# Absolute matrix order ideals in absolute Matrix order unit spaces

Amit Kumar  Discipline of Mathematics, School of Basic Sciences, Indian
Institute of Technology Bhubaneswar, Argul, Bhubaneswar, Pin - 752050, Odisha
(State), India<EMAIL_ADDRESS>
###### Abstract.

In this paper, we define and study absolute matrix order ideals in absolute
matrix order unit spaces. As an application of absolute matrix order unit
property, we construct some kinds of absolute matrix order ideals in absolute
matrix order unit spaces. Later, we show that the Grothendieck group of a such
kind of absolute matrix order ideal for order projections is a subgroup of
Grothendieck group of corresponding absolute matrix order unit space for order
projections.

###### Key words and phrases:

Absolute matrix order unit space, absolute matrix order unit property,
absolute matrix order ideal, completely absolute value preserving map, order
projection, partial isometry, $\mathcal{K}_{0}$-group.

###### 2010 Mathematics Subject Classification:

Primary 46B40; Secondary 46L05, 46L30.

The author was financially supported by the Institute Post-doctoral Fellowship
of IIT Bhubaneswar, India.

##  1\. Introduction

The concept of Hereditary sub-$C^{*}$-algebras is well known in a
$C^{*}$-algebra theory. Let $\mathbb{A}$ be a $C^{*}$-algebra. A
sub-$C^{*}$-algebra $\mathbb{B}$ of $\mathbb{A}$ is said to be Hereditary
sub-$C^{*}$-algebra if it satisfies the following condition: $a\in\mathbb{B}$
for all $a\in\mathbb{A}^{+}$ and $b\in\mathbb{B}^{+}$ such that $a\leq b.$
This concept helped a lot in the study of certain notions in a
$C^{*}$-algebra. For example, the notions of minimal, maximal, purely infinite
and central projections in $C^{*}$-algebras are defined and studied in the
terms of Hereditary sub-$C^{*}$-algebras. In this regard, we refer to see [5,
6, 34, 38] and references therein.

A tiny version of the notion of Hereditary sub-$C^{*}$-algebras is also
defined and studied in real ordered vector spaces that is known as the notion
of order ideals. Let $\mathbb{X}$ be a real ordered vector space. A vector
subpace $\mathbb{Y}$ of $\mathbb{X}$ is said to be an order ideal if it
satisfies the following condition: $x\in\mathbb{Y}$ for all
$x\in\mathbb{X}^{+}$ and $y\in\mathbb{Y}^{+}$ such that $x\leq y.$ Since the
self adjoint part $\mathbb{A}_{sa}$ of $\mathbb{A}$ is a real ordered vector
space, the notion of order ideals in ordered vector spaces is a genealization
of the notion of Hereditary sub-$C^{*}$-algebra in $C^{*}$-algebras. For
definition and more informations about order ideals, see [1, 15, 39].

Order strucure of $C^{*}$-algebras has its own importance. It is very rich
with certain properties and plays an important role in the classification of
certain kind of $C^{*}$-algebras. Therefore the study of order structure
becomes necessary. Many researchers started working in this direction. In this
regard, we refer to see [13, 17, 19, 2, 3, 4, 7, 8, 10, 12, 32, 33] and
references therein. The notions of matricially normed spaces defined and
studied by Ruan [11, 35], and matrix ordered and matrix order unit spaces
defined and studied by Choi and Effros [9], have high impact in this
direction. The unital $C^{*}$-algebras are examples of such spaces. However,
the non-unital versions of such spaces has been taken care by Karn [28, 29,
30, 31] and Schreiner [37]. Such spaces are called $L^{\infty}$-matricially
Riesz normed spaces and matrix regular operator spaces by Karn and Schreiner
respectively. The non-unital $C^{*}$-algebras are examples of such spaces.

More than two decades, Karn is also working in direction. In fact, he is
working on the order theoretic aspects of $C^{*}$-algebras. He has published
several papers in this direction. We refer to see [27, 28, 20, 21, 22, 23]. In
[27], he defined and studied order unit and matrix order unit properties in
$L^{\infty}$-matricially Riesz normed spaces. He found that only projections
satisfy order unit property in $C^{*}$-algebras [27, Theorem 3.4]. Thus, by
help of order unit property, he characterized hereditary sub-$C^{*}$-algebras
in $C^{*}$-algebras. He has also studied matrix order ideals in matrix order
unit spaces which is found to be an abstract version of hereditary
sub-$C^{*}$-algebras. He gave a characrterization of matrix order ideals in
terms of order unit property [27, Proposition 2.1]. In [23], he defined and
studied a non-commutative vector lattice structure called as absolute order
unit spaces. Unital $C^{*}$-algebras, unital $JB$-algebras and unital
$AM$-spaces are some examples of such spaces. He defined and studied order
unit property in absolute order unit spaces. He also defined and studied
absolute version of order unit property in absolute order unit spaces namely
absolute order unit property. Later, he introduced and studied the notion of
order projections in absolute order unit spaces and gave a characterization in
terms of absolute order unit property [23, Proposition 6.1].

The author in the collaboration with Karn has also been starting working in
this direction (see [24, 25, 26]). In [24], he defined and studied the notion
of absolute matrix order unit spaces. This notion is a matricial version of
absolute order unit spaces. The unital $C^{*}$-algebras and $JB$-algebras are
primary examples of these spaces. He defined and studied the notion of
absolute value preserving maps on these spaces. He found that under certain
conditions, these maps are isometries [24, Theorems 3.3 and 4.6]. He also
defined the notion of absolute order ideals and found that kernels of absolute
value preserving maps forms absolute order ideals [24, Theorem 2.7]. In [25],
he generalized the notion of order projections in absolute matrix order unit
spaces and, defined and characterized some variants of order projections [25,
Theorems 5.1 and 5.2]. In [26], he defined and studied some equivalences on
order projections in absolute matrix order unit spaces. By help of these
equivalences, he formed $\mathcal{K}_{0}$-group, the Grothendieck group for
order projections [26, Theorem 4.8]. He proved that $\mathcal{K}_{0}$-group is
an ordered abelian and a covariant functor [26, Corollaries 4.11 and 5.3].

We start our research work in the continuation of our earlier work [24, 25,
26]. In this paper, we define and study absolute matrix order unit property in
absolute matrix order unit spaces (Definition 3.1 and Proposition 3.2). The
absolute matrix order unit property in absolute matrix order unit spaces is a
generalization of absolute order unit property in absolute order unit spaces.
We also define the notion of absolute matrix order and absolute matrix order
unit ideals in absolute order unit spaces (Definition 3.3). As an application
of absolute matrix order unit propery, we construct absolute matrix order
ideals in absolute matrix order unit spaces (Theorem 3.5). Further, as an
application of absolute matrix order unit property, we also show that the
quotients spaces by kernels of completely absolutely preserving maps can also
be seen as absolute matrix order unit ideals (Theorem 3.9). Later, we prove
that under ceratin condition the Grothendieck group of absolue matrix order
unit ideal is subgroup of Grothendieck group of corresponding absolute matrix
order unit space (Theorem 4.6).

##  2\. Preliminaries

Throughout this paper, $\mathbb{X}$ stands for a complex vector space and
$\mathbb{M}_{l,m}(\mathbb{X})$ for the set of all the $l\times m$ matrices
over $\mathbb{X}.$ Then $\mathbb{M}_{l,m}(\mathbb{X})$ becomes a vector space
over $\mathbb{C}$ under the operations entry-wise addition and entry-wise
scalar multiplication. If $\mathbb{X}=\mathbb{C},$ then we write
$\mathbb{M}_{m,n}$ for $\mathbb{M}_{l,m}(\mathbb{C}).$ The zero element in
$\mathbb{M}_{m,n}$ is denoted by $0^{l,m}$ and for $l=m,$ the zero element in
$\mathbb{M}_{l}(\mathbb{X})$ is denoted by $0^{l}.$ Sometimes, we drop the
subscripts and write zero element in $\mathbb{M}_{l,m}$ by $0$ only. In fact,
$\mathbb{M}_{l}(\mathbb{X})$ is $\mathbb{M}_{l}$-bimodule under the following
matrix multiplications: for $x=[x_{i,j}]\in\mathbb{M}_{m,n}(\mathbb{X}),$ we
have $\varsigma
x=\begin{bmatrix}\displaystyle\sum_{l=1}^{m}\varsigma_{i,l}x_{l,j}\end{bmatrix}\in\mathbb{M}_{l,n}(\mathbb{X})$
if $\varsigma=[\varsigma_{i,j}]\in\mathbb{M}_{l,m}$ and
$x\varsigma=\begin{bmatrix}\displaystyle\sum_{l=1}^{n}x_{i,l}\varsigma_{l,j}\end{bmatrix}\in\mathbb{M}_{m,s}(\mathbb{X})$
if $\varsigma\in\mathbb{M}_{n,s}.$ Given
$x\in\mathbb{M}_{l,m}(\mathbb{X}),y\in\mathbb{M}_{n,s}(\mathbb{X}),$ we write:

$x\circledast y=\begin{bmatrix}x&0\\\
0&y\end{bmatrix}\in\mathbb{M}_{l+n,m+s}(\mathbb{X}).$

We also write: $x^{l}=x\circledast x\circledast\cdots\circledast
x\in\mathbb{M}_{lm}(\mathbb{X})$ for every $x\in\mathbb{M}_{m}(\mathbb{X}).$

For origin of such matricial notions, we refer to see [35].

We start with the definition of absolutely matrix ordered spaces, a matricial
version of absolutely ordered spaces. Absolutely ordered spaces were
introduced and studied by Karn in [23]. It is very near to a lattice
structure.

######  Definition 2.1 ([24], Definition 4.1).

Let $(\mathbb{X},\\{\ \mathbb{M}_{l}(\mathbb{X})^{+}\\})$ be a matrix ordered
space and also let
$|\cdot|_{l,m}:\mathbb{M}_{l,m}(\mathbb{X})\to\mathbb{M}_{m}(\mathbb{X})^{+}$
be functions for each pair $l,m\in\mathbb{N}$. For $l=m,$ we write
$|\cdot|_{l,m}=|\cdot|_{l}.$ Then
$\left(\mathbb{X},\\{\mathbb{M}_{m}(\mathbb{X})^{+}\\},\\{|\cdot|_{l,m}\\}\right)$
is said to be an _absolutely matrix ordered space_ , if the following
conditions are satisfied:

  1. (1)

$(\mathbb{M}_{m}(\mathbb{X})_{sa},\mathbb{M}_{m}(\mathbb{X})^{+},|\cdot|_{m})$
is an absolutely ordered space for every $m\in\mathbb{N}.$

  2. (2)

For all
$x\in\mathbb{M}_{m,n}(\mathbb{X}),y\in\mathbb{M}_{l,s}(\mathbb{X}),\varsigma_{1}\in\mathbb{M}_{l,m}$
and $\varsigma_{2}\in\mathbb{M}_{n,s},$ we have

    1. (a)

$|\varsigma_{1}x\varsigma_{2}|_{l,s}\leq\|\varsigma_{1}\|||x|_{m,n}\varsigma_{2}|_{n,s}.$

    2. (b)

$|x\circledast y|_{m+l,n+s}=|x|_{m,n}\circledast|y|_{l,s}.$

######  Example 2.2 ([24], Example 4.4).

Given a $C^{*}$-algebra $\mathbb{A},$ we know that
$\mathbb{M}_{m}(\mathbb{A})$ is also a $C^{*}$-algebra. Next, let
$\mathbb{M}_{m}(\mathbb{A})^{+}$ be the set of all the positive elements
contained in $\mathbb{M}_{m}(\mathbb{A}).$ Then
$(\mathbb{A},\\{\mathbb{M}_{m}(\mathbb{A})^{+}\\})$ becomes a matrix ordered
space and
$(\mathbb{A},\\{\mathbb{M}_{m}(\mathbb{A})^{+}\\},\\{|\cdot|_{l,m}\\})$
becomes an absolutely matrix ordered space, where each
$|\cdot|_{l,m}:\mathbb{M}_{l,m}(\mathbb{A})\to\mathbb{M}_{m}(\mathbb{A})^{+}$
is defined by $|x|_{l,m}=\sqrt{x^{*}x}$ for
$x\in\mathbb{M}_{l,m}(\mathbb{A}).$

The following result describes some properties of absolutely matrix ordered
spaces.

######  Proposition 2.3 ([24], Proposition 4.2).

In every absolutely matrix ordered space
$(\mathbb{X},\\{\mathbb{M}_{m}(\mathbb{X})^{+}\\},\\{|\cdot|_{l,m}\\}),$ for
each $l,m,n,s\in\mathbb{N}$ and
$x\in\mathbb{M}_{m,n}(\mathbb{X}),\varsigma\in\mathbb{M}_{l,m}$ with
$\varsigma^{*}\varsigma=\mathbb{I}_{m}~{}i.e.~{}\varsigma$ is an isometry, the
following hold:

  1. (1)

$|\varsigma x|_{l,n}=|x|_{m,n}.$

  2. (2)

$\left|\begin{bmatrix}0&x\\\
x^{*}&0\end{bmatrix}\right|_{m+n}=|x^{*}|_{n,m}\circledast|x|_{m,n}.$

  3. (3)

$\begin{bmatrix}|x^{*}|_{n,m}&x\\\
x^{*}&|x|_{m,n}\end{bmatrix}\in\mathbb{M}_{m+n}(\mathbb{X})^{+}.$

  4. (4)

$|x|_{m,n}=\left|\begin{bmatrix}x\\\ 0\end{bmatrix}\right|_{m+l,n}$ and
$|x|_{m,n}\circledast
0_{s}=\left|\begin{bmatrix}x&0\end{bmatrix}\right|_{m,n+s}.$

  5. (5)

$($[25], Lemma 3.1$).$ Moreover, if $l=m=n$ (in this case, $\varsigma$ is
unitary), we also have:
$|\varsigma^{*}x\varsigma|_{n}=\varsigma^{*}|x|_{n}\varsigma.$

Next, we write the definition of absolute matrix order unit spaces, a
matricial version of absolute order unit spaces. Absolute order unit spaces
were introduced and studied by Karn in [23].

######  Definition 2.4 ([24], Definition 4.3).

Let $(\mathbb{X},\\{\mathbb{M}_{m}(\mathbb{X})^{+}\\},e)$ be a matrix order
unit space and also let
$|\cdot|_{l,m}:\mathbb{M}_{l,m}(\mathbb{X})\to\mathbb{M}_{m}(\mathbb{X})^{+}$
be a function for each pair $l,m\in\mathbb{N}.$ If
$\left(\mathbb{X},\\{\mathbb{M}_{m}(\mathbb{X})^{+}\\},\\{|\cdot|_{l,m}\\}\right)$
forms an absolutely matrix ordered space and $\perp=\perp_{\infty}^{a}$ on
$\mathbb{M}_{m}(\mathbb{X})^{+}$ for all $m\in\mathbb{N},$ then we say that
$(\mathbb{X},\\{\mathbb{M}_{m}(\mathbb{X})^{+}\\},e)$ with
$\\{|\cdot|_{l,m}\\}$ is an _absolute matrix order unit space_. The absolute
matrix order unit space is denoted by
$(\mathbb{X},\\{\mathbb{M}_{m}(\mathbb{X})^{+}\\},\\{|\cdot|_{l,m}\\},e).$

######  Example 2.5 ([24], Example 4.4).

Given any $C^{*}$-algebra $\mathbb{A},$ we have $\perp=\perp_{\infty}^{a}$ on
$\mathbb{M}_{m}(\mathbb{A})^{+}$ for every $m\in\mathbb{N}.$ Therefore, every
unital $C^{*}$-algebra forms an absolute matrix order unit space.

Now, we recall the definitions of order unit property and absolute order unit
propertie introduced and studied by Karn in [23].

######  Definition 2.6 ([23], Definition 5.1).

Let $(\mathbb{X},e)$ be an absolute matrix order unit space and
$x\in\mathbb{M}_{l}(\mathbb{X})^{+}$. We write:
$\mathbb{M}_{l}(\mathbb{X})_{sa}^{x}=\\{y\in\mathbb{M}_{l}(\mathbb{X})_{sa}:\epsilon
x\pm y\in\mathbb{M}_{l}(\mathbb{X})^{+}~{}\textrm{for some
$\epsilon>0$}~{}\\}.$ Then $x\in\mathbb{M}_{l}(\mathbb{X})^{+}$ is said to
have the _order unit property_ in $\mathbb{M}_{l}(\mathbb{X})_{sa}$ provided
$\pm y\leq\|y\|x$ for every $y\in\mathbb{M}_{l}(\mathbb{X})_{sa}^{x}.$
Moreover, $x$ is said to have the _absolute order unit property_ in
$\mathbb{M}_{l}(\mathbb{X})_{sa}$ provided $|y|\leq\|y\|x$ for every
$y\in\mathbb{M}_{l}(\mathbb{X})_{sa}^{x}.$

The sets of all the elements having order unit property and absolute order
unit property in $\mathbb{M}_{l}(\mathbb{X})_{sa}$ are denoted by
$\mathcal{M}_{l}(\mathbb{X})$ and $|\mathcal{M}|_{l}(\mathbb{X})$
respectively. For $l=1,$ we write:
$\mathcal{M}_{1}(\mathbb{X})=\mathcal{M}(\mathbb{X})$ and
$|\mathcal{M}|_{1}(\mathbb{X})=|\mathcal{M}|(\mathbb{X}).$

We explain the notions of order projections and partial isometries in absolute
matrix order unit spaces. These notions are generalizations of the notions of
projections and partial isometries in a $C^{*}$-algebra respectively .

######  Definition 2.7 ([25], Definition 3.1).

Let $\mathbb{X}$ be an absolute matrix order unit space. We say that
$x\in\mathbb{M}_{n}(\mathbb{X})$ is an _order projection_ provided
$x\in\mathbb{M}_{n}(\mathbb{X})_{sa}$ with $|2x-e^{n}|_{n}=e^{n}.$ Next, if
$x\in\mathbb{M}_{m,n}(\mathbb{X}),$ then we say that $x$ is a _partial
isometry_ provided $|x|_{m,n}\in\mathbb{OP}_{n}(\mathbb{X})$ and
$|x^{*}|_{n,m}\in\mathbb{OP}_{m}(\mathbb{X}).$

The notations $\mathbb{OP}_{n}(\mathbb{X})$ and
$\mathbb{PI}_{m,n}(\mathbb{X})$ stand for the sets of all the order
projections and all the partial isometries in $\mathbb{M}_{n}(\mathbb{X})$ and
$\mathbb{M}_{m,n}(\mathbb{X})$ respectively. We write:
$\mathbb{PI}_{m,n}(\mathbb{X})=\mathbb{PI}_{n}(\mathbb{X})$ for $m=n.$ We also
write: $\mathbb{OP}_{1}(\mathbb{X})=\mathbb{OP}(\mathbb{X})$ and
$\mathbb{PI}_{1}(\mathbb{X})=\mathbb{PI}(\mathbb{X}).$

Next result recalls an elementary property of order projections.

######  Lemma 2.8.

[25, Proposition 3.2] In an absolute matrix order unit space $\mathbb{X},$ we
have: $p\circledast q\in\mathbb{OP}_{l+m}(\mathbb{X})$ if and only if
$p\in\mathbb{OP}_{l}(\mathbb{X})$ and $q\in\mathbb{OP}_{m}(\mathbb{X}).$

The following result provides a characterization of order projections in terms
of order unit property and absolute order unit property.

######  Proposition 2.9.

[23, Proposition 6.1] In an absolute matrix order unit space $(\mathbb{X},e),$
let $x\in\mathbb{M}_{l}(\mathbb{X})_{sa}$ such that $0\leq x\leq e^{l}.$ Then
the following statements are equivalent:

  1. (1)

$x\in\mathbb{OP}_{l}(\mathbb{X}).$

  2. (2)

$x$ and $e-x\in\mathcal{M}_{l}(\mathbb{X}).$

  3. (3)

$x$ and $e-x\in|\mathcal{M}|_{l}(\mathbb{X}).$

##  3\. Order ideals in absolute matrix order unit spaces

In this section, we recall matrix order unit property and define absolute
matrix order unit property. These are matricial versions of order unit
property and absolute order unit property which were introduced by Karn in
[23].

######  Definition 3.1.

[27] Given $x\in\mathbb{X}^{+}$ in an absolute matrix order unit space
$\mathbb{X},$ we say that $x$ has the matrix order unit property in
$\mathbb{X}$ provided $x^{l}$ has the order unit property in
$\mathbb{M}_{l}(\mathbb{X})_{sa}$ for any $l\in\mathbb{N}.$ Further, if
$x^{l}$ has the absolute order unit property in
$\mathbb{M}_{l}(\mathbb{X})_{sa}$ for any $l\in\mathbb{N},$ we say that $x$
has the absolute matrix order unit property in $\mathbb{X}.$

The sets of all the elements having matrix order unit property and absolute
matrix order unit property are denoted by $\mathcal{M}_{\infty}(\mathbb{X})$
and $|\mathcal{M}|_{\infty}(\mathbb{X})$ respectively.

The following result elaborates some properties enjoyed by matrix order and
absolute matrix order unit properties.

######  Proposition 3.2.

Given $x\in\mathbb{X}^{+}$ in an absolute matrix order unit space
$(\mathbb{X},e),$ the following statements hold:

  1. (1)

$x\in\mathcal{M}_{\infty}(\mathbb{X})$ if and only if, we have
$\begin{bmatrix}\|y\|_{l}x^{l}&y\\\
y^{*}&\|y\|_{l}x^{l}\end{bmatrix}\in\mathbb{M}_{2l}(\mathbb{X})^{+}$ whenever
$\begin{bmatrix}\epsilon x^{l}&y\\\ y^{*}&\epsilon
x^{l}\end{bmatrix}\in\mathbb{M}_{2l}(\mathbb{X})^{+}$ for some
$l\in\mathbb{N},\epsilon>0$ and $y\in\mathbb{M}_{l}(\mathbb{X}).$

  2. (2)

$x\in|\mathcal{M}|_{\infty}(\mathbb{X})$ if and only if, we have
$\|y\|_{l}x^{l}-|y|_{l}~{}\textrm{and}~{}\|y\|_{l}x^{l}-|y^{*}|_{l}\in\mathbb{M}_{l}(\mathbb{X})^{+}$
whenever $\begin{bmatrix}\epsilon x^{l}&y\\\ y^{*}&\epsilon
x^{l}\end{bmatrix}\in\mathbb{M}_{2l}(\mathbb{X})^{+}$ for some
$l\in\mathbb{N},\epsilon>0$ and $y\in\mathbb{M}_{l}(\mathbb{X}).$

Moreover, for $\|x\|\leq 1,$ we also have:

  1. (3)

$x$ and $e-x\in\mathcal{M}(\mathbb{X})$ if and only if $x$ and
$e-x\in\mathcal{M}_{\infty}(\mathbb{X}).$

  2. (4)

$x$ and $e-x\in|\mathcal{M}|(\mathbb{X})$ if and only if $x$ and
$e-x\in|\mathcal{M}|_{\infty}(\mathbb{X}).$

###### Proof.

  1. (1)

Let $x\in\mathcal{M}_{\infty}(\mathbb{X}).$ Assume that for
$y\in\mathbb{M}_{l}(\mathbb{X}),$ there exists $\epsilon>0$ satisfying
$\begin{bmatrix}\epsilon x^{l}&y\\\ y^{*}&\epsilon
x^{n}\end{bmatrix}\in\mathbb{M}_{2l}(\mathbb{X})^{+}~{}\textrm{for
some}~{}\epsilon\in\mathbb{R}$ with $\epsilon>0.$ In this case, we also have
$\begin{bmatrix}\epsilon x^{n}&-y\\\ -y^{*}&\epsilon
x^{n}\end{bmatrix}=\begin{bmatrix}-\mathbb{I}_{l}&0\\\
0&\mathbb{I}_{l}\end{bmatrix}\begin{bmatrix}\epsilon x^{l}&y\\\ y^{*}&\epsilon
x^{n}\end{bmatrix}\begin{bmatrix}-\mathbb{I}_{l}&0\\\
0&\mathbb{I}_{l}\end{bmatrix}\in\mathbb{M}_{2l}(\mathbb{X})^{+}.$ Since
$x^{2l}\in\mathcal{M}_{2l}(\mathbb{X})$, we get that $\pm\begin{bmatrix}0&y\\\
y^{*}&0\end{bmatrix}\leq\left\|\begin{bmatrix}0&y\\\
y^{*}&0\end{bmatrix}\right\|_{2l}\begin{bmatrix}x^{l}&0\\\
0&x^{l}\end{bmatrix}=\|y\|_{l}\begin{bmatrix}x^{l}&0\\\ 0&x^{l}\end{bmatrix}.$
Thus $\begin{bmatrix}\|y\|_{l}x^{l}&y\\\
y^{*}&\|y\|_{l}x^{l}\end{bmatrix}\in\mathbb{M}_{2l}(\mathbb{X})^{+}.$

Next, we prove the converse part. For that let
$y\in\mathbb{M}_{l}(\mathbb{X})_{sa}$ satisfying $\pm y\leq\epsilon
x^{l}~{}\textrm{for some}~{}\epsilon>0.$ Then $\begin{bmatrix}\epsilon
x^{l}&y\\\ y&\epsilon
x^{l}\end{bmatrix}=\frac{1}{2}\left(\begin{bmatrix}\mathbb{I}_{l}\\\
\mathbb{I}_{l}\end{bmatrix}(\epsilon
x^{l}+y)\begin{bmatrix}\mathbb{I}_{l}&\mathbb{I}_{l}\end{bmatrix}\right)+\frac{1}{2}\left(\begin{bmatrix}\mathbb{I}_{l}\\\
-\mathbb{I}_{l}\end{bmatrix}(\epsilon
x^{n}-y)\begin{bmatrix}\mathbb{I}_{l}&-\mathbb{I}_{l}\end{bmatrix}\right)\in\mathbb{M}_{2l}(\mathbb{X})^{+}.$
By assumption, we conclude that $\begin{bmatrix}\|y\|_{l}x^{l}&y\\\
y&\|y\|_{l}x^{l}\end{bmatrix}\in\mathbb{M}_{2l}(\mathbb{X})^{+}.$ In this
case, we also conclude that

$\|y\|x^{l}\pm
y=\frac{1}{2}\left(\begin{bmatrix}\mathbb{I}_{l}&\pm\mathbb{I}_{l}\end{bmatrix}\begin{bmatrix}\|y\|_{l}x^{l}&y\\\
y&\|y\|_{l}x^{l}\end{bmatrix}\begin{bmatrix}\mathbb{I}_{l}\\\
\pm\mathbb{I}_{l}\end{bmatrix}\right)\in\mathbb{X}^{+}.$

Thus $x^{l}\in\mathcal{M}_{l}(\mathbb{X})$ for every $l\in\mathbb{N}.$ Hence
$x\in\mathcal{M}_{\infty}(\mathbb{X}).$

  2. (2)

Let $x\in|\mathcal{M}|_{\infty}(\mathbb{X}).$ Also, let
$y\in\mathbb{M}_{l}(\mathbb{X})$ such that $\begin{bmatrix}\epsilon x^{l}&y\\\
y^{*}&\epsilon x^{n}\end{bmatrix}\in\mathbb{M}_{2l}(\mathbb{X})^{+}$ for some
$\epsilon>0.$ Then, we also have $\begin{bmatrix}\epsilon x^{n}&-y\\\
-y^{*}&\epsilon x^{n}\end{bmatrix}\in\mathbb{M}_{2l}(\mathbb{X})^{+}.$ Since
$x^{2l}\in|\mathcal{M}|_{2l}(\mathbb{X})$, we get that
$\begin{bmatrix}|y^{*}|_{l}&0\\\
0&|y|_{l}\end{bmatrix}=\left|\begin{bmatrix}0&y\\\
y^{*}&0\end{bmatrix}\right|_{2l}\leq\left\|\begin{bmatrix}0&y\\\
y^{*}&0\end{bmatrix}\right\|_{2l}\begin{bmatrix}x^{l}&0\\\
0&x^{l}\end{bmatrix}=\|y\|_{l}\begin{bmatrix}x^{l}&0\\\ 0&x^{l}\end{bmatrix}.$
Thus $\begin{bmatrix}\|y\|_{l}x^{l}-|y^{*}|_{l}&0\\\
0&\|y\|_{l}x^{l}-|y|_{l}\end{bmatrix}\in\mathbb{M}_{2l}(\mathbb{X})^{+}$ so
that

$\|y\|_{l}x^{l}-|y^{*}|_{l}=\begin{bmatrix}\mathbb{I}_{l}&0\end{bmatrix}\begin{bmatrix}\|y\|_{l}x^{l}-|y^{*}|_{l}&0\\\
0&\|y\|_{l}x^{l}-|y|_{l}\end{bmatrix}\begin{bmatrix}\mathbb{I}_{l}\\\
0\end{bmatrix}\in\mathbb{M}_{l}(\mathbb{X})^{+}$

and

$\|y\|_{l}x^{l}-|y|_{l}=\begin{bmatrix}0&\mathbb{I}_{l}\end{bmatrix}\begin{bmatrix}\|y\|_{l}x^{l}-|y^{*}|_{l}&0\\\
0&\|y\|_{l}x^{l}-|y|_{l}\end{bmatrix}\begin{bmatrix}0\\\
\mathbb{I}_{l}\end{bmatrix}\in\mathbb{M}_{l}(\mathbb{X})^{+}.$

Now, let’s prove the converse part. Let $y\in\mathbb{M}_{l}(\mathbb{X})_{sa}$
such that $\pm y\leq\epsilon x^{l}~{}\textrm{for some}~{}\epsilon>0.$ Then
$\begin{bmatrix}\epsilon x^{l}&y\\\ y&\epsilon
x^{l}\end{bmatrix}\in\mathbb{M}_{2l}(\mathbb{X})^{+}.$ By assumption, we get
that $\|y\|_{l}x^{l}-|y|_{l}\in\mathbb{M}_{l}(\mathbb{X})^{+}.$ Thus
$x^{l}\in|\mathcal{M}|_{l}(\mathbb{X})$ for every $l\in\mathbb{N}.$ Hence
$x\in|\mathcal{M}|_{\infty}(\mathbb{X}).$

  3. (3)

Suppose that $\|x\|\leq e.$ In this case, we have $0\leq x\leq e$ and
consequently $0\leq e-x\leq e.$ By Lemma 2.8 and Proposition 2.9, $x$ and
$e-x\in\mathcal{M}(\mathbb{X})$ if and only if $x\in\mathbb{OP}(\mathbb{X})$
if and only if $x^{l}\in\mathbb{OP}(\mathbb{X})$ if and only if $x^{l}$ and
$e^{l}-x^{l}\in\mathcal{M}_{l}(\mathbb{X})$ for every $l\in\mathbb{N}.$ Thus
$x$ and $e-x\in\mathcal{M}(\mathbb{X})$ if and only if $x$ and
$e-x\in\mathcal{M}_{\infty}(\mathbb{X}).$

Now, using Lemma 2.8 and Proposition 2.9, and repeating the proof of (3), we
can also prove (4). ∎

######  Definition 3.3.

Let
$\left(\mathbb{X},\\{\mathbb{M}_{m}(\mathbb{X})^{+}\\},\\{|\cdot|_{l,m}\\}\right)$
be an absolutely matrix ordered space and $\mathbb{Y}$ be a subspace of
$\mathbb{X}.$ For each $l\in\mathbb{N},$ we write:
$\mathbb{M}_{l}(\mathbb{Y})_{sa}=\mathbb{M}_{l}(\mathbb{X})_{sa}\cap\mathbb{M}_{l}(\mathbb{Y})~{}\textrm{and}~{}\mathbb{M}_{l}(\mathbb{Y})^{+}=\mathbb{M}_{l}(\mathbb{X})^{+}\cap\mathbb{M}_{l}(\mathbb{Y}).$
Then $\mathbb{Y}$ is said to be absolute matrix order ideal of $\mathbb{X},$
if the following two conditions are satisfied:

  1. (1)

$(\mathbb{M}_{l}(\mathbb{Y})_{sa},\mathbb{M}_{l}(\mathbb{Y})^{+})$ is an order
ideal of $(\mathbb{M}_{l}(\mathbb{X})_{sa},\mathbb{M}_{l}(\mathbb{X})^{+})$
for each $l\in\mathbb{N}.$

  2. (2)

$|\cdot|_{l,m}:\mathbb{M}_{l,m}(\mathbb{Y})\to\mathbb{M}(\mathbb{Y})_{m}^{+}$
is well defined for each pair of $l,m\in\mathbb{N}.$ In other words,
$\left(\mathbb{Y},\\{\mathbb{M}_{m}(\mathbb{Y})^{+}\\},\\{|\cdot|_{l,m}\\}\right)$
is itself an absolutely matrix ordered space.

Further, let $(\mathbb{X},e)$ be an absolute matrix order unit space. Then
$\mathbb{Y}$ is said to be absolute matrix order unit ideal, if along with
condition (1), the following additional condition is also satisfied:

  1. (3)

$(\mathbb{Y},\\{\mathbb{M}_{m}(\mathbb{Y})^{+}\\},\\{|\cdot|_{l,m}\\},y)$ is
an absolute matrix order unit space such that
$\|\cdot\|_{m}^{y}=\|\cdot\|_{m}$ on $\mathbb{M}_{m}(\mathbb{Y}),$ where
$\|\cdot\|_{m}~{}\textrm{and}~{}\|\cdot\|_{m}^{y}$ are the matrix norms
determined by the order units $e$ and $y$ respectively for each
$m\in\mathbb{N}.$

######  Example 3.4.

Let $\mathbb{A}$ be a unital $C^{*}$-algebra with unity element $1$ and $p$ be
a non-zero projection in $\mathbb{A}.$ Put
$\mathbb{A}_{p}=\\{pap:a\in\mathbb{A}\\}.$ Then $(\mathbb{A}_{p},p)$ is an
absolute matrix order unit ideal of $\mathbb{A}.$ To see this, we verify only
order ideal condition as other conditions are routine to verify $($we refer to
see [18]$).$ Now, let $a,b\in\mathbb{A}$ be such that $0\leq b\leq pap.$ Then
$(1-p)b(1-p)=0$ so that $\|\sqrt{b}(1-p)\|^{2}=0.$ In this case,
$\sqrt{b}(1-p)=0$ and we get that $b=bp$ and $b=b^{*}=(bp)^{*}=pb.$ Thus
$b=pbp\in\mathbb{A}_{p}.$

Next result generalize the last example in absolute matrix order unit spaces
for more general elements.

######  Theorem 3.5.

In an absolute matrix order unit space $\mathbb{X},$ let
$x\in|\mathcal{M}|_{\infty}(\mathbb{X})$ such that $\|x\|=1.$ We write:

$\mathbb{X}_{x}=\left\\{y\in\mathbb{X}:\begin{bmatrix}\epsilon x&y\\\
y^{*}&\epsilon x\end{bmatrix}\in\mathbb{M}_{2}(\mathbb{X})^{+}~{}\textrm{for
some}~{}\epsilon>0\right\\}$

and

$\mathbb{M}_{l,m}(\mathbb{X})_{x^{l},x^{m}}=\left\\{y\in\mathbb{M}_{l,m}(\mathbb{X}):\begin{bmatrix}\epsilon
x^{l}&y\\\ y^{*}&\epsilon
x^{m}\end{bmatrix}\in\mathbb{M}_{l+m}(\mathbb{X})^{+}~{}\textrm{for
some}~{}\epsilon>0\right\\}.$

For $l=m,$ write:
$\mathbb{M}_{l,m}(\mathbb{X})_{x^{l},x^{m}}=\mathbb{M}_{l}(\mathbb{X})_{x^{l}}.$
The following statements are true:

  1. (1)

$\mathbb{M}_{l,m}(\mathbb{X}_{x})=\mathbb{M}_{l,m}(\mathbb{X})_{x^{l},x^{m}}$.

  2. (2)

$\mathbb{M}_{l}(\mathbb{X}_{x})^{+}=\mathbb{M}_{l}(\mathbb{X}_{x})\bigcap\mathbb{M}_{l}(\mathbb{X})^{+}$
forms a proper cone.

  3. (3)

$x$ forms an order unit for $\mathbb{X}_{x}.$

  4. (4)

For every $y\in\mathbb{M}_{l}(\mathbb{X}_{x}),$ we write:

$\|y\|_{l}^{x}=\inf\left\\{\epsilon>0:\begin{bmatrix}\epsilon x^{l}&y\\\
y^{*}&\epsilon x^{l}\end{bmatrix}\in\mathbb{M}_{2l}(\mathbb{X})^{+}\right\\}.$

It turns out that $\|\cdot\|_{l}^{x}=\|\cdot\|_{l}$ on
$\mathbb{M}_{l}(\mathbb{X}_{x})$ for every $l\in\mathbb{N}.$

  5. (5)

$|y|_{l,m}\in\mathbb{M}_{m}(\mathbb{X}_{x})^{+}$ for every
$y\in\mathbb{M}_{l,m}(\mathbb{X}_{x}).$

In this case,
$(\mathbb{X}_{x},\\{\mathbb{M}_{m}(\mathbb{X}_{x})^{+}\\},\\{|\cdot|_{l,m}\\},x)$
forms an absolute matrix order unit ideal of $\mathbb{X}.$

###### Proof.

It is routine to verify (2). Now, let’s prove the other statements.

  1. (1)

Let $y=[y_{i,j}]\in\mathbb{M}_{l,m}(\mathbb{X}_{x}).$ There exists
$\epsilon_{i,j}>0$ satisfying $\begin{bmatrix}\epsilon_{i,j}x&y_{i,j}\\\
y_{i,j}^{*}&\epsilon_{i,j}x\end{bmatrix}\in\mathbb{M}_{2}(\mathbb{X})^{+}$ for
every $y_{i,j}\in\mathbb{X}_{p}.$ Put
$\epsilon_{0}=\max\\{\epsilon_{i,j}:1\leq i\leq l,1\leq j\leq m\\}.$ Then
$\epsilon_{0}>0$ and $\begin{bmatrix}\epsilon_{0}x&y_{i,j}\\\
y_{i,j}^{*}&\epsilon_{0}x\end{bmatrix}\in\mathbb{M}_{2}(\mathbb{X})^{+}$ for
every pair of $i$ and $j.$

Next, let $\varsigma_{i,j}\in\mathbb{M}_{l+m,2}$ such that

$\varsigma_{i,j}=\begin{cases}1&\text{at (i,1) and (l+j,2)}\\\
0&\text{elsewhere}.\end{cases}$

so that

$\begin{bmatrix}m\epsilon_{0}x^{l}&y\\\
y^{*}&l\epsilon_{0}x^{m}\end{bmatrix}=\displaystyle\sum_{i,j}\varsigma_{i,j}\begin{bmatrix}\epsilon_{0}x&y_{i,j}\\\
y_{i,j}^{*}&\epsilon_{0}x\end{bmatrix}\varsigma_{ij}^{*}\in\mathbb{M}_{l+m}(\mathbb{X})^{+}.$

For $\epsilon=\max\\{m,l\\}\epsilon_{0},$ we get that

$\begin{bmatrix}\epsilon x^{l}&y\\\ y^{*}&\epsilon
x^{m}\end{bmatrix}\in\mathbb{M}_{l+m}(\mathbb{X})^{+}.$

Thus $y\in\mathbb{M}_{l,m}(\mathbb{X})_{x^{l},x^{m}}.$

Now, we prove the converse part. For that let
$y\in\mathbb{M}_{l,m}(\mathbb{X})_{x^{l},x^{m}}.$ There exists $\epsilon>0$
satisfying $\begin{bmatrix}\epsilon x^{l}&y\\\ y^{*}&\epsilon
x^{m}\end{bmatrix}\in\mathbb{M}_{l+m}(\mathbb{X})^{+}.$ Then

$\begin{bmatrix}\epsilon x&y_{i,j}\\\ y_{i,j}^{*}&\epsilon
x\end{bmatrix}=\varsigma_{i,j}^{*}\begin{bmatrix}\epsilon x^{l}&y\\\
y^{*}&\epsilon
x^{m}\end{bmatrix}\varsigma_{i,j}\in\mathbb{M}_{2}(\mathbb{X})^{+}.$

Thus $y_{i,j}\in\mathbb{X}_{x}$ for every $i,j$ and consequently
$\mathbb{M}_{l,m}(\mathbb{X}_{x})=\mathbb{M}_{l,m}(\mathbb{X})_{x^{l},x^{m}}.$

  2. (3)

By (1), for every $y\in\mathbb{M}_{l}(\mathbb{X}_{x}),$ there exists
$\epsilon>0$ satisfying $\begin{bmatrix}\epsilon x^{l}&y\\\ y^{*}&\epsilon
x^{l}\end{bmatrix}\in\mathbb{M}_{2l}(\mathbb{X})^{+}.$ Thus $x$ forms an order
unit for $\mathbb{X}_{x}.$

  3. (4)

Let $y\in\mathbb{M}_{l}(\mathbb{X}).$ Since
$x\in|\mathcal{M}|_{\infty}(\mathbb{X}),$ we have

$\pm\begin{bmatrix}0&y\\\ y^{*}&0\end{bmatrix}\leq\left\|\begin{bmatrix}0&y\\\
y^{*}&0\end{bmatrix}\right\|_{2l}\begin{bmatrix}x^{l}&0\\\
0&x^{l}\end{bmatrix}=\|y\|_{l}\begin{bmatrix}x^{l}&0\\\ 0&x^{l}\end{bmatrix}.$

Thus $\begin{bmatrix}\|y\|_{l}x^{l}&y\\\
y^{*}&\|y\|_{l}x^{l}\end{bmatrix}\in\mathbb{M}_{2l}(\mathbb{X})^{+}$ so that
$\|y\|_{l}^{x}\leq\|y\|_{l}.$

Now, $x\in\mathbb{X}^{+}$ with $\|x\|=1.$ Then $\begin{bmatrix}x^{l}&0\\\
0&x^{l}\end{bmatrix}\in\mathbb{M}_{2l}(\mathbb{X})^{+}$ and
$\left\|\begin{bmatrix}x^{l}&0\\\ 0&x^{l}\end{bmatrix}\right\|=1$ so that
$\begin{bmatrix}x^{l}&0\\\ 0&x^{l}\end{bmatrix}\leq\begin{bmatrix}e^{l}&0\\\
0&e^{l}\end{bmatrix}.$ As $x$ is order unit for $\mathbb{X}_{x},$ we have

$\pm\begin{bmatrix}0&y\\\ y^{*}&0\end{bmatrix}\leq\left\|\begin{bmatrix}0&y\\\
y^{*}&0\end{bmatrix}\right\|_{2l}^{x}\begin{bmatrix}x^{l}&0\\\
0&x^{l}\end{bmatrix}=\|y\|_{l}^{x}\begin{bmatrix}x^{l}&0\\\
0&x^{l}\end{bmatrix}$

and consequently

$\pm\begin{bmatrix}0&y\\\
y^{*}&0\end{bmatrix}\leq\|y\|_{l}^{x}\begin{bmatrix}e^{l}&0\\\
0&e^{l}\end{bmatrix}.$

Thus $\begin{bmatrix}\|y\|_{l}^{x}e^{l}&y\\\
y^{*}&\|y\|_{l}^{x}e^{l}\end{bmatrix}\in\mathbb{M}_{2l}(\mathbb{X})^{+}$ so
that $\|y\|_{l}\leq\|y\|_{l}^{x}.$ Hence $\|y\|_{l}^{x}=\|y\|_{l}$ for every
$y\in\mathbb{M}_{l}(\mathbb{X}_{x}).$

  4. (5)

Let $y\in\mathbb{M}_{l,m}(\mathbb{X}_{x}).$ For
$x\in|\mathcal{M}|_{\infty}(\mathbb{X})$ and
$\epsilon=\left\|\begin{bmatrix}0&y\\\ y^{*}&0\end{bmatrix}\right\|_{m+n},$ we
have

$\begin{bmatrix}|y^{*}|_{m,l}&0\\\
0&|y|_{l,m}\end{bmatrix}=\left|\begin{bmatrix}0&y\\\
y^{*}&0\end{bmatrix}\right|_{l+m}\leq\epsilon\begin{bmatrix}x^{l}&0\\\
0&x^{m}\end{bmatrix}.$

Then $|y|_{l,m}\leq\epsilon x^{m}$ so that
$|y|_{l,m}\in\mathbb{M}_{m}(\mathbb{X}_{x})^{+}.$ Thus, the maps
$|\cdot|_{l,m}:\mathbb{M}_{l,m}(\mathbb{X}_{x})\to\mathbb{M}_{m}(\mathbb{X}_{x})^{+}$
are well defined.

It is routine to verify $\perp=\perp_{\infty}^{a}$ on
$\mathbb{M}_{l}(\mathbb{X}_{x})^{+}$ for every $l\in\mathbb{N}.$

Hence
$(\mathbb{X}_{x},\\{\mathbb{M}_{l}(\mathbb{X}_{x})^{+}\\},x,\\{|\cdot|_{l,m}\\})$
forms an absolute matrix order unit ideal of $\mathbb{X}.$ ∎

The following result shows that matrix norm in absolute matrix order unit
spaces takes two discrete values $0$ and $1$ whenever it is restricted to the
order projections.

######  Proposition 3.6.

In an absolute matrix order unit space $\mathbb{X},$ let
$p\in\mathbb{OP}_{l}(\mathbb{X})$ such that $p\neq 0.$ Then $\|p\|_{l}=1.$

###### Proof.

Let $p\in\mathbb{OP}_{l}(\mathbb{X}).$ Then $0\leq p\leq e^{l}$ so that
$\|p\|_{l}\leq\max\\{\|0\|_{l},\|e^{l}\|_{l}\\}=\|e^{l}\|_{l}=1.$ Next, by
Proposition 2.9, we also have $0\leq p\leq\|p\|_{l}p.$ Thus
$\|p\|_{l}\leq\max\\{\|0\|_{l},\|\|p\|_{l}p\|_{l}\\}=\|p\|_{l}^{2}$ and
consequently $1\leq\|p\|_{l}$ for $p\neq 0.$ Finally, we conclude that
$\|p\|_{l}=1.$ ∎

We generalize the example 3.4 for order projections in absolute matrix order
unit spaces by this immediate consequence.

######  Corollary 3.7.

Let $\mathbb{X}$ be an absolute matrix order unit space and
$p\in\mathbb{OP}(\mathbb{X})\setminus\\{0\\},$ then $\mathbb{X}_{p}$ forms an
absolute matrix order unit ideal of $\mathbb{X}.$

The notion of absolute and completely absolute value presrving maps have been
introduced and studied in [24]. The following result tells that kernels of
completely absolute value preserving maps are absolute matrix order ideals.

######  Theorem 3.8.

Let $\mathbb{X}$ and $\mathbb{Y}$ be absolutely matrix ordered spaces and
$\varphi:\mathbb{X}\to\mathbb{Y}$ be a completely absolute value preserving
map. Put $Ker(\varphi)=\\{x\in\mathbb{X}:\varphi(x)=0\\}.$ Then the following
statements are true:

  1. (1)

$\mathbb{M}_{l,m}(Ker(\varphi))=Ker(\varphi_{l,m})$ for every pair
$l,m\in\mathbb{N}.$

  2. (2)

$Ker(\varphi_{l})$ is self-adjoint for every $l\in\mathbb{N}.$

  3. (3)

$Ker(\varphi)$ is an absolutely matrix order ideal of $\mathbb{X}.$

  4. (4)

$\varphi=0$ if and only if $Ker(\varphi)=\mathbb{X}$ if and only if
$Ker(\varphi)^{+}=\mathbb{X}^{+}.$

Moreover, if $X$ is absolute matrix order unit space, then

  1. (5)

$\varphi=0$ if and only if $\varphi(e)=0.$

###### Proof.

  1. (1)

Let $[x_{i,j}]\in\mathbb{M}_{l,m}(Ker(\varphi)).$ Then $\varphi(x_{i,j})=0$
for every pair of $i$ and $j$ and consequently
$\varphi_{l,m}([x_{i,j}])=[\varphi(x_{i,j})]=0.$ Thus $[x_{i,j}]\in
Ker(\varphi_{l,m})$ so that $\mathbb{M}_{l,m}(Ker(\varphi))\subset
Ker(\varphi_{l,m}).$ Tracing back the proof, we also conclude that
$Ker(\varphi_{l,m})\subset\mathbb{M}_{l,m}(Ker(\varphi)).$ Hence
$\mathbb{M}_{l,m}(Ker(\varphi))=Ker(\varphi_{l,m}).$

  2. (2)

$\varphi$ is absolute value preserving, we have
$\varphi_{l}(x^{*})=\varphi_{l}(x)^{*}$ for every
$x\in\mathbb{M}_{l}(\mathbb{X})$ and $l\in\mathbb{N}.$ Now, the result
follows.

  3. (3)

Put $\overline{\varphi}_{l}(x)=\varphi_{l}(x)$ for all
$x\in\mathbb{M}_{l}(\mathbb{X})_{sa}.$ Then
$\overline{\varphi}_{l}:\mathbb{M}_{l}(\mathbb{X})_{sa}\to\mathbb{M}_{l}(\mathbb{Y})_{sa}$
defines an absolute value preserving map with
$Ker(\overline{\varphi}_{l})=Ker(\varphi_{l})_{sa}.$ By [24, Theorem 2.7],
$Ker(\overline{\varphi}_{l})$ is an order ideal of
$\mathbb{M}_{l}(\mathbb{X})_{sa}.$ Thus $Ker(\phi)$ forms a matrix order ideal
of $\mathbb{X}.$ Next, let $x=[x_{i,j}]\in\mathbb{M}_{l,m}(Ker(\varphi)).$
Without loss of genearlity, assume that $l\geq m$ so that
$[x,0]\in\mathbb{M}_{l-m}(Ker(\varphi)).$ Then $\varphi_{l}([x,0])=0$ and
consequently by Proposition 2.3(4), we have $\varphi_{m}(|x|_{l,m})\circledast
0_{l-m}=\varphi_{l}(|x|_{l,m}\circledast
0_{l-m})=\varphi_{l}(|[x,0]|_{l})=|\varphi_{l}([x,0])|_{l}=0.$ Thus
$|x|_{l,m}\in\mathbb{M}_{m}(Ker(\varphi))^{+}$ so that
$|\cdot|_{l,m}:\mathbb{M}_{l,m}(Ker(\varphi))\to\mathbb{M}_{m}(\varphi)^{+}$
is well defined for every pair of $l,m\in\mathbb{N}.$ Similarily, the case
$l<m$ can be taken care. Hence the result follows.

  4. (4)

Let $\varphi=0.$ Then $Ker(\varphi)=\mathbb{X}$ so that
$Ker(\varphi)^{+}=\mathbb{X}^{+}.$ Conversely, assume that
$Ker(\varphi)^{+}=\mathbb{X}^{+}.$ Let $x\in\mathbb{X}_{sa}$ and
$x=x^{+}-x^{-}$ be the orthogonal decomposition of $x.$ Since
$\varphi(x^{+})=\varphi(x^{-})=0,$ we get
$\varphi(x)=\varphi(x^{+})-\varphi(x^{-})=0.$ Next, let $x\in\mathbb{X}.$ Put
$x_{1}=\frac{x-x^{*}}{2}$ and $x_{2}=\frac{x-x^{*}}{2i}$ so that
$x_{1},x_{2}\in\mathbb{X}_{sa}$ with $x=x_{1}+ix_{2}.$ Then
$\varphi(x_{1})=\varphi(x_{2})=0$ and consequently
$\varphi(x)=\varphi(x_{1})+i\varphi(x_{2})=0.$ Hence $\varphi=0.$

If $X$ is an absolute matrix order unit space, then

  1. (5)

$\varphi=0$ implies $\varphi(e)=0.$ Conversely, assume that $\varphi(e)=0.$
Then $e\in Ker(\varphi).$ Now, let $x\in\mathbb{X}^{+}$ so that $0\leq
x\leq\|x\|e.$ Since $Ker(\varphi)$ is an order ideal, we get $x\in
Ker(\varphi)~{}i.e.~{}\varphi(x)=0.$ Thus $\phi(x)=0$ for all
$x\in\mathbb{X}^{+}$ so that $Ker(\varphi)^{+}=\mathbb{X}^{+}.$ Hence, by (4),
the result follows.

∎

Finally, we prove that $\frac{\mathbb{X}}{Ker(\varphi)},$ the quotient of
$\mathbb{X}$ by $Ker(\varphi),$ is always an absolutely matrix ordered space.
However, under the assumption of absolute matrix order unit property, it is
identified with $\varphi(\mathbb{X})$ as an absolute matrix order unit ideal
of $\mathbb{Y}.$

######  Theorem 3.9.

Let $\mathbb{X}$ and $\mathbb{Y}$ be absolutely matrix ordered spaces and
$\varphi:\mathbb{X}\to\mathbb{Y}$ be a completely absolute value preserving
map. Put $\mathbb{X}_{0}=\frac{\mathbb{X}}{Ker(\varphi)}.$ Then

  1. (1)

$\mathbb{M}_{l}(\mathbb{X}_{0})=\frac{\mathbb{M}_{l}(\mathbb{X})}{Ker(\varphi_{l})}.$

We write:
$\mathbb{M}_{l}(\mathbb{X}_{0})^{+}=\\{Ker(\varphi_{l})+x:x\in\mathbb{M}_{l}(\mathbb{X})^{+}\\}.$

  1. (2)

$Ker(\varphi_{l})+x\in\mathbb{M}_{l}(\mathbb{X}_{0})^{+}$ if and only if
$\varphi_{l}(x)\geq 0.$

  2. (3)

$|\cdot|_{l,m}^{0}:\mathbb{M}_{l,m}(\mathbb{X}_{0})\to\mathbb{M}_{l}(\mathbb{X}_{0})^{+}$
given by $Ker(\varphi_{l,m})+x\longmapsto Ker(\varphi_{l})+|x|_{l,m}$ are well
defined maps.

  3. (4)

$\left(\mathbb{X}_{0},\\{\mathbb{M}_{l}(X_{0})^{+}\\},\\{|\cdot|^{0}_{l,m}\\}\right)$
is an absolutely matrix ordered space identified with $\varphi(\mathbb{X}).$

Moreover, if $\mathbb{X}~{}\textrm{and}~{}\mathbb{Y}$ are absolute matrix
order unit spaces with $\|\varphi(e)\|=1$ and
$\varphi(e)\in|\mathcal{M}|_{\infty}(\mathbb{Y}),$ then we also have:

  1. (5)

$\mathbb{X}_{0}$ is an absolute matrix order ideal of $\mathbb{Y}$ as
identified with $\mathbb{Y}_{\varphi(e)}=\varphi(\mathbb{X}).$

###### Proof.

  1. (1)

Observe that

$\displaystyle\mathbb{M}_{l}(\mathbb{X}_{0})\ni\begin{bmatrix}Ker(\varphi)+x_{i,j}\end{bmatrix}$
$\displaystyle=$
$\displaystyle\\{\begin{bmatrix}z_{i,j}+x_{i,j}\end{bmatrix}:z_{i,j}\in
Ker(\varphi)\\}$ $\displaystyle=$
$\displaystyle\\{\begin{bmatrix}z_{i,j}\end{bmatrix}:z_{i,j}\in
Ker(\varphi)\\}+\begin{bmatrix}x_{i,j}\end{bmatrix}$ $\displaystyle=$
$\displaystyle
Ker(\varphi_{l})+\begin{bmatrix}x_{i,j}\end{bmatrix}\in\frac{\mathbb{M}_{l}(\mathbb{X})}{Ker(\varphi_{l})}.$

Thus
$\mathbb{M}_{l}(\mathbb{X}_{0})=\frac{\mathbb{M}_{l}(\mathbb{X})}{Ker(\varphi_{l})}$
so that
$\mathbb{M}_{l}(\mathbb{X}_{0})^{+}=\\{Ker(\varphi_{l})+x:x\in\mathbb{M}_{l}(\mathbb{X})^{+}\\}.$

  2. (2)

Let $Ker(\varphi_{l})+x\in\mathbb{M}_{l}(\mathbb{X}_{0})^{+}.$ Then there
exists $z\in\mathbb{M}_{l}(\mathbb{X})^{+}$ such that
$Ker(\varphi_{l})+x=Ker(\varphi_{l})+z.$ Thus $x-z\in Ker(\varphi_{l})$ and we
have $\varphi_{l}(x)=\varphi_{l}(z)\geq 0.$ Conversely, let
$\varphi_{l}(x)\geq 0.$ For $x_{1}=\frac{x-x^{*}}{2}$ and
$x_{2}=\frac{x-x^{*}}{2i}\in\mathbb{M}_{l}(\mathbb{X})_{sa}$ such that
$x_{1}+ix_{2}=x,$ we have $\varphi(x_{1})+i\varphi(x_{2})=\varphi(x)\geq 0.$
In this case, we get $\varphi_{l}(x_{2})=0$ so that
$Ker(\varphi_{l})+x=Ker(\varphi_{l})+x_{1}.$ Therefore, without loss of
generality, we assume that $x\in\mathbb{M}_{l}(\mathbb{X})_{sa}.$ Next, let
$x=x^{+}-x^{-}$ be the orthogonal decomposition of $x.$ By [24, Proposition
2.6], we get that $\varphi_{l}(x)^{+}=\varphi_{l}(x^{+})$ and
$\varphi_{l}(x)^{-}=\varphi_{l}(x^{-}).$ Since $\varphi_{l}(x)\geq 0$ and
orthogonal decomposition is always unique, we conclude that
$\varphi_{l}(x^{-})=0.$ Thus
$Ker(\varphi_{l})+x=Ker(\varphi_{l})+x^{+}\in\mathbb{M}_{l}(\mathbb{X}_{0})^{+}.$

  3. (3)

Let $Ker(\varphi_{l,m})+x=Ker(\varphi_{l,m})+y.$ Then $x-y\in
Ker(\varphi_{l,m})$ so that $\varphi_{l,m}(x)=\varphi_{l,m}(y).$ Without loss
of generality, we assume that $l\geq m.$ Since $\varphi_{l}$ is absolute value
preserving, by Proposition 2.3(4), we get

$\displaystyle\varphi_{m}(|x|_{l,m})\circledast 0$ $\displaystyle=$
$\displaystyle\varphi_{l}(|\begin{bmatrix}x&0\end{bmatrix}|_{l})$
$\displaystyle=$
$\displaystyle|\varphi_{l}(\begin{bmatrix}x&0\end{bmatrix})|_{l}$
$\displaystyle=$
$\displaystyle|\begin{bmatrix}\varphi_{l,m}(x)&0\end{bmatrix}|_{l}$
$\displaystyle=$
$\displaystyle|\begin{bmatrix}\varphi_{l,m}(y)&0\end{bmatrix}|_{l}$
$\displaystyle=$ $\displaystyle\varphi_{m}(|y|_{l,m})\circledast 0.$

and consequently $\varphi_{m}(|x|_{l,m})=\varphi_{m}(|y|_{l,m}).$ Thus
$|x|_{l,m}-|y|_{l,m}\in Ker(\varphi_{m})$ so that
$Ker(\varphi_{m})+|x|_{l,m}=Ker(\varphi_{m})+|y|_{l,m}.$ Hence the maps
$|\cdot|^{0}_{l,m}$ are well-defined.

  4. (4)

Let $x\in\mathbb{M}_{k,l}(\mathbb{X})$ and $y\in\mathbb{M}_{m,n}(\mathbb{X}).$
Then by 2.1(2)(b), we get

$\displaystyle|(Ker(\varphi_{l,m})+x)\circledast(Ker(\varphi_{l,m})+y)|^{0}_{k+m,l+n}$
$\displaystyle=$ $\displaystyle|Ker(\varphi_{k+m,l+n})+(x\circledast
y)|_{k+m,l+n}^{0}$ $\displaystyle=$ $\displaystyle
Ker(\varphi_{l+n})+(|x\circledast y|_{k+m,l+n})$ $\displaystyle=$
$\displaystyle Ker(\varphi_{l+n})+(|x|_{k,l}\circledast|y|_{m,n})$
$\displaystyle=$
$\displaystyle(Ker(\varphi_{l})+|x|_{k,l})\circledast(Ker(\varphi_{n})+|y|_{m,n})$
$\displaystyle=$
$\displaystyle|Ker(\varphi_{k,l})+x|_{k,l}\circledast|Ker(\varphi_{m,n})+y|_{m,n}.$

Next, let $\varsigma_{1}\in\mathbb{M}_{k,m}$ and
$\varsigma_{2}\in\mathbb{M}_{n,l}.$ Again by 2.1(2)(a), we have:

$\displaystyle|\varsigma_{1}(Ker(\varphi_{m,n})+y)\varsigma_{2}|_{k,l}^{0}$
$\displaystyle=$
$\displaystyle|Ker(\varphi_{k,l})+\varsigma_{1}y\varsigma_{2}|_{k,l}^{0}$
$\displaystyle=$ $\displaystyle
Ker(\varphi_{l})+|\varsigma_{1}y\varsigma_{2}|_{k,l}^{0}$ $\displaystyle\leq$
$\displaystyle
Ker(\varphi_{l})+\|\varsigma_{1}\|||y|_{m,n}\varsigma_{2}|_{n,l}^{0}$
$\displaystyle=$
$\displaystyle\|\varsigma_{1}\|||Ker(\varphi_{m,n})+y|_{m,n}^{0}\varsigma_{2}|_{n,l}^{0}.$

Thus $\mathbb{X}_{0}$ is an absolutely matrix ordered space. Now,
$\mathbb{X}_{0}$ is identified with $\varphi(\mathbb{X})$ is immediately
followed by first isomorphism theorem for vector spaces.

Assume that $\mathbb{X}~{}\textrm{and}~{}\mathbb{Y}$ are absolute matrix order
unit spaces with $\|\varphi(e)\|=1$ and
$\varphi(e)\in|\mathcal{M}|_{\infty}(\mathbb{Y}).$ Then

  1. (5)

For $x\in\mathbb{M}_{l}(\mathbb{X})_{sa},$ there exists $\epsilon>0$ such that
$\epsilon e^{l}\pm x\in\mathbb{M}_{l}(\mathbb{X})^{+}.$ Then
$\epsilon(Ker(\varphi)+e)^{l}\pm(Ker(\varphi_{l})+x)=\epsilon(Ker(\varphi_{l})+e^{l})+(Ker(\varphi_{l})\pm
x)=Ker(\varphi_{l})+(\epsilon e^{l}\pm
x)\in\mathbb{M}_{l}(\mathbb{X}_{0})^{+}.$ Thus $Ker(\varphi)+e$ is the order
unit for $\mathbb{X}_{0}.$

Let $Ker(\varphi_{l})\pm x\geq 0.$ By (2), we have
$\pm\varphi_{l}(x)=\varphi_{l}(\pm x)\geq 0.$ Since
$\mathbb{M}_{l}(\mathbb{X})^{+}$ is proper, we get that
$\varphi_{l}(x)=0~{}i.e.~{}Ker(\varphi_{l})+x=0.$ Thus
$\mathbb{M}_{l}(\mathbb{X}_{0})^{+}$ is proper for every $l\in\mathbb{N}.$

Let $x\geq 0$ with $\epsilon(Ker(\varphi_{l})+x)+(Ker(\varphi_{l})+z)\geq 0$
for all $\epsilon>0.$ Again, by (2), we have
$\epsilon\varphi_{l}(x)+\varphi_{l}(z)=\varphi_{l}(\epsilon x+z)\geq 0.$ Since
$\varphi_{l}(x)\geq 0~{}\textrm{and}~{}\mathbb{M}_{l}(\mathbb{X})^{+}$ is
Archimedean, we get that $\varphi_{l}(z)\geq 0.$ Then $Ker(\varphi_{l})+z\geq
0.$ Therefore $\mathbb{M}_{l}(\mathbb{X}_{0})^{+}$ is Archimedean for every
$l\in\mathbb{N}.$

As $\mathbb{M}_{l}(\mathbb{X})_{0}^{+}$ is proper and Archimedean for all
$l\in\mathbb{N}.$ Therefore order unit $Ker(\varphi)+e$ determines a matrix
norm $\\{\|\cdot\|_{l}^{0}\\}$ on $\mathbb{X}_{0}$ defined in the following
way:

$\displaystyle\|Ker(\varphi_{l})+x\|_{l}^{0}$ $\displaystyle=$
$\displaystyle\inf\left\\{\epsilon>0:Ker(\varphi_{2l})+\left(\begin{bmatrix}\epsilon
e^{l}&\pm x\\\ \pm x^{*}&\epsilon e^{l}\end{bmatrix}\right)\geq 0\right\\}$
$\displaystyle=$
$\displaystyle\inf\left\\{\epsilon>0:\begin{bmatrix}\epsilon\varphi(e)^{l}&\pm\varphi_{l}(x)\\\
\pm\varphi_{l}(x)^{*}&\epsilon\varphi(e)^{l}\end{bmatrix}\geq 0\right\\}$

for every $l\in\mathbb{N}.$

Finally, assume that $\varphi(e)\in|\mathcal{M}|_{\infty}(Y)$ such that
$\|\varphi(e)\|=1.$ In this case, we show that
$\mathbb{X}_{0}\cong\mathbb{Y}_{\varphi(x)}=\varphi(\mathbb{X}).$ For that let
$Ker(\varphi_{l})+x\geq 0.$ Then by Theorem 3.5, we get that

$\displaystyle\|Ker(\varphi_{l})+x\|_{l}^{0}$ $\displaystyle=$
$\displaystyle\inf\\{\epsilon>0:\epsilon(Ker(\varphi)+e)\pm(Ker(\varphi)+x)\geq
0\\}$ $\displaystyle=$ $\displaystyle\inf\\{\epsilon>0:Ker(\varphi)+\epsilon
e\pm x\geq 0\\}$ $\displaystyle=$
$\displaystyle\inf\\{\epsilon>0:\epsilon\varphi(e)\pm\varphi(x)\geq 0\\}$
$\displaystyle=$ $\displaystyle\|\varphi(x)\|_{l}^{\varphi(e)}$
$\displaystyle=$ $\displaystyle\|\varphi(x)\|_{l}.$

Hence the result follows. ∎

##  4\. Relation between $\mathcal{K}_{0}(\mathbb{X}_{x})$ and
$\mathcal{K}_{0}(\mathbb{X})$

In this section, we derive a relation between
$\mathcal{K}_{0}(\mathbb{X}_{x})$ and $\mathcal{K}_{0}(\mathbb{X}).$ For that,
we start with the following characterization of order projections in
$\mathbb{X}_{x}$ in terms of order projections in $\mathbb{X}.$

######  Lemma 4.1.

In an absolute matrix order unit space $\mathbb{X},$ let $x\in\mathbb{X}^{+}$
such that $\|x\|=1.$ Then
$\mathbb{OP}_{l}(\mathbb{X}_{x})=\\{p\in\mathbb{OP}_{l}(\mathbb{X}):p\leq
x^{l}\\}=\mathbb{OP}_{l}(\mathbb{X})\cap\mathbb{M}_{l}(\mathbb{X}_{x})$ for
every $l\in\mathbb{N}.$

###### Proof.

Let $p\in\mathbb{OP}_{l}(\mathbb{X})\cap\mathbb{M}_{l}(\mathbb{X}_{x}).$ By
proposition 3.6, we get that $0\leq p\leq x^{l}.$ For $\|x\|=1,$ we also get
that $0\leq x^{l}\leq e^{l}.$ Then $0\leq x^{l}-p\leq e^{l}-p.$ Since
$p\in\mathbb{OP}_{l}(\mathbb{X}),$ we have $e^{l}\perp e^{l}-p.$ By [23,
Definition 3.4(4)], we conclude $p\perp x^{l}-p.$ Thus
$p\in\mathbb{OP}_{l}(\mathbb{X}_{x}).$ Hence the reslut follows. ∎

######  Corollary 4.2.

In an absolute matrix order unit space $\mathbb{X},$ we have:
$\mathbb{OP}_{l}(\mathbb{X}_{p})=\\{q\in\mathbb{OP}_{l}(\mathbb{X}):q\leq
p^{l}\\}=\mathbb{OP}_{l}(\mathbb{X})\cap\mathbb{M}_{l}(\mathbb{X}_{p})$ for
every $p\in\mathbb{OP}(\mathbb{X})\setminus\\{0\\}$ and $l\in\mathbb{N}.$

The partial isometric equivalence (denoted by $\sim$) on order projections is
defined in [25]. Now, we characterize $\sim$ in $\mathbb{X}_{x}$ in terms of
$\sim$ in $\mathbb{X}.$

######  Proposition 4.3.

In an absolute matrix order unit space $\mathbb{X},$ let
$p,q\in\mathbb{OP}_{\infty}(\mathbb{X}_{x})$ for some $x\in\mathbb{X}^{+}$
such that $\|x\|=1.$ Then $p\sim q$ in $\mathbb{X}_{x}$ if and only if $p\sim
q$ in $\mathbb{X}.$

###### Proof.

Suppose that $p\sim q$ in $\mathbb{X}$ for some
$p\in\mathbb{OP}_{l}(\mathbb{X}_{x})$ and
$q\in\mathbb{OP}_{m}(\mathbb{X}_{x}).$ There exists
$y\in\mathbb{M}_{l,m}(\mathbb{X})$ satisfying $p=|y^{*}|_{m,l}$ and
$q=|y|_{l,m}.$ For $\|p\|\leq 1$ and $\|q\|\leq 1,$ we have $p\leq x^{l}$ and
$q\leq x^{m}.$ Then

$\displaystyle\begin{bmatrix}x^{l}&0\\\ 0&x^{m}\end{bmatrix}$
$\displaystyle\geq$ $\displaystyle\begin{bmatrix}p&0\\\ 0&q\end{bmatrix}$
$\displaystyle=$ $\displaystyle\begin{bmatrix}|y^{*}|_{m,l}&0\\\
0&|y|_{l,m}\end{bmatrix}$ $\displaystyle=$
$\displaystyle\left|\begin{bmatrix}0&y\\\ y^{*}&0\end{bmatrix}\right|_{l+m}$
$\displaystyle\geq$ $\displaystyle\pm\begin{bmatrix}0&y\\\
y^{*}&0\end{bmatrix}$

so that $\begin{bmatrix}x^{l}&y\\\
y^{*}&x^{m}\end{bmatrix}\in\mathbb{M}_{l+m}(\mathbb{X})^{+}.$ Thus
$y\in\mathbb{M}_{l,m}(\mathbb{X}_{x})$ and consequently $p\sim q$ in
$\mathbb{X}_{x}.$

Next, $p\sim q$ in $\mathbb{X}_{x}$ implies $p\sim q$ in $\mathbb{X}$ follows
trivially. ∎

The corresponding characterization for partial isometrises of $\mathbb{X}_{x}$
is also given in terms of partial isometrises of $\mathbb{X}$ by the following
result.

######  Corollary 4.4.

In an absolute matrix order unit space $\mathbb{X},$ let $x\in\mathbb{X}^{+}$
such that $\|x\|=1.$ If $y\in\mathbb{PI}_{l,m}(\mathbb{X})$ such that
$|y|_{l,m}\in\mathbb{OP}_{m}(\mathbb{X}_{x})$ and
$|y^{*}|_{m,l}\in\mathbb{OP}_{l}(\mathbb{X}_{x}),$ then
$x\in\mathbb{PI}_{l,m}(\mathbb{X}_{x}).$ In this case,
$\mathbb{PI}_{l,m}(\mathbb{X}_{x})=\mathbb{M}_{l,m}(\mathbb{X}_{x})\cap\mathbb{PI}_{l,m}(\mathbb{X}).$
In particular,
$\mathbb{PI}_{l,m}(\mathbb{X}_{p})=\mathbb{M}_{l,m}(\mathbb{X}_{p})\cap\mathbb{PI}_{l,m}(\mathbb{X})$
for every $p\in\mathbb{OP}(\mathbb{X})\setminus\\{0\\}.$

###### Proof.

Let $y\in\mathbb{PI}_{l,m}(\mathbb{X})$ such that
$|y|_{l,m}\in\mathbb{OP}_{m}(\mathbb{X}_{x})$ and
$|y^{*}|_{m,l}\in\mathbb{OP}_{l}(\mathbb{X}_{x}).$ Put $p=|y^{*}|_{m,l}$ and
$q=|y|_{l,m}.$ Then $p\sim q$ in $\mathbb{X}_{x}.$ Now the result follows
immediately from the Proposition 4.3. ∎

The condition $(T)$ is also defined in [25]. Under $(T),$ the relation $\sim$
becomes an equivalence relation on order projections. In the next result, it
is shown that $(T)$ is tranfered on $\mathbb{X}_{x}$ as a heredity.

######  Proposition 4.5.

Let $\mathbb{X}$ an absolute matrix order unit space and let
$x\in\mathbb{X}^{+}$ such that $\|x\|=1.$ Then (T) holds in $\mathbb{X}$
implies (T) holds in $\mathbb{X}_{x}.$ In particular, (T) holds in
$\mathbb{X}$ implies (T) holds in $\mathbb{X}_{p}$ for every
$p\in\mathbb{OP}(\mathbb{X})\setminus\\{0\\}.$

###### Proof.

Assume that (T) holds in $\mathbb{X}.$ Let
$y\in\mathbb{PI}_{m,n}(\mathbb{X}_{x})$ and
$z\in\mathbb{PI}_{l,n}(\mathbb{X}_{x})$ satisfying $|y|_{m,n}=|z|_{l,n}.$ Then
there exists $w\in\mathbb{PI}_{m,l}(\mathbb{X})$ such that
$|w^{*}|_{l,m}=|y^{*}|_{n,m}$ and $|w|_{m,l}=|z^{*}|_{n,l}.$ Note that
$|w|_{m,l}\in\mathbb{OP}_{l}(\mathbb{X}_{x})$ and
$|w^{*}|_{l,m}\in\mathbb{OP}_{m}(\mathbb{X}_{x}).$ By Corollary 4.4, we get
$w\in\mathbb{PI}_{m,l}(\mathbb{X}_{x}).$ Thus (T) also holds in
$\mathbb{X}_{x}.$ ∎

Finally, we derive a relation between $\mathcal{K}_{0}(\mathbb{X}_{x})$ and
$\mathcal{K}_{0}(\mathbb{X})$ by the following result.

######  Theorem 4.6.

Let $(T)$ holds in an absolute matrix order unit space $\mathbb{X}$ and
$x\in\mathbb{X}^{+}$ such that $\|x\|=1.$ Then $[(p,q)]\longmapsto[(p,q)]$
defines a group homomorphism $\mathcal{K}_{0}(\mathbb{X}_{x})$ to
$\mathcal{K}_{0}(\mathbb{X}).$ Moreover, if $p\approx q$ in
$\mathbb{OP}_{\infty}(\mathbb{X})$ implies $p\approx q$ in
$\mathbb{OP}_{\infty}(\mathbb{X}_{x})$ for every pair of
$p,q\in\mathbb{OP}_{\infty}(\mathbb{X}_{x}),$ then this group homomorphism
turns out to be an injection.

###### Proof.

Consider the inclusion map $i:\mathbb{X}_{x}\hookrightarrow\mathbb{X}.$ By
[26, Theorem 5.2], there exists a unique group homomorphism
$\mathcal{K}_{0}(i):\mathcal{K}_{0}(\mathbb{X}_{x})\to\mathcal{K}_{0}(\mathbb{X})$
satisfying the following commutative diagram:

${\mathbb{OP}_{\infty}(\mathbb{X}_{x})}$${\mathbb{OP}_{\infty}(\mathbb{X})}$${\mathcal{K}_{0}(\mathbb{X}_{x})}$${\mathcal{K}_{0}(\mathbb{X})}$$\scriptstyle{i}$$\scriptstyle{\chi_{\mathbb{X}_{x}}}$$\scriptstyle{\chi_{\mathbb{X}}}$$\scriptstyle{\mathcal{K}_{0}(i)}$

so that

$\mathcal{K}_{0}(i)(\chi_{\mathbb{X}_{x}}(p))=\chi_{\mathbb{X}}(p)~{}\textrm{for
every}~{}p\in\mathbb{OP}_{\infty}(\mathbb{X}_{x}).$

Now, for $[(p,q)]\in\mathcal{K}_{0}(\mathbb{X}_{x}),$ we have:

$\displaystyle\mathcal{K}_{0}(i)([(p,q)])$ $\displaystyle=$
$\displaystyle\mathcal{K}_{0}(i)([(p,0)])-[(0,q)])$ $\displaystyle=$
$\displaystyle\mathcal{K}_{0}(i)(\chi_{\mathbb{X}_{x}}(p)-\chi_{\mathbb{X}_{x}}(q))$
$\displaystyle=$
$\displaystyle\mathcal{K}_{0}(i)(\chi_{\mathbb{X}_{x}}(p))-\mathcal{K}_{0}(i)(\chi_{\mathbb{X}_{x}}(q))$
$\displaystyle=$ $\displaystyle\chi_{\mathbb{X}}(p)-\chi_{\mathbb{X}}(q)$
$\displaystyle=$ $\displaystyle[(p,q)].$

Thus $[(p,q)]\longmapsto[(p,q)]$ defines a group homomorphism from
$\mathcal{K}_{0}(\mathbb{X}_{x})$ to $\mathcal{K}_{0}(\mathbb{X})$.

Next, let $p,q\in\mathbb{OP}_{\infty}(\mathbb{X}_{x})$ such that $[(p,q)]=0$
in $\mathbb{X}.$ In this case, we have $p\oplus 0\approx 0\oplus q$ in
$\mathbb{X}.$ By assumption, we conclude that $p\oplus 0\approx 0\oplus q$ in
$\mathbb{X}_{x}.$ Thus $[(p,q)]=0$ in $\mathbb{X}_{x}.$ Hence the group
homomorphism is injective. ∎

######  Corollary 4.7.

Let $(T)$ holds in an absolute matrix order unit space $\mathbb{X}$ and
$p\in\mathbb{OP}(\mathbb{X})\setminus\\{0\\}.$ Then
$[(q,r)]\longmapsto[(q,r)]$ defines a group homomorphism
$\mathcal{K}_{0}(\mathbb{X}_{p})$ to $\mathcal{K}_{0}(\mathbb{X}).$ Moreover,
if $q\approx r$ in $\mathbb{X}$ implies $q\approx r$ in $\mathbb{X}_{p}$ for
every pair $q,r\in\mathbb{OP}_{\infty}(\mathbb{X}_{p}),$ then this group
homomorphism turns out to be an injection.

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Preface This textbook is intended for use by students of physics, physical
chemistry, and theoretical chemistry. The reader is presumed to have a basic
knowledge of atomic and quantum physics at the level provided, for example, by
the first few chapters in our book The Physics of Atoms and Quanta. The
student of physics will find here material which should be included in the
basic education of every physicist. This book should furthermore allow
students to acquire an appreciation of the breadth and variety within the
field of molecular physics and its future as a fascinating area of research.

For the student of chemistry, the concepts introduced in this book will
provide a theoretical framework for that entire field of study. With the help
of these concepts, it is at least in principle possible to reduce the enormous
body of empirical chemical knowledge to a few basic principles: those of
quantum mechanics. In addition, modern physical methods whose fundamentals are
introduced here are becoming increasingly important in chemistry and now
represent indispensable tools for the chemist. As examples, we might mention
the structural analysis of complex organic compounds, spectroscopic
investigation of very rapid reaction processes or, as a practical application,
the remote detection of pollutants in the air.

April 1995 Walter Olthoff
Program Chair
ECOOP’95

## Organization

ECOOP’95 is organized by the department of Computer Science, Univeristy of
Århus and AITO (association Internationa pour les Technologie Object) in
cooperation with ACM/SIGPLAN.

### Executive Commitee

Conference Chair: |  Ole Lehrmann Madsen (Århus University, DK)
---|---
Program Chair: |  Walter Olthoff (DFKI GmbH, Germany)
Organizing Chair: |  Jørgen Lindskov Knudsen (Århus University, DK)
Tutorials: |  Birger Møller-Pedersen (Norwegian Computing Center, Norway)
Workshops: |  Eric Jul (University of Kopenhagen, Denmark)
Panels: |  Boris Magnusson (Lund University, Sweden)
Exhibition: |  Elmer Sandvad (Århus University, DK)
Demonstrations: |  Kurt Nørdmark (Århus University, DK)

### Program Commitee

Conference Chair: |  Ole Lehrmann Madsen (Århus University, DK)
---|---
Program Chair: |  Walter Olthoff (DFKI GmbH, Germany)
Organizing Chair: |  Jørgen Lindskov Knudsen (Århus University, DK)
Tutorials: |  Birger Møller-Pedersen (Norwegian Computing Center, Norway)
Workshops: |  Eric Jul (University of Kopenhagen, Denmark)
Panels: |  Boris Magnusson (Lund University, Sweden)
Exhibition: |  Elmer Sandvad (Århus University, DK)
Demonstrations: |  Kurt Nørdmark (Århus University, DK)

### Referees

V. Andreev
Bärwolff
E. Barrelet
H.P. Beck
G. Bernardi
E. Binder
P.C. Bosetti
Braunschweig
F.W. Büsser
T. Carli
A.B. Clegg
G. Cozzika
S. Dagoret
Del Buono
P. Dingus
H. Duhm
J. Ebert
S. Eichenberger
R.J. Ellison
Feltesse
W. Flauger
A. Fomenko
G. Franke
J. Garvey
M. Gennis
L. Goerlich
P. Goritchev
H. Greif
E.M. Hanlon
R. Haydar
R.C.W. Henderso
P. Hill
H. Hufnagel
A. Jacholkowska
Johannsen
S. Kasarian
I.R. Kenyon
C. Kleinwort
T. Köhler
S.D. Kolya
P. Kostka
U. Krüger
J. Kurzhöfer
M.P.J. Landon
A. Lebedev
Ch. Ley
F. Linsel
H. Lohmand
Martin
S. Masson
K. Meier
C.A. Meyer
S. Mikocki
J.V. Morris
B. Naroska
Nguyen
U. Obrock
G.D. Patel
Ch. Pichler
S. Prell
F. Raupach
V. Riech
P. Robmann
N. Sahlmann
P. Schleper
Schöning
B. Schwab
A. Semenov
G. Siegmon
J.R. Smith
M. Steenbock
U. Straumann
C. Thiebaux
P. Van Esch
from Yerevan Ph
L.R. West
G.-G. Winter
T.P. Yiou
M. Zimmer

### Sponsoring Institutions

Ivar Ekeland 1Princeton University, Princeton NJ 08544, USA 1 Roger Temam
Jeffrey Dean 2Université de Paris-Sud, Laboratoire d’Analyse Numérique,
Bâtiment 425,
F-91405 Orsay Cedex, France2 David Grove  Craig Chambers  Kim B. Bruce  Elsa
Bertino 1Princeton University, Princeton NJ 08544, USA 12Université de Paris-
Sud, Laboratoire d’Analyse Numérique, Bâtiment 425,
F-91405 Orsay Cedex, France2

Bernauer-Budiman Inc., Reading, Mass.
The Hofmann-International Company, San Louis Obispo, Cal.
Kramer Industries, Heidelberg, Germany

###### Contents

  1. 1 Fixed-Period Problems: The Sublinear Case
    1. 1.1 Autonomous Systems
      1. 1.1.1 The General Case: Nontriviality.
  2. 2 Fixed-Period Problems: The Sublinear Case
    1. 2.1 Autonomous Systems
      1. 2.1.1 The General Case: Nontriviality.

11institutetext: Princeton University, Princeton NJ 08544, USA,
11email<EMAIL_ADDRESS>
WWW home page: http://users/ iekeland/web/welcome.html 22institutetext:
Université de Paris-Sud, Laboratoire d’Analyse Numérique, Bâtiment 425,
F-91405 Orsay Cedex, France

### Hamiltonian Mechanics unter besonderer Berücksichtigung der höhreren
Lehranstalten

Ivar Ekeland 1Princeton University, Princeton NJ 08544, USA 1 Roger Temam
Jeffrey Dean 2Université de Paris-Sud, Laboratoire d’Analyse Numérique,
Bâtiment 425,
F-91405 Orsay Cedex, France2 David Grove  Craig Chambers  Kim B. Bruce  Elsa
Bertino 1Princeton University, Princeton NJ 08544, USA 12Université de Paris-
Sud, Laboratoire d’Analyse Numérique, Bâtiment 425,
F-91405 Orsay Cedex, France2

###### Abstract

The abstract should summarize the contents of the paper using at least 70 and
at most 150 words. It will be set in 9-point font size and be inset 1.0 cm
from the right and left margins. There will be two blank lines before and
after the Abstract. …

###### Abstract

The abstract should summarize the contents of the paper using at least 70 and
at most 150 words. It will be set in 9-point font size and be inset 1.0 cm
from the right and left margins. There will be two blank lines before and
after the Abstract. …

###### Keywords:

computational geometry, graph theory, Hamilton cycles

###### Keywords:

graph transformations, convex geometry, lattice computations, convex polygons,
triangulations, discrete geometry

###  1 Fixed-Period Problems: The Sublinear Case

With this chapter, the preliminaries are over, and we begin the search for
periodic solutions to Hamiltonian systems. All this will be done in the convex
case; that is, we shall study the boundary-value problem

$\displaystyle\dot{x}$ $\displaystyle=$ $\displaystyle JH^{\prime}(t,x)$
$\displaystyle x(0)$ $\displaystyle=$ $\displaystyle x(T)$

with $H(t,\cdot)$ a convex function of $x$, going to $+\infty$ when
$\left\|x\right\|\to\infty$.

####  1.1 Autonomous Systems

In this section, we will consider the case when the Hamiltonian $H(x)$ is
autonomous. For the sake of simplicity, we shall also assume that it is
$C^{1}$.

We shall first consider the question of nontriviality, within the general
framework of $\left(A_{\infty},B_{\infty}\right)$-subquadratic Hamiltonians.
In the second subsection, we shall look into the special case when $H$ is
$\left(0,b_{\infty}\right)$-subquadratic, and we shall try to derive
additional information.

#####  1.1.1 The General Case: Nontriviality.

We assume that $H$ is $\left(A_{\infty},B_{\infty}\right)$-subquadratic at
infinity, for some constant symmetric matrices $A_{\infty}$ and $B_{\infty}$,
with $B_{\infty}-A_{\infty}$ positive definite. Set:

$\displaystyle\gamma:$ $\displaystyle=$ $\displaystyle{\rm smallest\
eigenvalue\ of}\ \ B_{\infty}-A_{\infty}$ (1) $\displaystyle\lambda:$
$\displaystyle=$ $\displaystyle{\rm largest\ negative\ eigenvalue\ of}\ \
J\frac{d}{dt}+A_{\infty}\ .$ (2)

Theorem 1.1 tells us that if $\lambda+\gamma<0$, the boundary-value problem:

$\begin{array}[]{rcl}\dot{x}&=&JH^{\prime}(x)\\\ x(0)&=&x(T)\end{array}$ (3)

has at least one solution $\overline{x}$, which is found by minimizing the
dual action functional:

$\psi(u)=\int_{o}^{T}\left[\frac{1}{2}\left(\Lambda_{o}^{-1}u,u\right)+N^{\ast}(-u)\right]dt$
(4)

on the range of $\Lambda$, which is a subspace $R(\Lambda)_{L}^{2}$ with
finite codimension. Here

$N(x):=H(x)-\frac{1}{2}\left(A_{\infty}x,x\right)$ (5)

is a convex function, and

$N(x)\leq\frac{1}{2}\left(\left(B_{\infty}-A_{\infty}\right)x,x\right)+c\ \ \
\forall x\ .$ (6)

###### Proposition 1

Assume $H^{\prime}(0)=0$ and $H(0)=0$. Set:

$\delta:=\liminf_{x\to 0}2N(x)\left\|x\right\|^{-2}\ .$ (7)

If $\gamma<-\lambda<\delta$, the solution $\overline{u}$ is non-zero:

$\overline{x}(t)\neq 0\ \ \ \forall t\ .$ (8)

###### Proof

Condition (7) means that, for every $\delta^{\prime}>\delta$, there is some
$\varepsilon>0$ such that

$\left\|x\right\|\leq\varepsilon\Rightarrow
N(x)\leq\frac{\delta^{\prime}}{2}\left\|x\right\|^{2}\ .$ (9)

It is an exercise in convex analysis, into which we shall not go, to show that
this implies that there is an $\eta>0$ such that

$f\left\|x\right\|\leq\eta\Rightarrow
N^{\ast}(y)\leq\frac{1}{2\delta^{\prime}}\left\|y\right\|^{2}\ .$ (10)

Figure 1: This is the caption of the figure displaying a white eagle and a
white horse on a snow field

Since $u_{1}$ is a smooth function, we will have
$\left\|hu_{1}\right\|_{\infty}\leq\eta$ for $h$ small enough, and inequality
(10) will hold, yielding thereby:

$\psi(hu_{1})\leq\frac{h^{2}}{2}\frac{1}{\lambda}\left\|u_{1}\right\|_{2}^{2}+\frac{h^{2}}{2}\frac{1}{\delta^{\prime}}\left\|u_{1}\right\|^{2}\
.$ (11)

If we choose $\delta^{\prime}$ close enough to $\delta$, the quantity
$\left(\frac{1}{\lambda}+\frac{1}{\delta^{\prime}}\right)$ will be negative,
and we end up with

$\psi(hu_{1})<0\ \ \ \ \ {\rm for}\ \ h\neq 0\ \ {\rm small}\ .$ (12)

On the other hand, we check directly that $\psi(0)=0$. This shows that 0
cannot be a minimizer of $\psi$, not even a local one. So $\overline{u}\neq 0$
and $\overline{u}\neq\Lambda_{o}^{-1}(0)=0$. ∎

###### Corollary 1

Assume $H$ is $C^{2}$ and $\left(a_{\infty},b_{\infty}\right)$-subquadratic at
infinity. Let $\xi_{1},\allowbreak\dots,\allowbreak\xi_{N}$ be the equilibria,
that is, the solutions of $H^{\prime}(\xi)=0$. Denote by $\omega_{k}$ the
smallest eigenvalue of $H^{\prime\prime}\left(\xi_{k}\right)$, and set:

$\omega:={\rm Min\,}\left\\{\omega_{1},\dots,\omega_{k}\right\\}\ .$ (13)

If:

$\frac{T}{2\pi}b_{\infty}<-E\left[-\frac{T}{2\pi}a_{\infty}\right]<\frac{T}{2\pi}\omega$
(14)

then minimization of $\psi$ yields a non-constant $T$-periodic solution
$\overline{x}$.

We recall once more that by the integer part $E[\alpha]$ of $\alpha\in\bbbr$,
we mean the $a\in\bbbz$ such that $a<\alpha\leq a+1$. For instance, if we take
$a_{\infty}=0$, Corollary 2 tells us that $\overline{x}$ exists and is non-
constant provided that:

$\frac{T}{2\pi}b_{\infty}<1<\frac{T}{2\pi}$ (15)

or

$T\in\left(\frac{2\pi}{\omega},\frac{2\pi}{b_{\infty}}\right)\ .$ (16)

###### Proof

The spectrum of $\Lambda$ is $\frac{2\pi}{T}\bbbz+a_{\infty}$. The largest
negative eigenvalue $\lambda$ is given by $\frac{2\pi}{T}k_{o}+a_{\infty}$,
where

$\frac{2\pi}{T}k_{o}+a_{\infty}<0\leq\frac{2\pi}{T}(k_{o}+1)+a_{\infty}\ .$
(17)

Hence:

$k_{o}=E\left[-\frac{T}{2\pi}a_{\infty}\right]\ .$ (18)

The condition $\gamma<-\lambda<\delta$ now becomes:

$b_{\infty}-a_{\infty}<-\frac{2\pi}{T}k_{o}-a_{\infty}<\omega-a_{\infty}$ (19)

which is precisely condition (14).∎

###### Lemma 1

Assume that $H$ is $C^{2}$ on $\bbbr^{2n}\setminus\\{0\\}$ and that
$H^{\prime\prime}(x)$ is non-degenerate for any $x\neq 0$. Then any local
minimizer $\widetilde{x}$ of $\psi$ has minimal period $T$.

###### Proof

We know that $\widetilde{x}$, or $\widetilde{x}+\xi$ for some constant
$\xi\in\bbbr^{2n}$, is a $T$-periodic solution of the Hamiltonian system:

$\dot{x}=JH^{\prime}(x)\ .$ (20)

There is no loss of generality in taking $\xi=0$. So
$\psi(x)\geq\psi(\widetilde{x})$ for all $\widetilde{x}$ in some neighbourhood
of $x$ in $W^{1,2}\left(\bbbr/T\bbbz;\bbbr^{2n}\right)$.

But this index is precisely the index $i_{T}(\widetilde{x})$ of the
$T$-periodic solution $\widetilde{x}$ over the interval $(0,T)$, as defined in
Sect. 2.6. So

$i_{T}(\widetilde{x})=0\ .$ (21)

Now if $\widetilde{x}$ has a lower period, $T/k$ say, we would have, by
Corollary 31:

$i_{T}(\widetilde{x})=i_{kT/k}(\widetilde{x})\geq
ki_{T/k}(\widetilde{x})+k-1\geq k-1\geq 1\ .$ (22)

This would contradict (21), and thus cannot happen.∎

###### Notes and Comments.

The results in this section are a refined version of [1]; the minimality
result of Proposition 14 was the first of its kind.

To understand the nontriviality conditions, such as the one in formula (16),
one may think of a one-parameter family $x_{T}$,
$T\in\left(2\pi\omega^{-1},2\pi b_{\infty}^{-1}\right)$ of periodic solutions,
$x_{T}(0)=x_{T}(T)$, with $x_{T}$ going away to infinity when $T\to
2\pi\omega^{-1}$, which is the period of the linearized system at 0.

Table 1: This is the example table taken out of The TeXbook, p. 246 Year | World population
---|---
8000 B.C. | 5,000,000 |
50 A.D. | 200,000,000 |
1650 A.D. | 500,000,000 |
1945 A.D. | 2,300,000,000 |
1980 A.D. | 4,400,000,000 |

######  Theorem 1.1 (Ghoussoub-Preiss)

Assume $H(t,x)$ is $(0,\varepsilon)$-subquadratic at infinity for all
$\varepsilon>0$, and $T$-periodic in $t$

$H(t,\cdot)\ \ \ \ \ {\rm is\ convex}\ \ \forall t$ (23) $H(\cdot,x)\ \ \ \ \
{\rm is}\ \ T{\rm-periodic}\ \ \forall x$ (24) $H(t,x)\geq
n\left(\left\|x\right\|\right)\ \ \ \ \ {\rm with}\ \ n(s)s^{-1}\to\infty\ \
{\rm as}\ \ s\to\infty$ (25) $\forall\varepsilon>0\ ,\ \ \ \exists c\ :\
H(t,x)\leq\frac{\varepsilon}{2}\left\|x\right\|^{2}+c\ .$ (26)

Assume also that $H$ is $C^{2}$, and $H^{\prime\prime}(t,x)$ is positive
definite everywhere. Then there is a sequence $x_{k}$, $k\in\bbbn$, of
$kT$-periodic solutions of the system

$\dot{x}=JH^{\prime}(t,x)$ (27)

such that, for every $k\in\bbbn$, there is some $p_{o}\in\bbbn$ with:

$p\geq p_{o}\Rightarrow x_{pk}\neq x_{k}\ .$ (28)

∎

######  Example 1 (External forcing)

Consider the system:

$\dot{x}=JH^{\prime}(x)+f(t)$ (29)

where the Hamiltonian $H$ is $\left(0,b_{\infty}\right)$-subquadratic, and the
forcing term is a distribution on the circle:

$f=\frac{d}{dt}F+f_{o}\ \ \ \ \ {\rm with}\ \ F\in
L^{2}\left(\bbbr/T\bbbz;\bbbr^{2n}\right)\ ,$ (30)

where $f_{o}:=T^{-1}\int_{o}^{T}f(t)dt$. For instance,

$f(t)=\sum_{k\in\bbbn}\delta_{k}\xi\ ,$ (31)

where $\delta_{k}$ is the Dirac mass at $t=k$ and $\xi\in\bbbr^{2n}$ is a
constant, fits the prescription. This means that the system
$\dot{x}=JH^{\prime}(x)$ is being excited by a series of identical shocks at
interval $T$.

###### Definition 1

Let $A_{\infty}(t)$ and $B_{\infty}(t)$ be symmetric operators in
$\bbbr^{2n}$, depending continuously on $t\in[0,T]$, such that
$A_{\infty}(t)\leq B_{\infty}(t)$ for all $t$.

A Borelian function $H:[0,T]\times\bbbr^{2n}\to\bbbr$ is called
$\left(A_{\infty},B_{\infty}\right)$-subquadratic at infinity if there exists
a function $N(t,x)$ such that:

$H(t,x)=\frac{1}{2}\left(A_{\infty}(t)x,x\right)+N(t,x)$ (32) $\forall t\ ,\ \
\ N(t,x)\ \ \ \ \ {\rm is\ convex\ with\ respect\ to}\ \ x$ (33) $N(t,x)\geq
n\left(\left\|x\right\|\right)\ \ \ \ \ {\rm with}\ \ n(s)s^{-1}\to+\infty\ \
{\rm as}\ \ s\to+\infty$ (34) $\exists c\in\bbbr\ :\ \ \
H(t,x)\leq\frac{1}{2}\left(B_{\infty}(t)x,x\right)+c\ \ \ \forall x\ .$ (35)

If $A_{\infty}(t)=a_{\infty}I$ and $B_{\infty}(t)=b_{\infty}I$, with
$a_{\infty}\leq b_{\infty}\in\bbbr$, we shall say that $H$ is
$\left(a_{\infty},b_{\infty}\right)$-subquadratic at infinity. As an example,
the function $\left\|x\right\|^{\alpha}$, with $1\leq\alpha<2$, is
$(0,\varepsilon)$-subquadratic at infinity for every $\varepsilon>0$.
Similarly, the Hamiltonian

$H(t,x)=\frac{1}{2}k\left\|k\right\|^{2}+\left\|x\right\|^{\alpha}$ (36)

is $(k,k+\varepsilon)$-subquadratic for every $\varepsilon>0$. Note that, if
$k<0$, it is not convex.

###### Notes and Comments.

The first results on subharmonics were obtained by Rabinowitz in [5], who
showed the existence of infinitely many subharmonics both in the subquadratic
and superquadratic case, with suitable growth conditions on $H^{\prime}$.
Again the duality approach enabled Clarke and Ekeland in [2] to treat the same
problem in the convex-subquadratic case, with growth conditions on $H$ only.

Recently, Michalek and Tarantello (see [3] and [4]) have obtained lower bound
on the number of subharmonics of period $kT$, based on symmetry considerations
and on pinching estimates, as in Sect. 5.2 of this article.

## References

  * [1] Clarke, F., Ekeland, I.: Nonlinear oscillations and boundary-value problems for Hamiltonian systems. Arch. Rat. Mech. Anal. 78, 315–333 (1982)
  * [2] Clarke, F., Ekeland, I.: Solutions périodiques, du période donnée, des équations hamiltoniennes. Note CRAS Paris 287, 1013–1015 (1978)
  * [3] Michalek, R., Tarantello, G.: Subharmonic solutions with prescribed minimal period for nonautonomous Hamiltonian systems. J. Diff. Eq. 72, 28–55 (1988)
  * [4] Tarantello, G.: Subharmonic solutions for Hamiltonian systems via a $\bbbz_{p}$ pseudoindex theory. Annali di Matematica Pura (to appear)
  * [5] Rabinowitz, P.: On subharmonic solutions of a Hamiltonian system. Comm. Pure Appl. Math. 33, 609–633 (1980)

Hamiltonian Mechanics2

Ivar Ekeland Roger Temam

###  2 Fixed-Period Problems: The Sublinear Case

With this chapter, the preliminaries are over, and we begin the search for
periodic solutions to Hamiltonian systems. All this will be done in the convex
case; that is, we shall study the boundary-value problem

$\displaystyle\dot{x}$ $\displaystyle=$ $\displaystyle JH^{\prime}(t,x)$
$\displaystyle x(0)$ $\displaystyle=$ $\displaystyle x(T)$

with $H(t,\cdot)$ a convex function of $x$, going to $+\infty$ when
$\left\|x\right\|\to\infty$.

####  2.1 Autonomous Systems

In this section, we will consider the case when the Hamiltonian $H(x)$ is
autonomous. For the sake of simplicity, we shall also assume that it is
$C^{1}$.

We shall first consider the question of nontriviality, within the general
framework of $\left(A_{\infty},B_{\infty}\right)$-subquadratic Hamiltonians.
In the second subsection, we shall look into the special case when $H$ is
$\left(0,b_{\infty}\right)$-subquadratic, and we shall try to derive
additional information.

#####  2.1.1 The General Case: Nontriviality.

We assume that $H$ is $\left(A_{\infty},B_{\infty}\right)$-subquadratic at
infinity, for some constant symmetric matrices $A_{\infty}$ and $B_{\infty}$,
with $B_{\infty}-A_{\infty}$ positive definite. Set:

$\displaystyle\gamma:$ $\displaystyle=$ $\displaystyle{\rm smallest\
eigenvalue\ of}\ \ B_{\infty}-A_{\infty}$ (37) $\displaystyle\lambda:$
$\displaystyle=$ $\displaystyle{\rm largest\ negative\ eigenvalue\ of}\ \
J\frac{d}{dt}+A_{\infty}\ .$ (38)

Theorem 21 tells us that if $\lambda+\gamma<0$, the boundary-value problem:

$\begin{array}[]{rcl}\dot{x}&=&JH^{\prime}(x)\\\ x(0)&=&x(T)\end{array}$ (39)

has at least one solution $\overline{x}$, which is found by minimizing the
dual action functional:

$\psi(u)=\int_{o}^{T}\left[\frac{1}{2}\left(\Lambda_{o}^{-1}u,u\right)+N^{\ast}(-u)\right]dt$
(40)

on the range of $\Lambda$, which is a subspace $R(\Lambda)_{L}^{2}$ with
finite codimension. Here

$N(x):=H(x)-\frac{1}{2}\left(A_{\infty}x,x\right)$ (41)

is a convex function, and

$N(x)\leq\frac{1}{2}\left(\left(B_{\infty}-A_{\infty}\right)x,x\right)+c\ \ \
\forall x\ .$ (42)

###### Proposition 2

Assume $H^{\prime}(0)=0$ and $H(0)=0$. Set:

$\delta:=\liminf_{x\to 0}2N(x)\left\|x\right\|^{-2}\ .$ (43)

If $\gamma<-\lambda<\delta$, the solution $\overline{u}$ is non-zero:

$\overline{x}(t)\neq 0\ \ \ \forall t\ .$ (44)

###### Proof

Condition (43) means that, for every $\delta^{\prime}>\delta$, there is some
$\varepsilon>0$ such that

$\left\|x\right\|\leq\varepsilon\Rightarrow
N(x)\leq\frac{\delta^{\prime}}{2}\left\|x\right\|^{2}\ .$ (45)

It is an exercise in convex analysis, into which we shall not go, to show that
this implies that there is an $\eta>0$ such that

$f\left\|x\right\|\leq\eta\Rightarrow
N^{\ast}(y)\leq\frac{1}{2\delta^{\prime}}\left\|y\right\|^{2}\ .$ (46)

Figure 2: This is the caption of the figure displaying a white eagle and a
white horse on a snow field

Since $u_{1}$ is a smooth function, we will have
$\left\|hu_{1}\right\|_{\infty}\leq\eta$ for $h$ small enough, and inequality
(46) will hold, yielding thereby:

$\psi(hu_{1})\leq\frac{h^{2}}{2}\frac{1}{\lambda}\left\|u_{1}\right\|_{2}^{2}+\frac{h^{2}}{2}\frac{1}{\delta^{\prime}}\left\|u_{1}\right\|^{2}\
.$ (47)

If we choose $\delta^{\prime}$ close enough to $\delta$, the quantity
$\left(\frac{1}{\lambda}+\frac{1}{\delta^{\prime}}\right)$ will be negative,
and we end up with

$\psi(hu_{1})<0\ \ \ \ \ {\rm for}\ \ h\neq 0\ \ {\rm small}\ .$ (48)

On the other hand, we check directly that $\psi(0)=0$. This shows that 0
cannot be a minimizer of $\psi$, not even a local one. So $\overline{u}\neq 0$
and $\overline{u}\neq\Lambda_{o}^{-1}(0)=0$. ∎

###### Corollary 2

Assume $H$ is $C^{2}$ and $\left(a_{\infty},b_{\infty}\right)$-subquadratic at
infinity. Let $\xi_{1},\allowbreak\dots,\allowbreak\xi_{N}$ be the equilibria,
that is, the solutions of $H^{\prime}(\xi)=0$. Denote by $\omega_{k}$ the
smallest eigenvalue of $H^{\prime\prime}\left(\xi_{k}\right)$, and set:

$\omega:={\rm Min\,}\left\\{\omega_{1},\dots,\omega_{k}\right\\}\ .$ (49)

If:

$\frac{T}{2\pi}b_{\infty}<-E\left[-\frac{T}{2\pi}a_{\infty}\right]<\frac{T}{2\pi}\omega$
(50)

then minimization of $\psi$ yields a non-constant $T$-periodic solution
$\overline{x}$.

We recall once more that by the integer part $E[\alpha]$ of $\alpha\in\bbbr$,
we mean the $a\in\bbbz$ such that $a<\alpha\leq a+1$. For instance, if we take
$a_{\infty}=0$, Corollary 2 tells us that $\overline{x}$ exists and is non-
constant provided that:

$\frac{T}{2\pi}b_{\infty}<1<\frac{T}{2\pi}$ (51)

or

$T\in\left(\frac{2\pi}{\omega},\frac{2\pi}{b_{\infty}}\right)\ .$ (52)

###### Proof

The spectrum of $\Lambda$ is $\frac{2\pi}{T}\bbbz+a_{\infty}$. The largest
negative eigenvalue $\lambda$ is given by $\frac{2\pi}{T}k_{o}+a_{\infty}$,
where

$\frac{2\pi}{T}k_{o}+a_{\infty}<0\leq\frac{2\pi}{T}(k_{o}+1)+a_{\infty}\ .$
(53)

Hence:

$k_{o}=E\left[-\frac{T}{2\pi}a_{\infty}\right]\ .$ (54)

The condition $\gamma<-\lambda<\delta$ now becomes:

$b_{\infty}-a_{\infty}<-\frac{2\pi}{T}k_{o}-a_{\infty}<\omega-a_{\infty}$ (55)

which is precisely condition (50).∎

###### Lemma 2

Assume that $H$ is $C^{2}$ on $\bbbr^{2n}\setminus\\{0\\}$ and that
$H^{\prime\prime}(x)$ is non-degenerate for any $x\neq 0$. Then any local
minimizer $\widetilde{x}$ of $\psi$ has minimal period $T$.

###### Proof

We know that $\widetilde{x}$, or $\widetilde{x}+\xi$ for some constant
$\xi\in\bbbr^{2n}$, is a $T$-periodic solution of the Hamiltonian system:

$\dot{x}=JH^{\prime}(x)\ .$ (56)

There is no loss of generality in taking $\xi=0$. So
$\psi(x)\geq\psi(\widetilde{x})$ for all $\widetilde{x}$ in some neighbourhood
of $x$ in $W^{1,2}\left(\bbbr/T\bbbz;\bbbr^{2n}\right)$.

But this index is precisely the index $i_{T}(\widetilde{x})$ of the
$T$-periodic solution $\widetilde{x}$ over the interval $(0,T)$, as defined in
Sect. 2.6. So

$i_{T}(\widetilde{x})=0\ .$ (57)

Now if $\widetilde{x}$ has a lower period, $T/k$ say, we would have, by
Corollary 31:

$i_{T}(\widetilde{x})=i_{kT/k}(\widetilde{x})\geq
ki_{T/k}(\widetilde{x})+k-1\geq k-1\geq 1\ .$ (58)

This would contradict (57), and thus cannot happen.∎

###### Notes and Comments.

The results in this section are a refined version of [1980]; the minimality
result of Proposition 14 was the first of its kind.

To understand the nontriviality conditions, such as the one in formula (52),
one may think of a one-parameter family $x_{T}$,
$T\in\left(2\pi\omega^{-1},2\pi b_{\infty}^{-1}\right)$ of periodic solutions,
$x_{T}(0)=x_{T}(T)$, with $x_{T}$ going away to infinity when $T\to
2\pi\omega^{-1}$, which is the period of the linearized system at 0.

Table 2: This is the example table taken out of The TeXbook, p. 246 Year | World population
---|---
8000 B.C. | 5,000,000 |
50 A.D. | 200,000,000 |
1650 A.D. | 500,000,000 |
1945 A.D. | 2,300,000,000 |
1980 A.D. | 4,400,000,000 |

######  Theorem 2.1 (Ghoussoub-Preiss)

Assume $H(t,x)$ is $(0,\varepsilon)$-subquadratic at infinity for all
$\varepsilon>0$, and $T$-periodic in $t$

$H(t,\cdot)\ \ \ \ \ {\rm is\ convex}\ \ \forall t$ (59) $H(\cdot,x)\ \ \ \ \
{\rm is}\ \ T{\rm-periodic}\ \ \forall x$ (60) $H(t,x)\geq
n\left(\left\|x\right\|\right)\ \ \ \ \ {\rm with}\ \ n(s)s^{-1}\to\infty\ \
{\rm as}\ \ s\to\infty$ (61) $\forall\varepsilon>0\ ,\ \ \ \exists c\ :\
H(t,x)\leq\frac{\varepsilon}{2}\left\|x\right\|^{2}+c\ .$ (62)

Assume also that $H$ is $C^{2}$, and $H^{\prime\prime}(t,x)$ is positive
definite everywhere. Then there is a sequence $x_{k}$, $k\in\bbbn$, of
$kT$-periodic solutions of the system

$\dot{x}=JH^{\prime}(t,x)$ (63)

such that, for every $k\in\bbbn$, there is some $p_{o}\in\bbbn$ with:

$p\geq p_{o}\Rightarrow x_{pk}\neq x_{k}\ .$ (64)

∎

######  Example 2 (External forcing)

Consider the system:

$\dot{x}=JH^{\prime}(x)+f(t)$ (65)

where the Hamiltonian $H$ is $\left(0,b_{\infty}\right)$-subquadratic, and the
forcing term is a distribution on the circle:

$f=\frac{d}{dt}F+f_{o}\ \ \ \ \ {\rm with}\ \ F\in
L^{2}\left(\bbbr/T\bbbz;\bbbr^{2n}\right)\ ,$ (66)

where $f_{o}:=T^{-1}\int_{o}^{T}f(t)dt$. For instance,

$f(t)=\sum_{k\in\bbbn}\delta_{k}\xi\ ,$ (67)

where $\delta_{k}$ is the Dirac mass at $t=k$ and $\xi\in\bbbr^{2n}$ is a
constant, fits the prescription. This means that the system
$\dot{x}=JH^{\prime}(x)$ is being excited by a series of identical shocks at
interval $T$.

###### Definition 2

Let $A_{\infty}(t)$ and $B_{\infty}(t)$ be symmetric operators in
$\bbbr^{2n}$, depending continuously on $t\in[0,T]$, such that
$A_{\infty}(t)\leq B_{\infty}(t)$ for all $t$.

A Borelian function $H:[0,T]\times\bbbr^{2n}\to\bbbr$ is called
$\left(A_{\infty},B_{\infty}\right)$-subquadratic at infinity if there exists
a function $N(t,x)$ such that:

$H(t,x)=\frac{1}{2}\left(A_{\infty}(t)x,x\right)+N(t,x)$ (68) $\forall t\ ,\ \
\ N(t,x)\ \ \ \ \ {\rm is\ convex\ with\ respect\ to}\ \ x$ (69) $N(t,x)\geq
n\left(\left\|x\right\|\right)\ \ \ \ \ {\rm with}\ \ n(s)s^{-1}\to+\infty\ \
{\rm as}\ \ s\to+\infty$ (70) $\exists c\in\bbbr\ :\ \ \
H(t,x)\leq\frac{1}{2}\left(B_{\infty}(t)x,x\right)+c\ \ \ \forall x\ .$ (71)

If $A_{\infty}(t)=a_{\infty}I$ and $B_{\infty}(t)=b_{\infty}I$, with
$a_{\infty}\leq b_{\infty}\in\bbbr$, we shall say that $H$ is
$\left(a_{\infty},b_{\infty}\right)$-subquadratic at infinity. As an example,
the function $\left\|x\right\|^{\alpha}$, with $1\leq\alpha<2$, is
$(0,\varepsilon)$-subquadratic at infinity for every $\varepsilon>0$.
Similarly, the Hamiltonian

$H(t,x)=\frac{1}{2}k\left\|k\right\|^{2}+\left\|x\right\|^{\alpha}$ (72)

is $(k,k+\varepsilon)$-subquadratic for every $\varepsilon>0$. Note that, if
$k<0$, it is not convex.

###### Notes and Comments.

The first results on subharmonics were obtained by Rabinowitz in [1985], who
showed the existence of infinitely many subharmonics both in the subquadratic
and superquadratic case, with suitable growth conditions on $H^{\prime}$.
Again the duality approach enabled Clarke and Ekeland in [1981] to treat the
same problem in the convex-subquadratic case, with growth conditions on $H$
only.

Recently, Michalek and Tarantello (see Michalek, R., Tarantello, G. [1982] and
Tarantello, G. [1983]) have obtained lower bound on the number of subharmonics
of period $kT$, based on symmetry considerations and on pinching estimates, as
in Sect. 5.2 of this article.

## References

  * [1980] Clarke, F., Ekeland, I.: Nonlinear oscillations and boundary-value problems for Hamiltonian systems. Arch. Rat. Mech. Anal. 78, 315–333 (1982)
  * [1981] Clarke, F., Ekeland, I.: Solutions périodiques, du période donnée, des équations hamiltoniennes. Note CRAS Paris 287, 1013–1015 (1978)
  * [1982] Michalek, R., Tarantello, G.: Subharmonic solutions with prescribed minimal period for nonautonomous Hamiltonian systems. J. Diff. Eq. 72, 28–55 (1988)
  * [1983] Tarantello, G.: Subharmonic solutions for Hamiltonian systems via a $\bbbz_{p}$ pseudoindex theory. Annali di Matematica Pura (to appear)
  * [1985] Rabinowitz, P.: On subharmonic solutions of a Hamiltonian system. Comm. Pure Appl. Math. 33, 609–633 (1980)

Author Index

## Index

Subject Index