\hat{e}_{\alpha}=\mathbf{1}$. Therefore,
by Theorem 4.1, $a^{\prime}b^{\prime}$ is algebraic under the polynomial

$q(\lambda)\coloneqq\prod_{\alpha\in K^{2}}(\lambda-\alpha).$

∎

It turns out that when $a$ is algebraic, we have a few additional equivalent
statements to the list in Corollary 2.4. This then mostly completes
generalizing [5, Theorem 4.3] to algebraic elements.

######  Theorem 4.3.

Let $A$ be a semisimple Banach algebra and suppose that $a\in E(p)$. Then the
following are equivalent:

  1. (i)

$a$ is central;

  2. (ii)

there exists a polynomial $q(\lambda)$ such that $aE(p)\subseteq E(q)$;

  3. (iii)

there exists a polynomial $q(\lambda)$ such that
$a\operatorname{orb}(a)\subseteq E(q)$;

  4. (iv)

there exists a polynomial $q(\lambda)$ such that $a\circ E(p)\subseteq E(q)$;

  5. (v)

there exists a polynomial $q(\lambda)$ such that
$a\circ\operatorname{orb}(a)\subseteq E(q)$.

###### Proof.

If $a$ is central, then applying an identical argument as the one given in the
proof for Proposition 4.2 shows that (ii) and (iii) are true. Since $a\circ
b=\mathbf{1}-(\mathbf{1}-a)(\mathbf{1}-b)$, it follows that (i) is also
sufficient for (iv) and (v), for the subsequent reasons: Notice that if $a$ is
algebraic under the polynomial $p$, then so too is $\mathbf{1}-a$, under

$\lambda\mapsto\prod_{i=1}^{n}(1-\lambda_{i}-\lambda).$

Therefore, when $b\in E(p)$, the element $\mathbf{1}-b$ is algebraic as well
and hence, via commutation of $a$ and $b$ (as well as the proof of Proposition
4.2), it follows that $\mathbf{1}-(\mathbf{1}-a)(\mathbf{1}-b)$ is algebraic
under a polynomial $q(\lambda)$. A similar case can be made showing
(i)$\Rightarrow$(v).
If (ii) or (iii) is true, then obviously we must have that

$\sup_{r\in\operatorname{orb}(a)}\rho(ar)<\infty,$

whence Corollary 2.5 implies that $a$ is central. Similarly, if (iv) or (v)
holds, then

$\sup_{r\in\operatorname{orb}(a)}\rho(a\circ r)<\infty,$

so that again $a$ is central. ∎

For the purposes of the next results, define the following:

$t_{1}\coloneqq\sup_{r\in\operatorname{orb}(b)}\rho(ar),\quad
t_{2}\coloneqq\sup_{r\in\operatorname{orb}(b)}\rho(a+r),\quad\kappa\coloneqq\inf\left\\{\lvert\lambda\rvert\colon\lambda\in
K^{2}\right\\}.$

######  Theorem 4.4.

Let $A$ be a semisimple Banach algebra and suppose that $a,b\in E(p)$. If we
have that

$\\#\left\\{\lambda\in K^{2}\colon\lvert\lambda\rvert=\kappa\right\\}=1,$

then there exist $\mu\geq 0$ and $\delta>0$ such that if $t_{1}<\delta$, then

$\operatorname{orb}(a)\operatorname{orb}(b)=\operatorname{orb}(b)\operatorname{orb}(a)=\\{\mu\mathbf{1}\\}.$

In particular, if $a\not\in G(A)$ then

$\operatorname{orb}(a)\operatorname{orb}(b)=\operatorname{orb}(b)\operatorname{orb}(a)=\\{0\\}.$

###### Proof.

Choose $\delta$ to be the real number

$\inf\\{\lvert\lambda\rvert\colon\lambda\in
K^{2},\lvert\lambda\rvert>\kappa\\}.$

Hence if $t_{1}<\delta$, Corollary 2.5 implies that
$a^{\prime}b^{\prime}=b^{\prime}a^{\prime}$ for
$a^{\prime}\in\operatorname{orb}(a)$ and $b^{\prime}\in\operatorname{orb}(b)$.
Furthermore, $a^{\prime}b^{\prime}$ is algebraic and
$\sigma(a^{\prime}b^{\prime})\subseteq K^{2}$. Let $\mu$ be the complex number
in $K^{2}$ satisfying $\lvert\mu\rvert=\kappa$. By Theorem 4.1, if
$a^{\prime}b^{\prime}\neq\mu\mathbf{1}$, it must be the case that
$\sigma(a^{\prime}b^{\prime})\cap K^{2}\neq\\{\mu\\}$—implying that for some
$r\in\operatorname{orb}(b)$ (depending on $a^{\prime}$ and $b^{\prime}$) we
will have that $\rho(ar)\geq\delta$, which is a contradiction.
If $a\notin G(A)$ then $0\in\sigma(a)$, so that $0\in K^{2}$ and hence
$\kappa=\mu=0$. Obviously then $\\#\left\\{\lambda\in
K^{2}\colon\lvert\lambda\rvert=\kappa\right\\}=1$ so by the first part the
orthogonality of $\operatorname{orb}(a)$ and $\operatorname{orb}(b)$ is
apparent. ∎

In the special case for idempotents, we have that $K^{2}=K$, $\delta=1$ and
$\mu=\kappa=0$; that is, $\operatorname{orb}(a)$ and $\operatorname{orb}(b)$
are orthogonal when $t_{1}<1$.

######  Corollary 4.5.

Let $A$ be a semisimple Banach algebra and suppose that $p$ and $q$ are
idempotents in $A$. Then the following statements are equivalent:

  1. (i)

$\displaystyle\operatorname{orb}(p)\operatorname{orb}(q)=\operatorname{orb}(q)\operatorname{orb}(p)=\\{0\\}$,

  2. (ii)

$\displaystyle t_{1}=\sup_{r\in\operatorname{orb}(q)}\rho(pr)<1$,

  3. (iii)

$\displaystyle t_{2}=\sup_{r\in\operatorname{orb}(q)}\rho(p+r)<2$.

###### Proof.

Observe that from Theorem 4.4, we immediately have that

$\operatorname{orb}(p)\operatorname{orb}(q)=\operatorname{orb}(q)\operatorname{orb}(p)=\\{0\\}\quad\iff\quad
t_{1}<1.$

Hence it suffices to show $t_{2}<2$ is equivalent to $t_{1}<1$. If $t_{2}<2$
then Corollary 2.5 implies that every element of $\operatorname{orb}(p)$
commutes with every element of $\operatorname{orb}(q)$. Suppose to the
contrary that $t_{1}\not<1$. By commutation of $p$ and $r$, the element $pr$
is an idempotent for all $r\in\operatorname{orb}(q)$. But not all such
elements are $0$, so fix $r\in\operatorname{orb}(q)$ such that
$\sigma(pr)=\\{0,1\\}$. Using the following result from [2, Theorem 2.2],

(9)
$\lambda\not\in\\{0,1\\}\Rightarrow\left[\lambda\in\sigma(p+r)\iff(\lambda-1)^{2}\in\sigma(pr)\right],$

and due to the fact that $2\not\in\\{0,1\\}$ and $(2-1)^{2}\in\sigma(pr)$, it
follows that $2\in\sigma(p+r)$—which is obviously a contradiction to the
hypothesis. Thus, $t_{1}<1$. Conversely, suppose that $t_{1}<1$. Then by the
first part it follows that $\operatorname{orb}(p)$ and $\operatorname{orb}(q)$
are orthogonal—in which case, $p+r$ is an idempotent for all
$r\in\operatorname{orb}(q)$. Therefore, $t_{2}=1<2$. ∎

######  Remark.

In the proof of Theorem 4.4, the chosen $\delta$ is a sharp upper bound for
$t_{1}$. In Corollary 4.5, $1$ and $2$ are sharp upper bounds for $t_{1}$ and
$t_{2}$, respectively. Of course, by Corollary 2.5, if $t_{1}$ or $t_{2}$ is
finite then the best we can do is infer that the elements of
$\operatorname{orb}(p)$ commute with the elements of $\operatorname{orb}(q)$.

For idempotents $p$ and $q$ in $A$, we have a very strong localization
principle for similarity orbits. We need not even know what $\sigma(pr)$
contains, just that it is a singleton for all $r$ in a small open subset of
$\operatorname{orb}(q)$—in which case—$p$ and $q$ are the identity or their
similarity orbits are orthogonal.

######  Corollary 4.6.

Let $A$ be a semisimple Banach algebra and suppose $p$ and $q$ are idempotents
of $A$ such that $\\#\sigma(pr)=1$ for all $r$ in an open subset $N$ of
$\operatorname{orb}(q)$. Then either $p=q=\mathbf{1}$ or
$\operatorname{orb}(p)\operatorname{orb}(q)=\operatorname{orb}(q)\operatorname{orb}(p)=\\{0\\}$.

###### Proof.

We first claim that the hypothesis can be globalized in the following sense:
$\\#\sigma(pr)=1$ for all $r$ in $\operatorname{orb}(q)$. To see why this is
so, fix a member $e^{x_{1}}\cdots e^{x_{n}}qe^{-x_{n}}\cdots e^{-x_{1}}$ in
$N$ and let

$r=e^{y_{1}}\cdots e^{y_{m}}qe^{-y_{m}}\cdots e^{-y_{1}}$

be arbitrary in $\operatorname{orb}(q)$. Without loss of generality, we may
assume that $m=n$. Define the entire function $f\colon\mathbb{C}\to A$ by:

$f(\lambda)\coloneqq e^{\lambda x_{1}+(1-\lambda)y_{1}}\cdots e^{\lambda
x_{n}+(1-\lambda)y_{n}}.$

Hence by continuity of the map $g\colon\lambda\mapsto
f(\lambda)q\left[f(\lambda)\right]^{-1}$, we also have that there is some disk
$B(1,\delta)$ in $\mathbb{C}$ such that $\lambda\in B(1,\delta)$ implies
$g(\lambda)\in N$— so that $\\#\sigma(pg(\lambda))=1$ for all $\lambda\in
B(1,\delta)$. By the Scarcity Theorem, this must hold for all
$\lambda\in\mathbb{C}$, and in particular, $1=\\#\sigma(pg(0))=\\#\sigma(pr)$.
Hence, $\\#\sigma(pr)=1$ for all $r\in\operatorname{orb}(q)$.
If there is some $r\in\operatorname{orb}(q)$ such that $0\not\in\sigma(pr)$,
then $pr$ is invertible. Moreover, $0=(\mathbf{1}-p)pr$ implies that
$p=\mathbf{1}$. But then, $r\in G(A)$ and hence $r=q=p=\mathbf{1}$. Hence if
$p\neq q$, then

$\sigma(pwqw^{-1})=\\{0\\}\text{ for all }w\in G_{\mathbf{1}}(A).$

Now observe that the value of $t_{1}$ is $0$ in Corollary 4.5, so the
components $\operatorname{orb}(p)$ and $\operatorname{orb}(q)$ are orthogonal,
as advertised. ∎

######  Remarks.

A couple of remarks related to the above corollary are in order.

  1. (i)

The second part of Corollary 4.6 can be directly proven without the use of
Corollary 4.5 with some basic representation theory. A brief outline of the
proof is as follows: Begin by assuming $pq\neq 0$, to obtain a representation
$\pi$ on a Banach space $X$ such that $\pi(pq)\eta\neq 0$, for some $\eta\in
X$. Then set $\xi=\pi(q)\eta$ and argue as to why $\\{\pi(p)\xi,\pi(q)\xi\\}$
is a linearly independent set. Thereafter a simple application of Sinclair’s
Density Theorem to this set will yield the necessary contradiction.

  2. (ii)

If in the hypothesis of the corollary we assumed that $\\#\sigma(pr)\leq n>1$
for all $r\in\operatorname{orb}(q)$, then the result would be invalid. A
simple example illustrating the case for $n=2$ can be obtained from
$M_{2}(\mathbb{C})$. Set

$p=\begin{bmatrix}1&0\\\ 0&0\end{bmatrix},\quad q=\begin{bmatrix}1&0\\\
0&1\end{bmatrix}.$

Then $\\#\sigma(pr)=2$ for all $r\in\operatorname{orb}(q)$ but $p\neq q$ and
$\operatorname{orb}(p)\operatorname{orb}(q)=\operatorname{orb}(p)\neq\\{0\\}$.
One may construct similar examples in $M_{n}(\mathbb{C})$ for larger values of
$n$.

The final result gives a localization principle for algebraic elements, and,
expectedly, its conclusion is stronger than that of Theorem 3.1, in that we
have yet another characterization of central algebraic elements. Recall that
in Theorem 3.1, we required in the premise that $\sigma(b^{\prime}a^{\prime})$
is finite for some pair
$(a^{\prime},b^{\prime})\in\operatorname{orb}(a)\times\operatorname{orb}(b)$.
Here we are looking at just $\operatorname{orb}(a)$, and obviously then
$\sigma(a^{2})$ is finite since $\sigma(a)$ is finite.

######  Corollary 4.7.

Let $A$ be a semisimple Banach algebra and suppose that $a\in A$ is algebraic.
Then $a$ is central if and only if there exist a nonempty countable set
$C\subseteq\mathbb{C}$ and a nonempty open subset $N$ of
$\operatorname{orb}(a)$ such that $\sigma(ra)\subseteq C$ for all $r\in N$.

###### Proof.

If $a$ is central then $\operatorname{orb}(a)=\\{a\\}$ and hence $C$ may be
taken as the finite set $\sigma\left(a^{2}\right)$ to obtain the forward
implication.
For the converse, we take the pair
$(a,a)\in\operatorname{orb}(a)\times\operatorname{orb}(a)$ in Theorem 3.1 to
get that $a$ commutes with all elements of its orbit, so that an identical
argument as the one given in Corollary 2.4 implies the centrality of $a$. ∎

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