can be formulated for vector spaces without modifications.

######  Definition 6.2.1.

Let $\mathcal{V}$ be a vector space and $T:\mathcal{V}\to\mathcal{V}$ be a
linear map. The map $T$ is said to be a shift if
$\cap_{n=0}^{\infty}T^{n}(\mathcal{V})=\\{0\\}$.

######  Theorem 6.2.2.

(Wold decomposition for vector spaces) Let $T$ be an injective linear map on a
vector space $\mathcal{V}$. Then $\mathcal{V}$ decomposes as
$\mathcal{V}=\mathcal{V}_{b}\oplus\mathcal{V}_{s}$, where $\mathcal{V}_{b}$ is
a $T$-invariant subspace of $\mathcal{V}$,
$T_{|\mathcal{V}_{b}}:\mathcal{V}_{b}\to\mathcal{V}_{b}$ is a bijection and
$T_{|\mathcal{V}_{s}}:\mathcal{V}_{s}\to\mathcal{V}$ is a shift.

###### Proof.

Define $\mathcal{V}_{b}\coloneqq\cap_{n=0}^{\infty}T^{n}(\mathcal{V})$ and let
$\mathcal{V}_{s}$ be a vector space complement of $\mathcal{V}_{b}$ in
$\mathcal{V}$. We clearly have
$\mathcal{V}=\mathcal{V}_{b}\oplus\mathcal{V}_{s}$. Now
$T(\mathcal{V}_{b})=T(\cap_{n=0}^{\infty}T^{n}(\mathcal{V}))\subseteq\cap_{n=0}^{\infty}T^{n}(\mathcal{V})=\mathcal{V}_{b}$.
Thus $\mathcal{V}_{b}$ is a $T$-invariant subspace of $\mathcal{V}$. We now
try to show that $T_{|\mathcal{V}_{b}}$ is a bijection. Since $T$ is already
injective, it suffices to show that $T_{|\mathcal{V}_{b}}$ is surjective. Let
$y\in\mathcal{V}_{b}$. Then there exists a sequence
$\\{x_{n}\\}_{n=1}^{\infty}$ in $\mathcal{V}$ such that
$y=Tx_{1}=T^{2}x_{2}=T^{3}x_{3}=\cdots.$ Since $T$ is injective, we then have
$x_{1}=Tx_{2}=T^{2}x_{2}=\cdots$. Therefore $y=Tx_{1}$ and
$x_{1}\in\mathcal{V}_{b}$. Thus $T_{|\mathcal{V}_{b}}$ is surjective. We are
now left with proving that $T_{|\mathcal{V}_{s}}$ is a shift. Let
$y\in\cap_{n=0}^{\infty}(T_{|\mathcal{V}_{s}})^{n}(\mathcal{V}_{s})\subseteq(\cap_{n=0}^{\infty}T^{n}(\mathcal{V}))\cap\mathcal{V}_{s}=\mathcal{V}_{b}\cap\mathcal{V}_{s}$.
Hence $y=0$ which completes the proof. ∎

Since vector space complements are not unique, we do not have uniqueness in
Wold decomposition for vector spaces. We next derive Halmos dilation for
vector spaces.

######  Theorem 6.2.3.

(Halmos dilation for vector spaces) Let $\mathcal{V}$ be a vector space and
$T:\mathcal{V}\to\mathcal{V}$ be a linear map. Then the operator

$\displaystyle U\coloneqq\begin{pmatrix}T&I\\\ I&0\\\ \end{pmatrix}$

is invertible on $\mathcal{V}\oplus\mathcal{V}$. In other words,

$\displaystyle T=P_{\mathcal{V}}U_{|\mathcal{V}},$

where
$P_{\mathcal{V}}:\mathcal{V}\oplus\mathcal{V}\to\mathcal{V}\oplus\mathcal{V}$
is the first coordinate projection onto $\mathcal{V}$.

###### Proof.

It suffices to produce inverse map for $U$. A direct calculation says that

$\displaystyle V\coloneqq\begin{pmatrix}0&I\\\ I&-T\\\ \end{pmatrix}$

is the inverse of $U$. ∎

In the sequel, any invertible operator of the form

$\displaystyle\begin{pmatrix}T&B\\\ C&D\\\ \end{pmatrix},$

where $B,C,D:\mathcal{V}\to\mathcal{V}$ are linear operators, will be called
as a Halmos dilation of $T$. Now we observe that Halmos dilation for vector
spaces is not unique. Using the theory of block matrices (Lu and Shiou,
(2002)) we can produce a variety of Halmos dilations for a given operator.
Following are some classes of Halmos dilations.

  1. (i)

If $T:\mathcal{V}\to\mathcal{V}$ is an invertible linear map and the linear
operators $B,C,D:\mathcal{V}\to\mathcal{V}$ are such that $D-CT^{-1}B$ is
invertible, then the operator

$\displaystyle U\coloneqq\begin{pmatrix}T&B\\\ C&D\\\ \end{pmatrix}\text{ is a
Halmos dilation of $T$ on $\mathcal{V}\oplus\mathcal{V}$ whose inverse is }$
$\displaystyle\begin{pmatrix}T^{-1}+T^{-1}B(D-CT^{-1}B)^{-1}&-T^{-1}B(D-CT^{-1}B)^{-1}\\\
-(D-CT^{-1}B)^{-1}CT^{-1}&(D-CT^{-1}B)^{-1}\\\ \end{pmatrix}.$

  2. (ii)

$D:\mathcal{V}\to\mathcal{V}$ is an invertible linear map and the linear
operators $B,C:\mathcal{V}\to\mathcal{V}$ are such that $T-BD^{-1}C$ is
invertible, then the operator

$\displaystyle\begin{pmatrix}T&B\\\ C&D\\\ \end{pmatrix}\text{ is a Halmos
dilation of $T$ on $\mathcal{V}\oplus\mathcal{V}$ whose inverse is }$
$\displaystyle\begin{pmatrix}(T-BD^{-1}C)^{-1}&-(T-BD^{-1}C)^{-1}BD^{-1}\\\
-D^{-1}C(T-BD^{-1}C)^{-1}&D^{-1}+D^{-1}C(T-BD^{-1}C)^{-1}BD^{-1}\\\
\end{pmatrix}.$

  3. (iii)

$B:\mathcal{V}\to\mathcal{V}$ is an invertible linear map and the linear
operators $C,D:\mathcal{V}\to\mathcal{V}$ are such that $C-DB^{-1}T$ is
invertible, then the operator

$\displaystyle\begin{pmatrix}T&B\\\ C&D\\\ \end{pmatrix}\text{ is a Halmos
dilation of $T$ on $\mathcal{V}\oplus\mathcal{V}$ whose inverse is }$
$\displaystyle\begin{pmatrix}-(C-DB^{-1}T)^{-1}DB^{-1}&(C-DB^{-1}T)^{-1}\\\
B^{-1}+B^{-1}T(C-DB^{-1}T)^{-1}DB^{-1}&-B^{-1}T(C-DB^{-1}T)^{-1}\\\
\end{pmatrix}.$

  4. (iv)

$C:\mathcal{V}\to\mathcal{V}$ is an invertible linear map and the linear
operators $B,D:\mathcal{V}\to\mathcal{V}$ are such that $B-TC^{-1}D$ is
invertible, then the operator

$\displaystyle\begin{pmatrix}T&B\\\ C&D\\\ \end{pmatrix}\text{ is a Halmos
dilation of $T$ on $\mathcal{V}\oplus\mathcal{V}$ whose inverse is }$
$\displaystyle\begin{pmatrix}-C^{-1}D(B-TC^{-1}D)^{-1}&C^{-1}+C^{-1}D(B-TC^{-1}D)^{-1}TC^{-1}\\\
(B-TC^{-1}D)^{-1}&-(B-TC^{-1}D)^{-1}TC^{-1}\\\ \end{pmatrix}.$

Recently, Bhat and Mukherjee, (2020) proved that there is certain kind of
uniqueness of Halmos dilation for strict contractions in Hilbert spaces, as
shown below.

######  Theorem 6.2.4.

(Bhat and Mukherjee, (2020)) Let $\mathcal{H}$ be a finite dimensional Hilbert
space and $T:\mathcal{H}\to\mathcal{H}$ be a strict contraction (i.e.,
$\|T\|<1$). Then Halmos dilation of $T$ on $\mathcal{H}\oplus\mathcal{H}$ is
unitarily equivalent to

$\displaystyle\begin{pmatrix}T&-\sqrt{I-TT^{*}}W\\\ \sqrt{I-T^{*}T}&T^{*}W\\\
\end{pmatrix},\quad\text{ for some unitary operator
}W:\mathcal{H}\to\mathcal{H}.$

We next derive a negative result to Theorem 6.2.4 for Halmos dilation in
vector spaces.

######  Theorem 6.2.5.

Let $\mathcal{V}$ be a finite dimensional vector space and
$T:\mathcal{V}\to\mathcal{V}$ be a linear operator with nonzero trace. Then
there are Halmos dilations of $T$ which are not similar.

###### Proof.

Note that

$\displaystyle\begin{pmatrix}T&T-I\\\ T+I&T\\\ \end{pmatrix}$

is an invertible operator and hence is a Halmos dilation of $T$. It is now
enough to show that the matrices

$\displaystyle\begin{pmatrix}T&T-I\\\ T+I&T\\\ \end{pmatrix}\quad\text{ and
}\quad\begin{pmatrix}T&I\\\ I&0\\\ \end{pmatrix}$

are not similar. Since $\mathcal{V}$ is finite dimensional, we can use the
property of trace map to conclude that these matrices are not similar. ∎

Theorem 6.2.3 can be generalized which gives vector space version of Theorem
6.1.6.

######  Theorem 6.2.6.

(N-dilation for vector spaces) Let $\mathcal{V}$ be a vector space and
$T:\mathcal{V}\to\mathcal{V}$ be a linear map. Let $N$ be a natural number.
Then the operator

$\displaystyle U\coloneqq\begin{pmatrix}T&0&0&\cdots&0&I\\\
I&0&0&\cdots&0&0\\\ 0&I&0&\cdots&0&0\\\ 0&0&I&\cdots&0&0\\\
\vdots&\vdots&\vdots&&\vdots&\vdots\\\ 0&0&0&\cdots&0&0\\\ 0&0&0&\cdots&I&0\\\
\end{pmatrix}_{(N+1)\times(N+1)}$

is invertible on $\oplus_{k=1}^{N+1}\mathcal{V}$ and

$\displaystyle T^{k}=P_{\mathcal{V}}U_{|\mathcal{V}}^{k},\quad\forall
k=1,\dots,N,$ (6.2.1)

where
$P_{\mathcal{V}}:\oplus_{k=1}^{N+1}\mathcal{V}\to\oplus_{k=1}^{N+1}\mathcal{V}$
is the first coordinate projection onto $\mathcal{V}$.

###### Proof.

A direct calculation of power of $U$ gives Equation (6.2.1). To complete the
proof, now we need show that $U$ is invertible. Define

$\displaystyle V\coloneqq\begin{pmatrix}0&I&0&0&\cdots&0&0\\\
0&0&I&0&\cdots&0&0\\\ 0&0&0&I&\cdots&0&0\\\ 0&0&0&0&\cdots&0&0\\\
\vdots&\vdots&\vdots&\vdots&&\vdots&\vdots\\\ 0&0&0&0&\cdots&0&I\\\
I&-T&0&0&\cdots&0&0\\\ \end{pmatrix}_{(N+1)\times(N+1)}.$

Then $UV=VU=I$. Thus $V$ is the inverse of $U$. ∎

Note that the Equation (6.2.1) holds only upto $N$ and not for $N+1$ and
higher natural numbers. We next derive vector space version of Theorem 6.1.5.
In the following theorem, $\oplus_{n=-\infty}^{\infty}\mathcal{V}$ is the
vector space defined by

$\displaystyle\oplus_{n=-\infty}^{\infty}\mathcal{V}\coloneqq\left\\{\\{x_{n}\\}_{n=-\infty}^{\infty},x_{n}\in\mathcal{V},\forall
n\in\mathbb{Z},x_{n}\neq 0\text{ only for finitely many
}n^{\prime}\text{s}\right\\}$

with respect to natural operations.

######  Theorem 6.2.7.

(Sz. Nagy dilation for vector spaces) Let $\mathcal{V}$ be a vector space and
$T:\mathcal{V}\to\mathcal{V}$ be a linear map. Let
$U\coloneqq[u_{n,m}]_{-\infty\leq n,m\leq\infty}$ be the operator defined on
$\oplus_{n=-\infty}^{\infty}\mathcal{V}$ given by the infinite matrix defined
as follows:

$\displaystyle u_{0,0}\coloneqq T,\quad u_{n,n+1}\coloneqq I,\quad\forall
n\in\mathbb{Z},\quad u_{n,m}\coloneqq 0\quad\text{ otherwise},$

i.e.,

$\displaystyle U=\begin{pmatrix}&\vdots&\vdots&\vdots&\vdots&\vdots&\\\
\cdots&0&I&0&0&0&\cdots&\\\ \cdots&0&0&I&0&0&\cdots&\\\
\cdots&0&0&\underline{T}&I&0&\cdots&\\\ \cdots&0&0&0&0&I&\cdots&\\\
\cdots&0&0&0&0&0&\cdots&\\\ &\vdots&\vdots&\vdots&\vdots&\vdots&\\\
\end{pmatrix}_{\infty\times\infty}$

where $T$ is in the $(0,0)$ position (which is underlined), which is
invertible on $\oplus_{n=-\infty}^{\infty}\mathcal{V}$ and

$\displaystyle T^{n}=P_{\mathcal{V}}U_{\mathcal{V}}^{n},\quad\forall
n\in\mathbb{N},$ (6.2.2)

where
$P_{\mathcal{V}}:\oplus_{n=-\infty}^{\infty}\mathcal{V}\to\oplus_{n=-\infty}^{\infty}\mathcal{V}$
is the first coordinate projection onto $\mathcal{V}$.

###### Proof.

We get Equation (6.2.2) by calculation of powers of operator $U$. The matrix
$V\coloneqq[v_{n,m}]_{-\infty<n,m<\infty}$ defined by

$\displaystyle v_{0,0}\coloneqq 0,\quad v_{1,-1}\coloneqq-T,\quad
v_{n,n-1}\coloneqq I,\quad\forall n\in\mathbb{Z},\quad v_{n,m}\coloneqq
0\quad\text{ otherwise},$

i.e.,

$\displaystyle V=\begin{pmatrix}&\vdots&\vdots&\vdots&\vdots&\vdots&\\\
\cdots&I&0&0&0&0&\cdots&\\\ \cdots&0&I&\underline{0}&0&0&\cdots&\\\
\cdots&0&-T&I&0&0&\cdots&\\\ \cdots&0&0&0&I&0&\cdots&\\\
\cdots&0&0&0&0&I&\cdots&\\\ &\vdots&\vdots&\vdots&\vdots&\vdots&\\\
\end{pmatrix}_{\infty\times\infty}$

where $0$ is in the $(0,0)$ position (which is underlined), satisfies
$UV=VU=I$ and hence $U$ is invertible which completes the proof. ∎

###  6.3 MINIMAL DILATION, INTERTWINING LIFTING THEOREM AND VARIANT OF ANDO
DILATION FOR VECTOR SPACES

An important observation associated with Theorems 6.2.5, 6.2.6 and 6.2.7 is
that the dilation is not optimal, i.e., even if the given operator is
invertible, then also $U$ is not same as $T$. To overcome this, next we move
on with the definition of dilation given by Bhat, De, and Rakshit (Bhat et
al., (2021)). Set theoretic definition of dilation, given in Definition 6.1.12
motivated Bhat, De, and Rakshit, to introduce the dilation of linear maps on
vector spaces.

######  Definition 6.3.1.

(Bhat et al., (2021)) Let $\mathcal{V}$ be a vector space and
$T:\mathcal{V}\to\mathcal{V}$ be a linear map. A linear injective dilation of
$T$ is a quadruple $(\mathcal{W},I,U,P)$, where $\mathcal{W}$ is a vector
space, and $I:\mathcal{V}\to\mathcal{W}$ is an injective linear map,
$U:\mathcal{W}\to\mathcal{W}$ is an injective linear map,
$P:\mathcal{W}\to\mathcal{W}$ is an idempotent linear map such that
$P(\mathcal{W})=I(\mathcal{W})$ and

$\displaystyle\text{({Dilation equation})}\quad IT^{n}x=PU^{n}Ix,\quad\forall
n\in\mathbb{Z}_{+},\forall x\in\mathcal{V}.$

A dilation $(\mathcal{W},I,U,P)$ of $T$ is said to be minimal if

$\displaystyle\mathcal{W}=\operatorname{span}\\{U^{n}Ix:n\in\mathbb{Z}_{+},x\in\mathcal{V}\\}.$

An easier way to remember the dilation equation is the following commutative
diagram.

${\mathcal{W}}$${\mathcal{W}}$${\mathcal{W}}$${\mathcal{V}}$${\mathcal{V}}$$\scriptstyle{U^{n}}$$\scriptstyle{P}$$\scriptstyle{I}$$\scriptstyle{T^{n}}$$\scriptstyle{I}$

Following result is the vector space version of Theorem 6.1.16.

######  Theorem 6.3.2.

(Bhat et al., (2021)) (Minimal Sz. Nagy dilation for sets) Every linear map
$T:\mathcal{V}\to\mathcal{V}$ admits minimal injective linear dilation.

###### Proof.

We reproduce the proof given by Bhat et al., (2021) for the sake of future
use. Define

$\displaystyle\mathcal{W}\coloneqq\left\\{(x_{n})_{n=0}^{\infty}:x_{n}\in\mathcal{V},\forall
n\in\mathbb{Z}_{+},x_{n}\neq 0\text{ only for finitely many
}n^{\prime}\text{s}\right\\}.$

Clearly $\mathcal{W}$ is a vector space w.r.t. natural operations. Now define

$\displaystyle I:\mathcal{V}\ni x\mapsto(x,0,\dots)\in\mathcal{W},$
$\displaystyle
U:\mathcal{W}\ni(x_{n})_{n=0}^{\infty}\mapsto(0,x_{0},\dots)\in\mathcal{W},$
$\displaystyle
P:\mathcal{W}\ni(x_{n})_{n=0}^{\infty}\mapsto\sum_{n=0}^{\infty}IT^{n}x_{n}\in\mathcal{W}.$

Then $(\mathcal{W},I,U,P)$ is a minimal injective linear dilation of $T$. ∎

We call the dilation $(\mathcal{W},I,U,P)$ constructed in Theorem 6.3.2 as the
standard dilation of $T$. We next consider inter-twining lifting theorem.

######  Theorem 6.3.3.

(Inter-twining lifting theorem for vector spaces) Let $\mathcal{V}_{1}$,
$\mathcal{V}_{2}$ be vector spaces, $T_{1}:\mathcal{V}_{1}\to\mathcal{V}_{1}$,
$T_{2}:\mathcal{V}_{2}\to\mathcal{V}_{2}$ be linear maps. Let
$(\mathcal{W}_{1},I_{1},U_{1},P_{1})$, $(\mathcal{W}_{2},I_{2},U_{2},P_{2})$
be standard dilations of $T_{1}$, $T_{2}$, respectively. If
$S:\mathcal{V}_{2}\to\mathcal{V}_{1}$ is a linear map such that
$T_{1}S=ST_{2}$, then there exists a linear map
$R:\mathcal{W}_{2}\to\mathcal{W}_{1}$ such that

$\displaystyle U_{1}R=RU_{2},\quad RP_{2}=P_{1}R,\quad RI_{2}=I_{1}S.$ (6.3.1)

Conversely if $R:\mathcal{W}_{2}\to\mathcal{W}_{1}$ is a linear map such that
$U_{1}R=RU_{2},RP_{2}=P_{1}R$, then there exists a linear map
$S:\mathcal{V}_{2}\to\mathcal{V}_{1}$ such that

$\displaystyle RI_{2}=I_{1}S,\quad T_{1}S=ST_{2}.$ (6.3.2)

###### Proof.

Define
$R:\mathcal{W}_{2}\ni(x_{n})_{n=0}^{\infty}\mapsto(Sx_{n})_{n=0}^{\infty}\in\mathcal{W}_{1}$.
We now verify three equalities in Equation (6.3.1). Let
$(x_{n})_{n=0}^{\infty}\in\mathcal{W}_{2}$. Then

$\displaystyle
U_{1}R(x_{n})_{n=0}^{\infty}=U_{1}(Sx_{n})_{n=0}^{\infty}=(0,Sx_{0},Sx_{1},\dots),$
$\displaystyle
RU_{2}(x_{n})_{n=0}^{\infty}=R(0,x_{0},x_{1},\dots)=(0,Sx_{0},Sx_{1},\dots),$
$\displaystyle RP_{2}(x_{n})_{n=0}^{\infty}$
$\displaystyle=R\left(\sum_{n=0}^{\infty}I_{2}T_{2}^{n}x_{n}\right)=\sum_{n=0}^{\infty}RI_{2}T_{2}^{n}x_{n}$
$\displaystyle=\sum_{n=0}^{\infty}R(T_{2}^{n}x_{n},0,0,\dots)=\sum_{n=0}^{\infty}(ST_{2}^{n}x_{n},0,0,\dots),$
$\displaystyle P_{1}R(x_{n})_{n=0}^{\infty}$
$\displaystyle=P_{1}(Sx_{n})_{n=0}^{\infty}=\sum_{n=0}^{\infty}I_{1}T_{1}^{n}Sx_{n}$
$\displaystyle=\sum_{n=0}^{\infty}I_{1}ST_{2}^{n}x_{n}=\sum_{n=0}^{\infty}(ST_{2}^{n}x_{n},0,0,\dots),$
$\displaystyle RI_{2}x=R(x,0,0,\dots)=(Sx,0,0,\dots),\quad
I_{1}Sx=(Sx,0,0,\dots).$

We now consider the converse part. For this, first we have to define linear
map $S$. Let $y\in\mathcal{V}_{2}$. Now
$RP_{2}(y,0,\dots)=P_{1}R(y,0,\dots)\in I_{1}(\mathcal{V}_{1})$ and $I_{1}$ is
injective implies that there exists a unique $x\in\mathcal{V}_{2}$ such that
$RP_{2}(y,0,\dots)=P_{1}R(y,0,\dots)=I_{1}(x)$. We now define $Sy\coloneqq x.$
Then $S$ is well-defined and linear. Let $y\in\mathcal{V}_{2}$ and
$x\in\mathcal{V}_{2}$ be such that $Sy=x$. Then
$I_{1}Sy=RP_{2}(y,0,\dots)=RI_{2}y$. Thus we verified first equality in
(6.3.2). We are left with verification of second equality. We now calculate

$\displaystyle RP_{2}U_{2}(x,0,\dots)=RP_{2}(0,x,0,\dots)=RI_{2}T_{2}x$
(6.3.3)

and

$\displaystyle P_{1}U_{1}R(x,0,\dots)$
$\displaystyle=P_{1}RU_{2}(x,0,\dots)=P_{1}R(0,x,0,\dots)$ (6.3.4)
$\displaystyle=RP_{2}(0,x,0,\dots)=RI_{2}T_{2}x,\quad\forall
x\in\mathcal{V}_{2}.$ (6.3.5)

Given conditions produce

$\displaystyle RP_{2}U_{2}=P_{1}RU_{2}=P_{1}U_{1}R.$ (6.3.6)

Equation (6.3.6) says that (6.3.3) and (6.3.4) are equal which completes the
proof. ∎

Following is a variant of Ando dilation for vector spaces.

######  Theorem 6.3.4.

(Ando like dilation for vector spaces) Let $\mathcal{V}$ be a vector space and
$T,S:\mathcal{V}\to\mathcal{V}$ be commuting linear maps. Then there are
dilations $(\mathcal{W},I,U_{1},P)$ and $(\mathcal{W},I,U_{2},P)$ of $T,S$
respectively, such that

$\displaystyle\begin{pmatrix}0_{c}&U\end{pmatrix}=\begin{pmatrix}0_{r}\\\
V\end{pmatrix}$

and

$\displaystyle\quad IT^{n}S^{m}x=PU^{n}V^{m}Ix,\quad\forall
n,m\in\mathbb{Z}_{+},\forall x\in\mathcal{V},$

where $0_{c}$ denotes the infinite column matrix of zero vectors and $0_{r}$
denotes the infinite row matrix of zero vectors.

###### Proof.

We extend the construction in the proof of Theorem 6.3.2. Define

$\displaystyle\mathcal{W}\coloneqq\bigg{\\{}\begin{pmatrix}x_{0,0}&x_{0,1}&x_{0,2}&\cdots\\\
x_{1,0}&x_{1,1}&x_{1,2}&\cdots\\\ x_{2,0}&x_{2,1}&x_{2,2}&\cdots\\\
\vdots&\vdots&\vdots&\ddots\end{pmatrix}_{\infty\times\infty}:x_{n,m}\in\mathcal{V},\forall
n,m\in\mathbb{Z}_{+},x_{n,m}\neq 0$ $\displaystyle\text{ only for finitely
many }(n,m)^{\prime}\text{s}\bigg{\\}}.$

Then $\mathcal{W}$ becomes a vector space with respect to natural operations.
We now define the following four linear maps:

$\displaystyle I:\mathcal{V}\ni x\mapsto\begin{pmatrix}x&0&0&\cdots\\\
0&0&0&\cdots\\\ 0&0&0&\cdots\\\
\vdots&\vdots&\vdots&\ddots\end{pmatrix}\in\mathcal{W}$ $\displaystyle
U:\mathcal{W}\ni\begin{pmatrix}x_{0,0}&x_{0,1}&x_{0,2}&\cdots\\\
x_{1,0}&x_{1,1}&x_{1,2}&\cdots\\\ x_{2,0}&x_{2,1}&x_{2,2}&\cdots\\\
\vdots&\vdots&\vdots&\ddots\end{pmatrix}\mapsto\begin{pmatrix}0&0&0\\\
x_{0,0}&x_{0,1}&x_{0,2}&\cdots\\\ x_{1,0}&x_{1,1}&x_{1,2}&\cdots\\\
\vdots&\vdots&\vdots&\ddots\end{pmatrix}\in\mathcal{W}$ $\displaystyle
V:\mathcal{W}\ni\begin{pmatrix}x_{0,0}&x_{0,1}&x_{0,2}&\cdots\\\
x_{1,0}&x_{1,1}&x_{1,2}&\cdots\\\ x_{2,0}&x_{2,1}&x_{2,2}&\cdots\\\
\vdots&\vdots&\vdots&\ddots\end{pmatrix}\mapsto\begin{pmatrix}0&x_{0,0}&x_{0,1}&\cdots\\\
0&x_{1,0}&x_{1,1}&\cdots\\\ 0&x_{2,0}&x_{2,1}&\cdots\\\
\vdots&\vdots&\vdots&\ddots\end{pmatrix}\in\mathcal{W}$ $\displaystyle
P:\mathcal{W}\ni\begin{pmatrix}x_{0,0}&x_{0,1}&x_{0,2}&\cdots\\\
x_{1,0}&x_{1,1}&x_{1,2}&\cdots\\\ x_{2,0}&x_{2,1}&x_{2,2}&\cdots\\\
\vdots&\vdots&\vdots&\ddots\end{pmatrix}\mapsto\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}IT^{n}S^{m}x_{n,m}\in\mathcal{W}.$

We then have

$\displaystyle\begin{pmatrix}0_{c}&U\end{pmatrix}=\begin{pmatrix}0&0&0&0&\cdots\\\
0&x_{0,0}&x_{0,1}&x_{0,2}&\cdots\\\ 0&x_{1,0}&x_{1,1}&x_{1,2}&\cdots\\\
0&x_{2,0}&x_{2,1}&x_{2,2}&\cdots\\\
0&\vdots&\vdots&\vdots&\ddots\end{pmatrix}=\begin{pmatrix}0_{r}\\\
V\end{pmatrix}.$

Now $PU^{n}Ix=IT^{n}x,$ $PV^{n}Ix=IS^{m}x$, $\forall x\in\mathcal{V}$,
$\forall n,m\in\mathbb{Z}_{+}$. Hence $(\mathcal{W},I,U_{1},P)$ and
$(\mathcal{W},I,U_{2},P)$ are dilations of $T,S$, respectively. A calculation
now shows that $IT^{n}S^{m}x=PU^{n}V^{m}Ix,\forall
n,m\in\mathbb{Z}_{+},\forall x\in\mathcal{V}$. ∎

Conclusion and future work :  In this appendix we derived some basic results
on dilation of linear maps. Since vector spaces are more general than Hilbert
spaces and tools of Hilbert spaces will not work in vector space, we are
interested to explore algebraic aspects of dilation theory.

APPENDIX B: COMMUTATORS CLOSE TO THE IDENTITY

###  6.4 C*-ALGEBRAS

Israel M. Gelfand defined abstractly the notion of a complete algebra
(Gelfand, (1941)). These are Banach spaces in which we can multiply the
elements and the multiplication enjoys continuity.

######  Definition 6.4.1.

(cf. Zhu, (1993)) A Banach space $\mathcal{A}$ over $\mathbb{C}$ is said to be
a unital Banach algebra if it is a unital algebra and the multiplication
satisfies the following:

  1. (i)

$\|xy\|\leq\|x\|\|y\|$, $\forall x,y\in\mathcal{A}.$

  2. (ii)

$\|e\|=1$, where $e$ is the multiplicative identity of $\mathcal{A}$.

######  Example 6.4.2.

(cf. Zhu, (1993))

  1. (i)

If $K$ is a compact Hausdorff space, then the space $\mathcal{C}(K)$ of all
complex-valued continuous functions on $K$ is a commutative unital Banach
algebra w.r.t. sup-norm and pointwise multiplication.

  2. (ii)

If $\mathcal{X}$ is a Banach space, then the collection
$\mathcal{B}(\mathcal{X})$ of all bounded linear operators on $\mathcal{X}$ is
a noncommutative unital Banach algebra w.r.t. operator-norm and operator
composition.

######  Proposition 6.4.3.

(cf. Allan, (2011)) Every unital Banach algebra $\mathcal{A}$ can be
isometrically embedded in $\mathcal{B}(\mathcal{A})$.

One of the most important notion associated with the study of Banach algebras
is the notion of spectrum.

######  Definition 6.4.4.

(cf. Zhu, (1993)) Let $\mathcal{A}$ be a unital Banach algebra with the
identity $e$. Spectrum of an element $x$ in $\mathcal{A}$ is the set of all
complex numbers $\lambda$ such that $\lambda e-x$ is not invertible.

######  Theorem 6.4.5.

(cf. Zhu, (1993)) Spectrum of every element of a unital Banach algebra is a
nonempty compact subset of $\mathbb{C}$.

Following is the first fundamental theorem in the study of Banach algebras
which characterizes Banach algebras using the information of spectrum.

######  Theorem 6.4.6.

(cf. Zhu, (1993)) (Gelfand-Mazur theorem) If every nonzero element of a Banach
algebra is invertible, then it is isometrically isomorphic to $\mathbb{C}$.

A subclass of Banach algebras known as C*-algebras allows to do most of the
things which hold good for complex numbers. Notion of C*-algebras, for first
time, appeared in the work of Gelfand and Neumark, (1943).

######  Definition 6.4.7.

(cf. Zhu, (1993)) A unital Banach algebra $\mathcal{A}$ is called a unital
C*-algebra if there exists a map $*:\mathcal{A}\ni x\mapsto
x^{*}\in\mathcal{A}$ such that following conditions hold.

  1. (i)

$((x)^{*})^{*}=x,\forall x\in\mathcal{A}.$

  2. (ii)

$(x+y)^{*}=x^{*}+y^{*},\forall x,y\in\mathcal{A}.$

  3. (iii)

$(\alpha x)^{*}=\overline{\alpha}x^{*}$, $\forall\alpha\in\mathbb{K},\forall
x\in\mathcal{A}$.

  4. (iv)

$(xy)^{*}=y^{*}x^{*},\forall x,y\in\mathcal{A}.$

  5. (v)

$\|x^{*}x\|=\|x\|^{2},\forall x\in\mathcal{A}.$

A map $*:\mathcal{A}\ni x\mapsto x^{*}\in\mathcal{A}$ satisfying (i)-(iii) is
called as involution.

Segal called the term C*-algebra; the letter ‘C’ stands for uniformly closed.
C*-algebras are also known as Gelfand-Naimark algebras (cf. Pietsch, (2007)).

######  Example 6.4.8.

(cf. Zhu, (1993))

  1. (i)

If $K$ is a compact Hausdorff space, then $\mathcal{C}(K)$ is a commutative
unital C*-algebra w.r.t. involution $f^{*}(x)\coloneqq\overline{f(x)},\forall
x\in K$.

  2. (ii)

If $\mathcal{H}$ is a Hilbert space, then $\mathcal{B}(\mathcal{H})$ is a
noncommutative unital C*-algebra w.r.t. operator adjoint.

  3. (iii)

If $\mathcal{H}$ is a Hilbert space, the the space $\mathcal{K}(\mathcal{H})$
of compact operators is a noncommutative C*-subalgebra of
$\mathcal{B}(\mathcal{H})$. If $\mathcal{H}$ is infinite dimensional, then
this algebra is non unital.

Following two results characterize unital C*-algebras.

######  Theorem 6.4.9.

(cf. Zhu, (1993)) (Gelfand-Naimark theorem) If $\mathcal{A}$ is a commutative
unital C*-algebra, then $\mathcal{A}$ is isometrically $*$-isomorphic to
$\mathcal{C}(K)$ for some compact Hausdorff space $K$.

######  Theorem 6.4.10.

(cf. Zhu, (1993)) (Gelfand-Naimark-Segal theorem) Let $\mathcal{A}$ be a
unital C*-algebra. Then there exists a Hilbert space $\mathcal{H}$ such that
$\mathcal{A}$ is isometrically $*$-isomorphic to a C*-subalgebra of
$\mathcal{B}(\mathcal{H})$.

######  Example 6.4.11.

(cf. Kadison and Ringrose, (1997)) Consider the unital C*-algebra
$\mathcal{C}[0,1]$. The map

$\displaystyle\pi:\mathcal{C}[0,1]\ni
f\mapsto\pi(f)\in\mathcal{B}(\mathcal{L}^{2}[0,1]);$
$\displaystyle\pi(f):\mathcal{L}^{2}[0,1]\ni g\mapsto(\pi(f))(g)\coloneqq
fg\in\mathcal{L}^{2}[0,1]$

is an isometric $*$-isomorphism to a C*-subalgebra of
$\mathcal{B}(\mathcal{L}^{2}[0,1])$.

###  6.5 COMMUTATORS CLOSE TO THE IDENTITY IN $\mathcal{B}(\mathcal{H})$

Let $n\in\mathbb{N}$ and $M_{n}(\mathbb{K})$ be the ring of $n$ by $n$
matrices over $\mathbb{K}$. Using the property of trace map we easily get that
there does not exist $D,X\in M_{n}(\mathbb{K})$ such that $DX-
XD=1_{M_{n}(\mathbb{K})}$ (Halmos, (1982)). This argument will not work for
bounded linear operators on infinite dimensional Hilbert space since the map
trace is not defined on the algebra $\mathcal{B}(\mathcal{H})$ of all bounded
linear operators on an infinite dimensional Hilbert space $\mathcal{H}$ (it is
defined for a proper subalgebra of $\mathcal{B}(\mathcal{H})$ known as the
trace class operators (Schatten, (1960))). Operators of the form $DX-XD$ are
called as commutator of $D$ and $X$ and are denoted by $[D,X]$. An operator
$T\in\mathcal{B}(\mathcal{H})$ is said to be a commutator if $T=[D,X]$, for
some $D,X\in\mathcal{B}(\mathcal{H})$.
Using the property of spectrum of bounded linear operator, Winter in 1947
proved that the following result.

######  Theorem 6.5.1.

(Wintner, (1947)) Let $\mathcal{H}$ be an infinite dimensional Hilbert space.
Then there does not exist $D,X\in\mathcal{B}(\mathcal{H})$ such that

$\displaystyle[D,X]=1_{\mathcal{B}(\mathcal{H})}.$ (6.5.1)

After two years, Wielandt, (1949) gave a simple proof for the failure of
Equation 6.5.1. We note that the boundedness of operators is crucial in
Theorem 6.5.1. Following example shows that Theorem 6.5.1 fails for unbounded
operators

######  Example 6.5.2.

(Halmos, (1982)) Let $\mathcal{H}\coloneqq\mathcal{L}^{2}(\mathbb{R})$ and
define

$\displaystyle(Df)(x)\coloneqq\frac{d}{dx}f,\quad(Xf)(x)\coloneqq xf(x)$

Then $[D,X]=1_{\mathcal{B}(\mathcal{H})}$.

Theorem 6.5.1 leads to the question that which operators on infinite
dimensional Hilbert spaces can be written as commutators of operators? First
partial answer was given by Halmos.

######  Theorem 6.5.3.

(cf. Putnam, (1967)) Let $\mathcal{H}$ be an infinite dimensional Hilbert
space. If $C\in\mathcal{B}(\mathcal{H})$ is compact, then
$I_{\mathcal{B}(\mathcal{H})}+C$ is not a commutator.

Brown and Pearcy, (1965) characterized the set of bounded operators which can
be written as commutators.

######  Theorem 6.5.4.

(Brown and Pearcy, (1965)) Let $\mathcal{H}$ be an infinite dimensional
separable Hilbert space. Then an operator in $\mathcal{B}(\mathcal{H})$ is a
commutator if and only if it is not of the form $\lambda
I_{\mathcal{B}(\mathcal{H})}+C$, where $\lambda$ is a nonzero scalar and $C$
is a compact operator.

Following the paper of Brown and Pearcy, (1965) there is a series of papers
devoted to the study of commutators on sequence spaces,
$\mathcal{L}^{p}$-spaces, Banach spaces, C*-algebras, von Neumann algebras,
Banach *-algebras etc (Schneeberger, (1971); Laustsen, (2002); Dykema et al.,
(2004); Marcoux, (2006); Stasinski, (2016); Dosev and Johnson, (2010); Kadison
et al., (2020); Yood, (2008); Dosev et al., (2013); Dykema and Skripka,
(2012); Dosev, (2009); Kaftal et al., (2014); Marcoux, (2010)).

It was Popa, (1982) who started a quantitative study of commutators close to
the identity operator. He gave the following quantitative bound given by Popa
for the product of norm of operators whenever the commutator is close to the
identity.

######  Theorem 6.5.5.

(Popa, (1982)) Let $\mathcal{H}$ be an infinite dimensional Hilbert space. Let
$D,X\in\mathcal{B}(\mathcal{H})$ be such that

$\displaystyle\|[D,X]-1_{\mathcal{B}(\mathcal{H})}\|\leq\varepsilon$

for some $\varepsilon>0$. Then

$\displaystyle\|D\|\|X\|\geq\frac{1}{2}\log\frac{1}{\varepsilon}.$

Now the problem in Theorem 6.5.5, is the existence of
$D,X\in\mathcal{B}(\mathcal{H})$ such that the commutator $[D,X]$ is close to
the identity operator. This was again obtained by Popa which is stated in the
following result. Given real $r$ and positive $s$, by $r=O(s)$ we mean that
there is positive $\gamma$ such that $|r|\leq\gamma s$.

######  Theorem 6.5.6.

(Popa, (1982); Tao, (2019)) Let $\mathcal{H}$ be an infinite dimensional
Hilbert space. Then for each $0<\varepsilon\leq 1$, there exist
$D,X\in\mathcal{B}(\mathcal{H})$ with

$\displaystyle\|[D,X]-1_{\mathcal{B}(\mathcal{H})}\|\leq\varepsilon$

and

$\displaystyle\|D\|\|X\|=O(\varepsilon^{-2}).$

Terence Tao improved Theorem 6.5.6 and obtained the following theorem.

######  Theorem 6.5.7.

(Tao, (2019)) Let $\mathcal{H}$ be an infinite dimensional Hilbert space. Then
for each $0<\varepsilon\leq 1/2$, there exist $D,X\in\mathcal{B}(\mathcal{H})$
with

$\displaystyle\|[D,X]-1_{\mathcal{B}(\mathcal{H})}\|\leq\varepsilon$

such that

$\displaystyle\|D\|\|X\|=O\left(\log^{5}\frac{1}{\varepsilon}\right).$

In (Popa, (1982)) there is another result about commutators. Let
$\mathcal{K}(\mathcal{H})$ be the ideal of compact operators in
$\mathcal{B}(\mathcal{H})$ and define

$\displaystyle\mathbb{C}+\mathcal{K}(\mathcal{H})\coloneqq\\{\lambda.1_{\mathcal{B}(\mathcal{H})}+T:\lambda\in\mathbb{C},T\in\mathcal{K}(\mathcal{H})\\}.$

######  Theorem 6.5.8.

(Popa, (1982)) If $K\in\mathcal{B}(\mathcal{H})$ is such that

$\displaystyle\|A\|=O(1),\quad\|A\|=O(\text{dist}(A,\mathbb{C}+\mathcal{K}(\mathcal{H}))^{\frac{2}{3}}),$

then there exist $D,X\in\mathcal{B}(\mathcal{H})$ with

$\displaystyle\|D\|\|X\|=O(1)\quad\text{ such that }\quad A=[D,X].$

###  6.6 COMMUTATORS CLOSE TO THE IDENTITY IN
UNITAL C*-ALGEBRAS

We recall fundamentals of matrices over unital C*-algebras as given in
(Murphy, (1990)).

Let $\mathcal{A}$ be a unital C*-algebra. For $n\in\mathbb{N}$,
$M_{n}(\mathcal{A})$ is defined as the set of all $n$ by $n$ matrices over
$\mathcal{A}$. It is clearly an algebra with respect to natural matrix
operations. We define the involution of an element $A\coloneqq[a_{i,j}]_{1\leq
i,j\leq n}\in M_{n}(\mathcal{A})$ as $A^{*}\coloneqq[a_{j,i}^{*}]_{1\leq
i,j\leq n}$. Then $M_{n}(\mathcal{A})$ is a *-algebra. From the Gelfand-
Naimark-Segal theorem (Theorem 6.4.10) there exists unique universal
representation $(\mathcal{H},\pi)$, where $\mathcal{H}$ is a Hilbert space,
$\pi:M_{n}(\mathcal{A})\to M_{n}(\mathcal{B}(\mathcal{H}))$ is an isometric
*-homomorphism. This gives a norm on $M_{n}(\mathcal{A})$ defined as

$\displaystyle\|A\|\coloneqq\|\pi(A)\|,\quad\forall A\in M_{n}(\mathcal{A}).$

This norm makes $M_{n}(\mathcal{A})$ as a C*-algebra.
In the sequel, $\mathcal{A}$ is a unital C*-algebra. We first derive a lemma
followed by a corollary for unital C*-algebras. Proof of the lemma is a direct
algebraic calculation.

######  Lemma 6.6.1.

(Commutator calculation) Let $u,v,b_{1},\dots,b_{n}\in\mathcal{A}$ and
$\delta>0$. Let

$\displaystyle X\coloneqq\begin{pmatrix}0&0&0&\cdots&0&\delta b_{1}\\\
1_{\mathcal{A}}&0&0&\cdots&0&\delta b_{2}\\\
0&1_{\mathcal{A}}&0&\cdots&0&\delta b_{3}\\\
\vdots&\vdots&\vdots&\cdots&\vdots&\vdots\\\ 0&0&0&\cdots&0&\delta b_{n-1}\\\
0&0&0&\cdots&1_{\mathcal{A}}&\delta b_{n}\end{pmatrix}\in M_{n}(\mathcal{A})$

and

$\displaystyle
D\coloneqq\begin{pmatrix}\frac{v}{\delta}&1_{\mathcal{A}}&0&\cdots&0&\delta
b_{1}u\\\ \frac{u}{\delta}&\frac{v}{\delta}&2.1_{\mathcal{A}}&\cdots&0&\delta
b_{2}u\\\ 0&\frac{u}{\delta}&\frac{v}{\delta}&\cdots&0&\delta b_{3}u\\\
\vdots&\vdots&\vdots&\cdots&\vdots&\vdots\\\
0&0&0&\cdots&\frac{v}{\delta}&(n-1)1_{\mathcal{A}}+\delta b_{n-1}u\\\
0&0&0&\cdots&\frac{u}{\delta}&\frac{v}{\delta}+\delta b_{n}u\end{pmatrix}\in
M_{n}(\mathcal{A}).$

Then

$\displaystyle[D,X]=1_{M_{n}(\mathcal{A})}+\begin{pmatrix}0&0&0&\cdots&0&[v,b_{1}]+0+\delta
b_{2}+\delta b_{1}[u,b_{n}]\\\ 0&0&0&\cdots&0&[v,b_{2}]+[u,b_{1}]+2\delta
b_{3}+\delta b_{2}[u,b_{n}]\\\ 0&0&0&\cdots&0&[v,b_{3}]+[u,b_{2}]+3\delta
b_{4}+\delta b_{3}[u,b_{n}]\\\ \vdots&\vdots&\vdots&\cdots&\vdots&\vdots\\\
0&0&0&\cdots&0&[v,b_{n-1}]+[u,b_{n-2}]+(n-1)\delta b_{n}+\delta
b_{n-1}[u,b_{n}]\\\ 0&0&0&\cdots&0&[v,b_{n}]+[u,b_{n-1}]+0+\delta
b_{n}[u,b_{n}]-n.1_{\mathcal{A}}\end{pmatrix}.$

######  Corollary 6.6.2.

Let $u,v,b_{1},\dots,b_{n}\in\mathcal{A}$. Assume that for some $\delta>0$, we
have equations

$\displaystyle[v,b_{i}]+[u,b_{i-1}]+i\delta b_{i+1}+\delta
b_{i}[u,b_{n}]=0,\quad\forall i=2,\dots,n-1$ (6.6.1)

and

$\displaystyle[v,b_{n}]+[u,b_{n-1}]+\delta b_{n}[u,b_{n}]=n\cdot
1_{{M_{n}(\mathcal{A})}}.$ (6.6.2)

Then for any $\mu>0$, there exist matrices $D_{\mu},X_{\mu}\in
M_{n}(\mathcal{A})$ such that

$\displaystyle\|D_{\mu}\|\leq\frac{\|u\|}{\mu^{2}\delta}+\frac{\|v\|}{\mu\delta}+(n-1)+\delta\sum_{i=1}^{n}\mu^{n-i-1}\|b_{i}\|\|u\|,$
$\displaystyle\|X_{\mu}\|\leq
1+\delta\sum_{i=1}^{n}\mu^{n-i+1}\|b_{i}\|\quad\text{ and }$
$\displaystyle\|[D_{\mu},X_{\mu}]-1_{M_{n}(\mathcal{A})}\|\leq\mu^{n-1}\|[v,b_{1}]+\delta
b_{2}+\delta b_{1}[u,b_{n}]\|.$

###### Proof.

Let $D$ and $X$ be as in Lemma 6.6.1. Define

$\displaystyle S_{\mu}$
$\displaystyle\coloneqq\begin{pmatrix}\mu^{n-1}&0&0&\cdots&0&0\\\
0&\mu^{n-2}&0&\cdots&0&0\\\ 0&0&\mu^{n-3}&\cdots&0&0\\\
\vdots&\vdots&\vdots&\cdots&\vdots&\vdots\\\ 0&0&0&\cdots&\mu&0\\\
0&0&0&\cdots&0&1\end{pmatrix}\in M_{n}(\mathbb{K}),$ $\displaystyle D_{\mu}$
$\displaystyle\coloneqq\frac{1}{\mu}S_{\mu}DS_{\mu}^{-1},\quad
X_{\mu}\coloneqq\mu S_{\mu}XS_{\mu}^{-1}.$

Then

$\displaystyle\|D_{\mu}\|$
$\displaystyle=\left\|\begin{pmatrix}\frac{v}{\mu\delta}&1_{\mathcal{A}}&0&\cdots&0&\mu^{n-2}\delta
b_{1}u\\\
\frac{u}{\mu^{2}\delta}&\frac{v}{\mu\delta}&2.1_{\mathcal{A}}&\cdots&0&\mu^{n-3}\delta
b_{2}u\\\
0&\frac{u}{\mu^{2}\delta}&\frac{v}{\mu\delta}&\cdots&0&\mu^{n-4}\delta
b_{3}u\\\ \vdots&\vdots&\vdots&\cdots&\vdots&\vdots\\\
0&0&0&\cdots&\frac{v}{\mu\delta}&(n-1)1_{\mathcal{A}}+\delta b_{n-1}u\\\
0&0&0&\cdots&\frac{u}{\mu^{2}\delta}&\frac{v}{\mu\delta}+\mu^{-1}\delta
b_{n}u\end{pmatrix}\right\|$
$\displaystyle\leq\left\|\frac{u}{\mu^{2}\delta}\right\|+\left\|\frac{v}{\mu\delta}\right\|+\|(n-1)1_{\mathcal{A}}\|+\left\|\begin{pmatrix}0&0&0&\cdots&0&\mu^{n-2}\delta
b_{1}u\\\ 0&0&0&\cdots&0&\mu^{n-3}\delta b_{2}u\\\
0&0&0&\cdots&0&\mu^{n-4}\delta b_{3}u\\\
\vdots&\vdots&\vdots&\cdots&\vdots&\vdots\\\ 0&0&0&\cdots&0&\delta b_{n-1}u\\\
0&0&0&\cdots&0&\mu^{-1}\delta b_{n}u\end{pmatrix}\right\|$
$\displaystyle\leq\frac{\|u\|}{\mu^{2}\delta}+\frac{\|v\|}{\mu\delta}+(n-1)+\delta\sum_{i=1}^{n}\mu^{n-i-1}\|b_{i}\|\|u\|$

and

$\displaystyle\|X_{\mu}\|$
$\displaystyle=\left\|\begin{pmatrix}0&0&0&\cdots&0&\mu^{n}\delta b_{1}\\\
1_{\mathcal{A}}&0&0&\cdots&0&\mu^{n-1}\delta b_{2}\\\
0&1_{\mathcal{A}}&0&\cdots&0&\mu^{n-2}\delta b_{3}\\\
\vdots&\vdots&\vdots&\cdots&\vdots&\vdots\\\ 0&0&0&\cdots&0&\mu^{2}\delta
b_{n-1}\\\ 0&0&0&\cdots&1_{\mathcal{A}}&\mu\delta b_{n}\end{pmatrix}\right\|$
$\displaystyle\leq\|1_{\mathcal{A}}\|+\left\|\begin{pmatrix}0&0&0&\cdots&0&\mu^{n}\delta
b_{1}\\\ 0&0&0&\cdots&0&\mu^{n-1}\delta b_{2}\\\
0&0&0&\cdots&0&\mu^{n-2}\delta b_{3}\\\
\vdots&\vdots&\vdots&\cdots&\vdots&\vdots\\\ 0&0&0&\cdots&0&\mu^{2}\delta
b_{n-1}\\\ 0&0&0&\cdots&0&\mu\delta b_{n}\end{pmatrix}\right\|$
$\displaystyle\leq 1+\delta\sum_{i=1}^{n}\mu^{n-i+1}\|b_{i}\|.$

Now using (6.6.1) and (6.6.2) we get

$\displaystyle\|[D_{\mu},X_{\mu}]-1_{M_{n}(\mathcal{A})}\|$
$\displaystyle=\left\|\begin{pmatrix}0&0&0&\cdots&0&\mu^{n-1}([v,b_{1}]+\delta
b_{2}+\delta b_{1}[v,b_{n}])\\\ 0&0&0&\cdots&0&0\\\ 0&0&0&\cdots&0&0\\\
\vdots&\vdots&\vdots&\cdots&\vdots&\vdots\\\ 0&0&0&\cdots&0&0\\\
0&0&0&\cdots&0&0\end{pmatrix}\right\|$
$\displaystyle\leq\mu^{n-1}\|[v,b_{1}]+\delta b_{2}+\delta b_{1}[u,b_{n}]\|.$

∎

Let $\mathcal{A}$ be a unital C*-algebra. Assume that there are isometries
$u,v\in\mathcal{A}$ such that

$\displaystyle u^{*}u=v^{*}v=uu^{*}+vv^{*}=1_{\mathcal{A}}\quad\text{and}\quad
u^{*}v=v^{*}u=0.$ (6.6.3)

Examples of such unital C*-algebras are $\mathcal{B}(\mathcal{H})$ (where
$\mathcal{H}$ is an infinite dimensional Hilbert space) as well as any unital
C*-algebra which contains the Cuntz algebra $\mathcal{O}_{2}$ (Cuntz, (1977)).
Note that whenever a unital C*-algebra admits a trace map there are no
isometries satisfying Equation (6.6.3). In particular, any finite dimensional
unital C*-algebra does not have such elements. It is also clear that no
commutative unital C*-algebra can have isometries satisfying Equation (6.6.3).

It is shown in Tao, (2019) that whenever $\mathcal{H}$ is an infinite
dimensional Hilbert space, then the Banach algebras $\mathcal{B}(\mathcal{H})$
and $M_{2}(\mathcal{B}(\mathcal{H}))$ are isometrically isomorphic. We now do
these results for C*-algebras whenever they have isometries satisfying
Equation (6.6.3). To do so we first need a result from the theory of
C*-algebras.

######  Theorem 6.6.3.

(cf. Takesaki, (2002); Pedersen, (2018))

  1. (i)

Every *-homomorphism between C*-algebras is norm decreasing.

  2. (ii)

If a *-homomorphism between C*-algebras is injective, then it is isometric.

######  Theorem 6.6.4.

Let $\mathcal{A}$ be a unital C*-algebra. If there are isometries
$u,v\in\mathcal{A}$ such that Equation (6.6.3) holds, then the map

$\displaystyle\phi:\mathcal{A}\ni x\mapsto\begin{pmatrix}u^{*}xu&u^{*}xv\\\
v^{*}xu&v^{*}xv\end{pmatrix}\in M_{2}(\mathcal{A})$ (6.6.4)

is a C*-algebra isomorphism with the inverse map

$\displaystyle\psi:M_{2}(\mathcal{A})\ni\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\mapsto uau^{*}+ubv^{*}+vcu^{*}+vdv^{*}\in\mathcal{A}.$
(6.6.5)

###### Proof.

Using Equation (6.6.3), a direct computation gives

$\displaystyle\phi\psi\begin{pmatrix}a&b\\\ c&d\end{pmatrix}$
$\displaystyle=\phi(uau^{*}+ubv^{*}+vcu^{*}+vdv^{*})$
$\displaystyle=\begin{pmatrix}u^{*}(uau^{*}+ubv^{*}+vcu^{*}+vdv^{*})u&u^{*}(uau^{*}+ubv^{*}+vcu^{*}+vdv^{*})v\\\
v^{*}(uau^{*}+ubv^{*}+vcu^{*}+vdv^{*})u&v^{*}(uau^{*}+ubv^{*}+vcu^{*}+vdv^{*})v\end{pmatrix}$
$\displaystyle=\begin{pmatrix}1_{\mathcal{A}}a1_{\mathcal{A}}+1_{\mathcal{A}}b0+0c1_{\mathcal{A}}+0d0&1_{\mathcal{A}}a0+1_{\mathcal{A}}b1_{\mathcal{A}}+0c0+0d1_{\mathcal{A}}\\\
0a1_{\mathcal{A}}+0b0+1_{\mathcal{A}}c1_{\mathcal{A}}+1_{\mathcal{A}}d0&0a0+0b1_{\mathcal{A}}+1_{\mathcal{A}}c0+1_{\mathcal{A}}d1_{\mathcal{A}}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}a&b\\\
c&d\end{pmatrix},\quad\forall\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in
M_{2}(\mathcal{A})$

and

$\displaystyle\psi\phi x$ $\displaystyle=\psi\begin{pmatrix}u^{*}xu&u^{*}xv\\\
v^{*}xu&v^{*}xv\end{pmatrix}$
$\displaystyle=u(u^{*}xu)u^{*}+u(u^{*}xv)v^{*}+v(v^{*}xu)u^{*}+v(v^{*}xv)v^{*}$
$\displaystyle=uu^{*}x(uu^{*}+vv^{*})+vv^{*}x(uu^{*}+vv^{*})$
$\displaystyle=uu^{*}x1_{\mathcal{A}}+vv^{*}x1_{\mathcal{A}}=(uu^{*}+vv^{*})x=x,\quad\forall
x\in\mathcal{A}.$

Further,

$\displaystyle(\phi(x))^{*}$ $\displaystyle=\begin{pmatrix}u^{*}xu&u^{*}xv\\\
v^{*}xu&v^{*}xv\end{pmatrix}^{*}=\begin{pmatrix}u^{*}x^{*}u&(v^{*}xu)^{*}\\\
(u^{*}xv)^{*}&v^{*}x^{*}v\end{pmatrix}$
$\displaystyle=\begin{pmatrix}u^{*}x^{*}u&u^{*}x^{*}v\\\
v^{*}x^{*}u&v^{*}x^{*}v\end{pmatrix}=\phi(x^{*}),\quad\forall
x\in\mathcal{A}.$

Hence $\phi$ is a *-isomorphism. Using Theorem 6.6.3, to show $\phi$ is a
C*-algebra isomorphism (i.e., isometric isomorphism), it suffices to show that
$\phi$ is injective. Let $x\in\mathcal{A}$ be such that $\phi x=0$. Then

$\displaystyle u^{*}xu=u^{*}xv=0,\quad v^{*}xv=v^{*}xu=0.$

Using the first equation we get $uu^{*}xuu^{*}=uu^{*}xvv^{*}=0$ which implies
$uu^{*}x=uu^{*}x(uu^{*}+vv^{*})=0$. Similarly using the second equation we get
$vv^{*}x=0$. Therefore $x=(uu^{*}+vv^{*})x=0$. Hence $\phi$ is injective which
completes the proof. ∎

Along with the lines of Theorem 6.6.4 we can easily derive the following
result.

######  Theorem 6.6.5.

Let $\mathcal{A}$ be a unital C*-algebra and $n\in\mathbb{N}$. If there are
isometries $u,v\in\mathcal{A}$ such that Equation (6.6.3) holds, then the map

$\displaystyle\phi:M_{n}(\mathcal{A})\ni
X\mapsto\begin{pmatrix}u^{*}Xu&u^{*}Xv\\\ v^{*}Xu&v^{*}Xv\end{pmatrix}\in
M_{2n}(\mathcal{A})$

is a C*-algebra isomorphism with the inverse map

$\displaystyle\psi:M_{2n}(\mathcal{A})\ni\begin{pmatrix}A&B\\\
C&D\end{pmatrix}\mapsto uAu^{*}+uBv^{*}+vCu^{*}+vDv^{*}\in
M_{n}(\mathcal{A}),$

where if $X\coloneqq[x_{i,j}]_{i,j}$ is a matrix, and $a,b\in\mathcal{A}$, by
$aXb$ we mean the matrix $[ax_{i,j}b]_{i,j}$. In particular, the C*-algebras
$\mathcal{A},M_{2}(\mathcal{A}),M_{4}(\mathcal{A}),\dots,M_{2n}(\mathcal{A}),\dots$
are all *-isometrically isomorphic.

In the rest of this chapter, we assume that unital C*-algebra $\mathcal{A}$
has isometries $u,v$ satisfying Equation (6.6.3). In the next result we use
the following notation. Given a vector $x\in\mathcal{A}^{n}$, $x_{i}$ means
its $i^{\text{th}}$ coordinate.

######  Proposition 6.6.6.

Let $n\geq 2$ and $T:\mathcal{A}^{n}\to\mathcal{A}^{n-1}$ be the bounded
linear operator defined by

$\displaystyle
T(b_{i})_{i=1}^{n}\coloneqq([v,b_{i}]+[u,b_{i-1}])_{i=2}^{n},\quad\forall(b_{i})_{i=1}^{n}\in\mathcal{A}^{n}.$

Then there exists a bounded linear right-inverse
$R:\mathcal{A}^{n-1}\to\mathcal{A}^{n}$ for $T$ such that

$\displaystyle\|Rb\|$ $\displaystyle=\sup_{1\leq i\leq n}\|(Rb)_{i}\|\leq
8\sqrt{2}n^{2}\sup_{2\leq i\leq n}\|b_{i}\|$ $\displaystyle\leq
8\sqrt{2}n^{2}\sup_{1\leq i\leq n}\|b_{i}\|=8\sqrt{2}n^{2}\|b\|,\quad\forall
b\in\mathcal{A}^{n}.$

###### Proof.

Define

$\displaystyle
L:\mathcal{A}^{n-1}\ni(x_{i})_{i=2}^{n}\mapsto\left(-\frac{1}{2}x_{i}v^{*}-\frac{1}{2}x_{i+1}u^{*}\right)_{i=1}^{n}\in\mathcal{A}^{n},\quad\text{
where }x_{1}\coloneqq 0,x_{n+1}\coloneqq 0$

and

$\displaystyle
E:\mathcal{A}^{n-1}\ni(x_{i})_{i=2}^{n}\mapsto\left(\frac{1}{2}(vx_{i}v^{*}+vx_{i+1}u^{*}+ux_{i-1}v^{*}+ux_{i}u^{*})\right)_{i=2}^{n}\in\mathcal{A}^{n-1}.$

Then

$\displaystyle
TL(x_{i})_{i=2}^{n}=T\left(-\frac{1}{2}x_{i}v^{*}-\frac{1}{2}x_{i+1}u^{*}\right)_{i=1}^{n}=-\frac{1}{2}(T(x_{i}v^{*})_{i=2}^{n}+T(x_{i+1}u^{*})_{i=2}^{n})$
$\displaystyle=-\frac{1}{2}(([v,x_{i}v^{*}]+[u,x_{i-1}v^{*}])_{i=2}^{n}+([v,x_{i+1}u^{*}]+[u,x_{i}u^{*}])_{i=2}^{n})$
$\displaystyle=-\frac{1}{2}(vx_{i}v^{*}-x_{i}v^{*}v+ux_{i-1}v^{*}-x_{i-1}v^{*}u+vx_{i+1}u^{*}-x_{i+1}u^{*}v+ux_{i}u^{*}-x_{i}u^{*}u)_{i=2}^{n}$
$\displaystyle=-\frac{1}{2}(vx_{i}v^{*}-x_{i}+ux_{i-1}v^{*}-x_{i-1}v^{*}u+vx_{i+1}u^{*}+ux_{i}u^{*}-x_{i})_{i=2}^{n}$
$\displaystyle=(x_{i})_{i=2}^{n}-\frac{1}{2}(vx_{i}v^{*}+vx_{i+1}u^{*}+ux_{i-1}v^{*}+ux_{i}u^{*})_{i=1}^{n}$
$\displaystyle=(1-E)(x_{i})_{i=2}^{n},\quad\forall(x_{i})_{i=2}^{n}\in\mathcal{A}^{n-1},\quad\text{
where }1(x_{i})_{i=2}^{n}\coloneqq(x_{i})_{i=2}^{n}.$

We next try to show that the operator $1-E$ is bounded invertible with the
help of Neumann series. First step is to change the norm on $\mathcal{A}^{n}$
to an equivalent norm so that invertibility property will not affect in both
norms. Define a new norm on $\mathcal{A}^{n-1}$ by

$\displaystyle\|(x_{i})_{i=2}^{n}\|^{\prime}\coloneqq\sup_{2\leq i\leq
n}\left(2-\frac{i^{2}}{n^{2}}\right)^{\frac{-1}{2}}\|x_{i}\|.$

Let $x=(x_{i})_{i=2}^{n}\in\mathcal{A}^{n-1}$ be such that
$\|(x_{i})_{i=2}^{n}\|^{\prime}\leq 1$. Then

$\displaystyle\left(2-\frac{i^{2}}{n^{2}}\right)^{\frac{-1}{2}}\|x_{i}\|\leq\sup_{2\leq
i\leq n}\left(2-\frac{i^{2}}{n^{2}}\right)^{\frac{-1}{2}}\|x_{i}\|\leq
1,\quad\forall 2\leq i\leq n.$

Hence $\|x_{i}\|\leq\left(2-\frac{i^{2}}{n^{2}}\right)^{\frac{1}{2}}$ for all
$2\leq i\leq n$. Using Theorem 6.6.4 we now get

$\displaystyle\|(Ex)_{i}\|$
$\displaystyle=\frac{1}{2}\|vx_{i}v^{*}+vx_{i+1}u^{*}+ux_{i-1}v^{*}+ux_{i}u^{*}\|$
$\displaystyle=\frac{1}{2}\left\|\begin{pmatrix}x_{i}&x_{i+1}\\\
x_{i-1}&x_{i}\end{pmatrix}\right\|\leq\frac{1}{2}\left\|\begin{pmatrix}\|x_{i}\|&\|x_{i+1}\|\\\
\|x_{i-1}\|&\|x_{i}\|\end{pmatrix}\right\|$
$\displaystyle\leq\frac{1}{2}\left(\|x_{i}\|^{2}+\|x_{i+1}\|^{2}+\|x_{i-1}\|^{2}+\|x_{i}\|^{2}\right)^{\frac{1}{2}}$
$\displaystyle\leq\frac{1}{2}\left(\left(2-\frac{i^{2}}{n^{2}}\right)+\left(2-\frac{(i+1)^{2}}{n^{2}}\right)+\left(2-\frac{(i-1)^{2}}{n^{2}}\right)+\left(2-\frac{i^{2}}{n^{2}}\right)\right)^{\frac{1}{2}}$
$\displaystyle=\left(2-\frac{i^{2}}{n^{2}}-\frac{1}{2n^{2}}\right)^{\frac{1}{2}}\leq\left(1-\frac{1}{8n^{2}}\right)^{\frac{1}{2}}\left(2-\frac{i^{2}}{n^{2}}\right)^{\frac{1}{2}}$
$\displaystyle\leq\left(1-\frac{1}{8n^{2}}\right)\left(2-\frac{i^{2}}{n^{2}}\right)^{\frac{1}{2}},\quad\forall
2\leq i\leq n.$

Hence

$\displaystyle\|Ex\|^{\prime}=\sup_{2\leq i\leq
n}\left(2-\frac{i^{2}}{n^{2}}\right)^{\frac{-1}{2}}\|(Ex)_{i}\|\leq\left(1-\frac{1}{8n^{2}}\right)\|x\|^{\prime},\quad\forall
x\in\mathcal{A}^{n-1}.$

Since $1-\frac{1}{8n^{2}}<1$, $1-E$ is invertible and
$\|(1-E)^{-1}x\|^{\prime}\leq 8n^{2}\|x\|^{\prime}$. Now going back to the
original norm, we get

$\displaystyle\frac{1}{\sqrt{2}}\|((1-E)^{-1}x)_{i}\|$
$\displaystyle\leq\sup_{2\leq i\leq
n}\left(2-\frac{i^{2}}{n^{2}}\right)^{\frac{-1}{2}}\|((1-E)^{-1}x)_{i}\|$
$\displaystyle=\|(1-E)^{-1}x\|^{\prime}\leq 8n^{2}\|x\|^{\prime}$
$\displaystyle=8n^{2}\sup_{2\leq i\leq
n}\left(2-\frac{i^{2}}{n^{2}}\right)^{\frac{-1}{2}}\|x_{i}\|$
$\displaystyle\leq 8n^{2}\sup_{2\leq i\leq
n}\|x_{i}\|=8n^{2}\|x\|,\quad\forall x\in\mathcal{A}^{n-1}.$

Define $R\coloneqq L(1-E)^{-1}$. Then $TR=TL(1-E)^{-1}=(1-E)(1-E)^{-1}=1$ and

$\displaystyle\|Rb\|$ $\displaystyle=\sup_{1\leq i\leq
n}\|(Rb)_{i}\|=\|L(1-E)^{-1}b\|\leq\|L\|\|(1-E)^{-1}b\|\leq\|(1-E)^{-1}b\|$
$\displaystyle=\sup_{2\leq i\leq n}\|((1-E)^{-1}b)_{i}\|\leq
8\sqrt{2}n^{2}\|b\|=8\sqrt{2}n^{2}\sup_{2\leq i\leq n}\|b_{i}\|,\quad\forall
b\in\mathcal{A}^{n-1}.$

∎

As given in Tao, (2019) we try to shift from the systems of equations (6.6.1)
and (6.6.2) to the solution of single equation. Let $n\geq 2$. Define
$a\coloneqq(0,\dots,n)\in\mathcal{A}^{n}$,

$\displaystyle
F:\mathcal{A}^{n}\ni(b_{i})_{i=1}^{n}\mapsto(-2b_{3},\dots,-(n-1)b_{n},0)\in\mathcal{A}^{n-1}$

and

$\displaystyle
G:\mathcal{A}^{n}\times\mathcal{A}^{n}\ni((b_{i})_{i=1}^{n},(c_{i})_{i=1}^{n})\mapsto(-b_{2}[u,c_{n}],\dots,-b_{n}[u,c_{n}])\in\mathcal{A}^{n-1}.$

We then have $\|F\|\leq n-1$ and $\|G\|\leq 2$.

######  Proposition 6.6.7.

Systems (6.6.1) and (6.6.2) have a solution $b$ if and only if

$\displaystyle Tb=a+\delta F(b)+\delta G(b,b).$ (6.6.6)

###### Proof.

Systems (6.6.1) and (6.6.2) have a solution $b$ if and only if

$\displaystyle[v,b_{i}]+[u,b_{i-1}]=-i\delta b_{i+1}-\delta
b_{i}[u,b_{n}],\quad\forall i=2,\dots,n-1$

and

$\displaystyle[v,b_{n}]+[u,b_{n-1}]=-\delta b_{n}[u,b_{n}]+n\cdot
1_{{M_{n}(\mathcal{A})}}$

if and only if

$\displaystyle([v,b_{i}]+[u,b_{i-1}])_{i=2}^{n}=$
$\displaystyle\quad(0,\dots,n)+\delta(-2b_{3},\dots,-(n-1)b_{n},0)+\delta(-b_{2}[u,b_{n}],\dots,-b_{n}[u,b_{n}])$

if and only if

$\displaystyle Tb=a+\delta F(b)+\delta G(b,b).$

∎

The above proposition reduces the work of solving systems (6.6.1) and (6.6.2)
to a single operator equation. To solve (6.6.6) we need an abstract lemma from
Tao, (2019).

######  Lemma 6.6.8.

(Tao, (2019)) Let $\mathcal{X}$, $\mathcal{Y}$ be Banach spaces,
$T,F:\mathcal{X}\to\mathcal{Y}$ be bounded linear operators, and let
$G:\mathcal{X}\times\mathcal{X}\to\mathcal{Y}$ be a bounded bilinear operator
with bound $r>0$ and let $a\in\mathcal{Y}$. Suppose that $T$ has a bounded
linear right inverse $R:\mathcal{Y}\to\mathcal{X}$. If $\delta>0$ is such that

$\displaystyle\delta(2\|F\|\|R\|+4r\|R\|^{2}\|a\|)<1,$ (6.6.7)

then there exists $b\in\mathcal{X}$ with $\|b\|\leq 2\|R\|\|a\|$ that solves
the equation

$\displaystyle Tb=a+\delta F(b)+\delta G(b,b).$

######  Theorem 6.6.9.

For each $n\geq 2$, there exists a solution $b$ to Equation (6.6.6) such that
$\|b\|\leq 16\sqrt{2}n^{3}$.

###### Proof.

We apply Lemma 6.6.8 for

$\displaystyle\delta\coloneqq\frac{1}{2000n^{5}}.$

Then using Proposition 6.6.6, we get

$\displaystyle\delta(2\|F\|\|R\|+4r\|R\|^{2}\|a\|)$
$\displaystyle\leq\frac{1}{2000n^{5}}(2(n-1)8\sqrt{2}n^{2}+4.2.128.n^{4}.n)$
$\displaystyle\leq\frac{1}{2000n^{5}}(16\sqrt{2}n^{3}+1024n^{5})<1.$

Lemma 6.6.8 now says that there exists a $b$ which satisfies (6.6.6). ∎

######  Theorem 6.6.10.

For each $n\geq 2$, let $b$ be an element satisfying Equation (6.6.6) and
$\|b\|\leq 16\sqrt{2}n^{3}$. Then for $\mu=\frac{1}{2}$, $D_{\mu},X_{\mu}\in
M_{n}(\mathcal{A})$ such that

$\displaystyle\|D_{\mu}\|=O(n^{5}),\quad\|X_{\mu}\|=O(1),\quad\|[D_{\mu},X_{\mu}]-1_{M_{n}(\mathcal{A})}\|=O(n^{3}2^{-n}).$

###### Proof.

Let $D_{\mu},X_{\mu}\in M_{n}(\mathcal{A})$ be as in Corollary 6.6.2. We then
have

$\displaystyle\|D_{\mu}\|\leq
4.2000n^{5}\|u\|+2.2000n^{5}\|v\|+(n-1)+\frac{1}{2000n^{5}}\sum_{i=1}^{n}\frac{1}{2^{n-i-1}}16\sqrt{2}n^{3}\|u\|$
$\displaystyle\quad\quad=O(n^{5}),$ $\displaystyle\|X_{\mu}\|\leq
1+\frac{1}{2000n^{5}}\sum_{i=1}^{n}\frac{1}{2^{n-i-1}}16\sqrt{2}n^{3}=O(1),$
$\displaystyle\|[D_{\mu},X_{\mu}]-1_{M_{n}(\mathcal{A})}\|\leq
2\mu^{n-1}(\|v\|\|b_{1}\|+\delta\|b_{2}\|+\delta\|b_{1}\|\|u\|\|b_{n}\|).$
$\displaystyle\quad\leq
2\frac{1}{2^{n-1}}(\|u\|16\sqrt{2}n^{3}+\frac{1}{2000n^{5}}16\sqrt{2}n^{3}+\frac{1}{2000n^{5}}16\sqrt{2}n^{3}\|u\|16\sqrt{2}n^{3})$
$\displaystyle\quad\leq
2\frac{n^{3}}{2^{n-1}}(\|v\|16\sqrt{2}+\frac{1}{2000n^{5}}16\sqrt{2}+\frac{1}{2000n^{5}}16\sqrt{2}n^{3}\|u\|16\sqrt{2})=O(n^{3}2^{-n}).$

∎

######  Theorem 6.6.11.

Let $0<\varepsilon\leq 1/2$. Then there exist an even integer $n$ and $D,X\in
M_{n}(\mathcal{A})$ with

$\displaystyle\|[D,X]-1_{M_{n}(\mathcal{A})}\|\leq\varepsilon$

such that

$\displaystyle\|D\|\|X\|=O\left(\log^{5}\frac{1}{\varepsilon}\right).$

###### Proof.

Let $D_{\mu},X_{\mu}\in M_{n}(\mathcal{A})$ be as in Corollary 6.6.2. Theorem
6.6.10 says that there are $\alpha,\beta,\gamma>0$ be such that

$\displaystyle\|D_{\mu}\|\leq\alpha
n^{5},\quad\|X_{\mu}\|\leq\beta,\quad\|[D_{\mu},X_{\mu}]-1_{M_{n}(\mathcal{A})}\|\leq\gamma
n^{3}2^{-n}.$

Since $2^{n}>n^{4}$ all but finitely many $n$’s, $\gamma
n^{3}2^{-n}<\varepsilon$ all but finitely many $n$’s. We now choose real $c$
such that $n=c\log\frac{1}{\varepsilon}$ is even $\gamma
n^{3}2^{-n}<\varepsilon$. We then have
$\|D_{\mu}\|=O(\log^{5}(\frac{1}{\varepsilon}))$ and
$\|[D_{\mu},X_{\mu}]-1_{M_{n}(\mathcal{A})}\|\leq\varepsilon$. ∎

Theorem 6.6.11 and Theorem 6.6.5 easily give the following.

######  Theorem 6.6.12.

Let $\mathcal{A}$ be a unital C*-algebra. Suppose there are isometries
$u,v\in\mathcal{A}$ such that Equation (6.6.3) holds. Then for each
$0<\varepsilon\leq 1/2$, there exist $d,x\in\mathcal{A}$ with

$\displaystyle\|[d,x]-1_{\mathcal{A}}\|\leq\varepsilon$

such that

$\displaystyle\|d\|\|x\|=O\left(\log^{5}\frac{1}{\varepsilon}\right).$

######  Remark 6.6.13.

Let $\mathcal{A}$ be a finite dimensional unital C*-algebra. From the
structure theory (Davidson, (1996)) we have

$\displaystyle\mathcal{A}\cong M_{n_{1}}(\mathbb{C})\oplus\cdots\oplus
M_{n_{r}}(\mathbb{C}),$

for unique (upto permutation) natural numbers $n_{1},\dots,n_{r}$. This result
says that normalized trace map (a trace map Tr such that
$\text{Tr}(1_{\mathcal{A}})=1$) exists on $\mathcal{A}$. Using this we make
the following two observations.

  1. (i)

$\mathcal{A}$ can not have isometries satisfying Equation (6.6.3). Suppose
that there are such isometries. Then

$\displaystyle
1=\text{Tr}(uu^{*}+vv^{*})=\text{Tr}(uu^{*})+\text{Tr}(vv^{*})=\text{Tr}(u^{*}u)+\text{Tr}(v^{*}v)=2$

which is impossible.

  2. (ii)

In (Tao, (2019)), Tao observed that if $\mathcal{H}$ is a finite dimensional
Hilbert space, then there are no $D,X\in\mathcal{B}(\mathcal{H})$ satisfying
$\|[D,X]-1_{\mathcal{B}(\mathcal{H})}\|<1$. We elaborate this for any finite
dimensional unital C*-algebra $\mathcal{A}$, namely, there do not exist
$d,x\in\mathcal{A}$ satisfying $\|[d,x]-1_{\mathcal{A}}\|<1$. In other words,
Theorem 6.6.12 fails for every finite dimensional unital C*-algebra. Let
$d,x\in\mathcal{A}$ be arbitrary. From the structure theory, we identify that
$d$ as $D$ and $x$ as $X$ for some matrices $D,X\in M_{n}(\mathbb{C})$ and for
some $n$. Using the commutativity of trace we then have $\text{Tr}([D,X])=0$.
Let $\lambda_{1},\dots,\lambda_{n}$ be eigenvalues of $[D,X]$. Then
$\sum_{j=1}^{n}\lambda_{j}=\text{Tr}([D,X])=0$. This gives

$\displaystyle
n=\left|\sum_{j=1}^{n}(\lambda_{j}-1)\right|\leq\sum_{j=1}^{n}|\lambda_{j}-1|.$

Previous inequality says that there is atleast one $j$ such that
$|\lambda_{j}-1|\geq 1$. We next see that all the eigenvalues of
$[D,X]-1_{M_{n}(\mathbb{C})}$ are $\lambda_{1}-1,\dots,\lambda_{n}-1$. Using
the property of operator norm we finally get

$\displaystyle\|[d,x]-1_{\mathcal{A}}\|=\|[D,X]-1_{M_{n}(\mathbb{C})}\|\geq\sup_{1\leq
j\leq n}|\lambda_{j}-1|\geq 1.$

Conclusion and future work :  In this appendix we showed that the result of
Tao’s is valid in more general spaces. One of the future objectives is to
improve the bounds in Theorem 6.6.12 and Theorem 6.5.8.

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LIST OF SYMBOLS AND ABBREVIATIONS

$\mathbb{N}$ | : | Set of natural numbers
---|---|---
$\mathbb{R}$ | : | Field of real numbers
$\mathbb{C}$ | : | Field of complex numbers
$\\{\cdot\\}_{n}$ | : | Collections/sequences indexed by $\mathbb{N}$
$\mathbb{M}$ | : | Subset of natural numbers
$\mathbb{M}^{c}$ | : | Complement of $\mathbb{M}$
$\mathbb{K}$ | : |  $\mathbb{R}$ or $\mathbb{C}$
$\alpha,\beta,\gamma$ | : | Elements of $\mathbb{K}$
$\mathcal{H}$, $\mathcal{H}_{0}$, … | : | Separable Hilbert spaces
$h$, $h_{0}$, … | : | Elements of Hilbert spaces
$\mathcal{H}\otimes\mathcal{H}_{0}$, … | : | Tensor product of $\mathcal{H}$ and $\mathcal{H}_{0}$
$\langle\cdot,\cdot\rangle$ | : | Inner product which is linear in first variable and conjugate
|  | linear in second variable
$\mathcal{X}$, $\mathcal{Y}$, … | : | Separable Banach spaces
$\|\cdot\|$ | : | Norm
$\mathcal{X}^{*}$ | : | Dual of Banach space $\mathcal{X}$ equipped with operator norm
$I_{\mathcal{X}}$ | : | Identity operator on $\mathcal{X}$
$\mathcal{B}(\mathcal{X},\mathcal{Y})$ | : | Banach space of bounded linear operators from $\mathcal{X}$ to $\mathcal{Y}$
|  | equipped with operator-norm
cf. | : | Cross reference (reference may not be the first reference
|  | where the notion/result arose)
ASF | : | Approximate Schauder frame
OVF | : | Operator-valued frame
p | : | A real number in $[1,\infty)$
$\ell^{p}(\mathbb{N})$ | : | $\\{\\{a_{n}\\}_{n}:a_{n}\in\mathbb{K},\forall n\in\mathbb{N},\sum_{n=1}^{\infty}|a_{n}|^{p}<\infty\\}$
$\ell^{\infty}(\mathbb{N})$ | : | $\\{\\{a_{n}\\}_{n}:a_{n}\in\mathbb{K},\forall n\in\mathbb{N},\sup_{n\in\mathbb{N}}|a_{n}|<\infty\\}$
$\mathcal{L}^{p}(\mathbb{R})$ | : | $\\{f:\mathbb{R}\in\mathbb{C},f\text{ measurable },\int_{\mathbb{R}}|f(x)|^{p}\,dx<\infty\\}$
$\\{e_{n}\\}_{n}$ | : | Standard Schauder basis for $\ell^{p}(\mathbb{N})$
$\mathcal{A}$ | : | C*-algebra
$\mathcal{M}$, $\mathcal{N}$ | : | Metric spaces
$d(\cdot,\cdot)$ | : | Metric on a metric space

$G$ | : | Locally compact group
---|---|---
$\mu_{G}$ | : | Left Haar measure on $G$
$\pi$ | : | Unitary representation of $G$
$\mathcal{U}$ | : | Group-like unitary system
$\mathcal{X}_{d}$ | : | BK-space
$\operatorname{Lip}(\cdot)$ | : | Lipschitz number
$(\mathcal{M},0)$ | : | Pointed metric space
$\|\cdot\|_{\operatorname{Lip}_{0}}$ | : | Lipschitz norm
$\operatorname{Lip}(\mathcal{M},\mathcal{X})$ | : | Space of Lipschitz functions from $\mathcal{M}$ to $\mathcal{X}$ equipped
|  | with Lipschitz number
$\operatorname{Lip}_{0}(\mathcal{M},\mathcal{X})$ | : | Banach space of base point preserving Lipschitz functions
|  | from $\mathcal{M}$ to $\mathcal{X}$ equipped with Lipschitz norm
$\operatorname{Lip}_{0}(\mathcal{X})$ |  | $\operatorname{Lip}_{0}(\mathcal{X},\mathcal{X})$
$\mathcal{F}(\mathcal{M})$ | : | Lipschitz free Banach space of $\mathcal{M}$
$c_{0}(\mathbb{N})$ | : | $\\{\\{a_{n}\\}_{n}:a_{n}\in\mathbb{K},\forall n\in\mathbb{N},\lim_{n\to\infty}a_{n}=0\\}$
$c(\mathbb{N})$ | : | $\\{\\{a_{n}\\}_{n}:a_{n}\in\mathbb{K},\forall n\in\mathbb{N},\\{a_{n}\\}_{n}\text{ converges in }\mathbb{K}\\}$
$T^{*}$ | : | Adjoint of the operator $T$
$\theta_{\tau}$ | : | Analysis operator for frame $\\{\tau_{n}\\}_{n}$ for Hilbert space
$\theta_{\tau}^{*}$ | : | Synthesis operator for frame $\\{\tau_{n}\\}_{n}$ for Hilbert space
$S_{\tau}$ | : | Frame operator for frame $\\{\tau_{n}\\}_{n}$ for Hilbert space
$\theta_{f}$ | : | Analysis operator for p-ASF $(\\{f_{n}\\}_{n},\\{\tau_{n}\\}_{n})$ for Banach space
$S_{f,\tau}$ | : | Frame operator for p-ASF $(\\{f_{n}\\}_{n},\\{\tau_{n}\\}_{n})$ for Banach space
$\theta_{A}$ | : | Analysis operator for OVF $\\{A_{n}\\}_{n}$
$\theta_{A}^{*}$ | : | Synthesis operator for OVF $\\{A_{n}\\}_{n}$
$S_{A}$ | : | Frame operator for OVF $\\{A_{n}\\}_{n}$
$S_{A,\Psi}$ | : | Frame operator for weak OVF $(\\{A_{n}\\}_{n},\\{\Psi_{n}\\}_{n})$
$M_{\lambda,f,\tau}$ | : | Multiplier
$\tau\otimes f$ | : | Map defined by $(\tau\otimes f)(x)\coloneqq f(x)\tau$
$f^{-1}$ | : | Inverse of map $f$
$\chi_{[0,1]}$ | : | Characteristic function on $[0,1]$
dim | : | Dimension of a subspace of a vector space
$P^{\perp}$ | : |  $I_{\mathcal{H}}-P$, whenever $P$ is a projection on $\mathcal{H}$
$[\cdot,\cdot]$ | : | Semi-inner product
$A^{\dagger}$ | : | Generalized adjoint of the operator $A$ in a semi-inner product space
$\mathcal{A}^{\prime}$ | : | Commutant of a set in an algebra

ker | : | Kernel of a linear operator
---|---|---
$W^{\perp}$ | : | Orthogonal complement of a subspace $W$ of a Hilbert space
$\mathbb{T}$ | : | Unit circle group centered at origin
group$(\cdot)$ | : | Group generated by a subset of a group
$\mathcal{B}(\mathcal{H})$ | : | $\mathcal{B}(\mathcal{H},\mathcal{H})$
$\mathcal{H}\oplus\mathcal{H}_{0}$ | : | Direct sum of Hilbert spaces $\mathcal{H}$ and $\mathcal{H}_{0}$
$\overline{\text{span}}~{}W$ | : | Closure of span of subset $W$ of a Banach space $\mathcal{X}$
$M_{n}(\mathcal{A})$ | : | C*-algebra of $n$ by $n$ matrices over unital C*-algebra $\mathcal{A}$
$[A,B]$ | : |  $AB-BA$, commutator of $A$ and $B$
$\mathcal{O}_{2}$ | : | Cuntz algebra generated by two isometries
$\mathcal{K}(\mathcal{H})$ | : | C*-algebra of compact operators in $\mathcal{B}(\mathcal{H})$
dist$(x,Y)$ | : | Distance between an element $x$ and a subset $Y$ of a metric space
$r=O(s)$ | : | Asymptotic notation
$\mathcal{C}(K)$ | : | C*-algebra of all complex-valued continuous functions
|  | on compact Hausdorff space $K$
$\mathscr{A}$ | : | Non empty set
$\mathcal{V}$ | : | Vector space
$\mathbb{J}$ | : | Index set
$\mathbb{Z}_{+}$ | : | $\\{0\\}\cup\mathbb{N}$
$\delta_{n,m}$ | : | Kronecker delta

### PUBLICATIONS

  1. 1.

K. Mahesh Krishna and P. Sam Johnson. Frames for metric spaces. Results in
Mathematics 77, 49 (2022).

https://doi.org/10.1007/s00025-021-01583-3.

  2. 2.

K. Mahesh Krishna and P. Sam Johnson. Towards characterizations of approximate
Schauder frame and its duals for Banach spaces.  Journal of Pseudo-
Differential Operators and Applications 12, 9 (2021).

https://doi.org/10.1007/s11868-021-00379-x.

  3. 3.

K. Mahesh Krishna and P. Sam Johnson. Dilation theorem for p-approximate
Schauder frames for separable Banach spaces. Palestine Journal of Mathematics
(accepted).

https://arXiv.org/2011.12188v1 [math.FA] 23 November 2020.

  4. 4.

K. Mahesh Krishna and P. Sam Johnson. Expansion of weak reconstruction
sequences to approximate Schauder frames for Banach spaces. Asian-European
Journal of Mathematics 2250060 (2022).

https://doi.org/10.1142/S1793557122500607.

  5. 5.

K. Mahesh Krishna and P. Sam Johnson. New Identity on Parseval p-Approxim-ate
Schauder Frames and Applications.  Journal of Interdisciplinary Mathematics
1-10 (2021).

https://doi.org/10.1080/09720502.2021.1891698.

  6. 6.

K. Mahesh Krishna and P. Sam Johnson. Perturbation of p-approximate Schau-der
frames for separable Banach spaces. Poincare Journal of Analysis and
Applications (accepted).

https://arXiv.org/2012.030054v1 [math.FA] 5 December 2020.

  7. 7.

K. Mahesh Krishna and P. Sam Johnson. Factorable weak operator-valued frames.
Annals of Functional Analysis, 13, 11 (2022).

https://doi.org/10.1007/s43034-021-00155-4.

  8. 8.

K. Mahesh Krishna and P. Sam Johnson. Commutators close to the identity in
unital C*-algebras. Proceedings - Mathematical Sciences (accepted).

https://arXiv.org/2104.02035v1 [math.FA] 5 April 2021.

  9. 9.

K. Mahesh Krishna and P. Sam Johnson. Dilations of linear maps on vector
spaces. Operators and Matrices (accepted).

https://arXiv.org/2104.07544v1 [math.FA] 14 April 2021.

  10. 10.

K. Mahesh Krishna and P. Sam Johnson. Multipliers for Lipschitz p-Bessel
sequences in metric spaces.

https://arXiv.org/2007.03209v1 [math.FA] 7 July 2020.