#  $K_{1}$ and $K$-groups of 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 describe the Grothendieck groups $K_{1}(V)$ and $K(V)$ of an
absolute matrix order unit space $V$ for unitary and partial unitary elements
respectively. For this purpose, we study some basic properties of unitary and
partial unitary elements, and define their path homotopy equivalence. The
construction of $K(V)$ follows in a almost similar manner as that of
$K_{1}(V).$ We prove that $K_{1}(V)$ and $K(V)$ are ordered abelian groups. We
also prove that $K_{1}(V)$ and $K(V)$ are functors from the category of
absolute matrix order unit spaces with morphisms as unital completely
$|\cdot|$-preserving maps to the category of ordered abelian groups. Later, we
show that under certain conditions, quotient of $K(V)$ is isomorphic to the
direct sum of $K_{0}(V)$ and $K_{1}(V),$ where $K_{0}(V)$ is the Grothendieck
group for order projections.

###### Key words and phrases:

Absolute oder unit space, absolute matrix order unit space, path homotopy,
order projection, partial isometry, unitary, partal unitary,
$K_{0},K_{1},K$-groups, unital $|\cdot|$-preserving map, functoriality, order
structue on $K_{0},K_{1},K$.

###### 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

Operator $K$-theory dwells in the heart of the area of non-commutative
topology. It is an active research area and a much used tool for the study of
C∗-algebras. $K$-theory was introduced in C∗-algebra theory in the early 1970s
through some specific applications. However, the germs of $K$-theory are found
in the Grothendieck’s work in algebraic geometry. $K$-theory was developed by
Atiyah and Hirzebruch in the 1960s based on Grothendieck’s this work. It plays
a major role in structure theory of C∗-algebras. Due to its applications and
appeal, operator $K$-theory has been central point of attraction for a wide
range of mathematicians. For such and more informations, see [5, 6, 30, 32]
and references therein.

Order structure is one of the basic component of C∗-algebra theory. In the
light of Gelfand-Neumark theorem [13] and Kakutani theorem [19], we see that
self-adjoint part of a commutative C∗-algebra forms a Banach lattice. However,
as a contrast, Kadison’s anti-lattice theorem [17] explains that the self-
adjoint part of a non-commutative C∗-algebra is not a vector lattice.
Therefore, the order structure of a C∗-algebra becomes an interesting area for
study. The corresponding theory gives a birth to study of orderd vector spaces
which do not have any vector lattice structure. Being motivated by works of
Kadison, Effros, Strømer and Pedersen, and many others (see [29] and
references therein) which describe various aspects of order structure of
C∗-algebras and encourage to expect a ‘non-commutative vector lattice’ or a
‘near lattice’ structure in it, Karn started working in this direction (see
[20, 21, 22, 23] etc.).

The significant contribution in this direction begins with Kadison’s
functional representation of operator (C∗-) algebras [16]. By this
represenation, the self-adjoint part of every unital C∗-algebra $A$ can be
expressed as the space of continuous affine functions on the state space
$S(A)$ of $A.$ It becomes clear that the order structure of a C∗-algebra is
rich with many properties. For more, we refer to [2, 3, 4, 7, 8, 10, 12, 28]
and references therein.

In few years, the author, in collaboration with Karn, has been working on the
order theoretic aspects of C∗-algebras (see [24, 25, 26]). Karn is already
working in this direction, in the collaboration with several others. In [23],
he introduced the notion of absolute order unit spaces. Self-adjoiint parts of
unital C∗-algebras and unital $M$-spaces are examples of absolute order unit
spaces. More generally, unital $JB$-algebras are examples of absolute order
unit spaces. It is shown that under an additional condition (see [22, Theorem
4.12]) absolute order unit spaces turn out to be vector lattices. Therefore an
absolute order unit space may be termed as a ‘non-commutative vector lattice’.

In [24], Karn with the author introduced and studied the notion of absolute
matrix order unit space which is a matricial version of absolute order unit
space. We also studied absolute value preseving (completely absolute value
preseving) maps between two absolute order unit spaces (absolute matrix order
unit spaces). We found that unital bijective absolute value preseving
(completely absolute value preseving) maps between two absolute order unit
spaces (absolute matrix order unit spaces) are isometries (complete
isometries) [24, Theorems 3.3 and 4.6].

The notion of order projections in absolute order unit spaces has been
introduced and studied by Karn in [23] and it generalizes order theoretically
projections in C∗-algebras. In [25] (continuation of [24]), Karn with the
author generalized the notion of order projections in absolute order unit
spaces to order projections in absolute matrix order unit spaces. We
introduced and studied the notion of partial isometries in absolute matrix
order unit spaces to study comparison of order projections. We also studied
some variants of order projections in absolute matrix order unit spaces.

Furher, in the continuation of [24] and [25], Karn with the author has worked
in this direction in [26]. In [26], we initiate a study of $K$-theory in
absolute matrix order unit spaces. We studied direct limit $M_{\infty}(V)$ and
order projections in $M_{\infty}(V)$ to describe Grothendieck group $K_{0}(V)$
of an absolute matrix order unit space $V$ following the same path as
Grothendieck group $K_{0}(A)$ of a unital C∗-algebra $A$ is defined via the
class of projections $\mathcal{P}_{\infty}(A)$ corresponding to the inductive
(direct) limit $M_{\infty}(A)$ of the family of C∗-algebras $\\{M_{n}(A)\\}$.
Later, we proved that $K_{0}(V)$ is a functor from the category of absolute
matrix order unit spaces with morphisms as unital completely absolute value
preserving maps to the category of ordered abelian groups [26, Corollary 5.3].

Let’s recall the notion of order projections in $M_{\infty}(V)$ introduced in
[26]. Let $\mathfrak{p}\in M_{\infty}(V)$ and let $o(\mathfrak{p})$ be the
order of $\mathfrak{p}$. Then $\mathfrak{p}$ is said to be order projection,
if $|\mathfrak{e}^{n}-2\mathfrak{p}|=\mathfrak{e}^{n}$ for all
$n\in\mathbb{N}$ such that $n\geq o(\mathfrak{p}).$ For an absolute matrix
order unit space $V,$ we denote the set of all order projections, unitaries
and partial unitaries in $M_{n}(V)$ by
$\mathcal{OP}_{n}(V),\mathcal{U}_{n}(V)$ and $\mathcal{PU}_{n}(V)$
respectively, for any $n\in\mathbb{N}$ (for definitions see [25, 3.1]). In
[26], it is discussed that under the identification
$M_{\infty}(V)=\displaystyle\bigcup_{n=1}^{\infty}M_{n}(V),$ the corresponding
set of order projections in $M_{\infty}(V)$ is identified with
$\mathcal{OP}_{\infty}(V)=\displaystyle\bigcup_{n=1}^{\infty}\mathcal{OP}_{n}(V).$
Similarly, we can also define the notion of unitaries and partial unitaries in
$M_{\infty}(V)$ in the following way: Let $\nu\in M_{\infty}(V).$ Then $\nu$
is said to be unitary, if $|\nu|=|\nu^{*}|=\mathfrak{e}^{o(\nu)}$ and it is
said to be partial unitary, if $|\nu|=|\nu^{*}|$ is an order projection.
Again, under the identification
$M_{\infty}(V)=\displaystyle\bigcup_{n=1}^{\infty}M_{n}(V),$ the corresponding
set of unitaries and partial unitaries in $M_{\infty}(V)$ can be identified
with
$\mathcal{U}_{\infty}(V)=\displaystyle\bigcup_{n=1}^{\infty}\mathcal{U}_{n}(V)$
and
$\mathcal{PU}_{\infty}(V)=\displaystyle\bigcup_{n=1}^{\infty}\mathcal{PU}_{n}(V)$
respectively.

In this paper, we continue our work from [26]. We study some basic properties
of unitaries and partial unitaries in an absolute matrix order unit space
(Propositions 3.2 and 4.2). We also study path homotopy equivalences of
unitary and partial unitaries (Propositions 3.4, 3.5, 4.4, 4.5 and Lemmas 3.6,
4.6). By help of path homotopy, we induce new equivalence relations for
unitary and partial unitaries, and study their properties (Corollaries 3.8,
3.12, 4.8, 4.12 and Propositions 3.9, 3.11, 3.13, 4.9, 4.11, 4.13). By such
equivalence relations, we describe $K_{1}(V)$ and $K(V)$ the Grothendieck
groups of an absoute matrix order unit space $V$ for unitary and partial
unitaries elements respectively (Theorems 3.14 and 4.14), which are
generalizations of $K_{1}(A)$ and $K(A)$ of a C∗-algebra $A$ for unitary and
partial unitaries. We show that $K_{1}(V)$ and $K(V)$ are ordered abelian
groups (Theorems 3.16 and 4.16). We also prove that $K_{1}(V)$ and $K(V)$ are
functors from the category of absolute matrix order unit spaces with morphisms
as unital completely $|\cdot|$-preserving maps to the category of ordered
abelian groups (Corollaries 3.19 and 4.19). Later, we find a relation of
$K_{1}(V)$ and $K(V)$ with $K_{0}(V)$ (Theorem 5.7).

##  2\. $K_{0}$-group corresponding to an absolute matrix order unit space

In this section, we recall the $K_{0}$-group of an absolute matrix order unit
space that has been described by Karn and the author in [26]. For that, we
need to recall some preliminaries. We start this section with the following
matricial notions:

Let $V$ be a complex vector space. We denote by $M_{m,n}(V)$ the vector space
of all the $m\times n$ matrices $v=[v_{i,j}]$ with entries $v_{i,j}\in V$ and
by $M_{m,n}$ the vector space of all the $m\times n$ matrices $a=[a_{i,j}]$
with entries $a_{i,j}\in\mathbb{C}$. We write $0_{m,n}$ for zero element in
$M_{m,n}(V)$. For $m=n$, we write $0_{m,n}=0_{n}$. We define
$av=\begin{bmatrix}\displaystyle\sum_{k=1}^{m}a_{i,k}v_{k,j}\end{bmatrix}$ and
$vb=\begin{bmatrix}\displaystyle\sum_{k=1}^{n}v_{i,k}b_{k,j}\end{bmatrix}$ for
$a\in M_{r,m},v\in M_{m,n}(V)$ and $b\in M_{n,s}$. We write

$v\oplus w=\begin{bmatrix}v&0\\\ 0&w\end{bmatrix}$ for $v\in M_{m,n}(V),w\in
M_{r,s}(V).$

Here $0$ denotes suitable rectangular matrix of zero entries from $V$ [31].

###  2.1. Matrix Order Unit Spaces

In this subsection, we recall some matrix theory of ordered vector spaces. For
this theory, we refer to [31, 9, 11, 27] and references therein.

A complex vector space with an involution is called a _$\ast$ -vector space_.
We write $V_{sa}=\\{v\in V:v=v^{*}\\}$. Then $V_{sa}$ is a real vector space
[9].

Next, we recall the definition of a matrix ordered space.

######  Definition 2.1 ([9]).

A _matrix ordered space_ is a $*$-vector space $V$ together with a sequence
$\\{M_{n}(V)^{+}\\}$ with $M_{n}(V)^{+}\subset M_{n}(V)_{sa}$ for each
$n\in\mathbb{N}$ satisfying the following conditions:

  1. (a)

$(M_{n}(V)_{sa},M_{n}(V)^{+})$ is a real ordered vector space, for each
$n\in\mathbb{N}$; and

  2. (b)

$a^{*}va\in M_{m}(V)^{+}$ for all $v\in M_{n}(V)^{+}$, $a\in M_{n,m}$ and
$n,m\in\mathbb{N}$.

It is denoted by $(V,\\{M_{n}(V)^{+}\\})$. If, in addition, $e\in V^{+}$ is an
order unit in $V_{sa}$ such that $V^{+}$ is proper and $M_{n}(V)^{+}$ is
Archimedean for all $n\in\mathbb{N}$, then $V$ is called a _matrix order unit
space_ and is denoted by $(V,\\{M_{n}(V)^{+}\\},e)$.

The first part of next result elaborates that each level in a matrix ordered
space is proper and generating provided it is proper and generating at ground
level respectively, and the second part elaborates that a matrix ordered space
has order unit at each level provided it has order unit at ground level.

######  Proposition 2.2.

Let $(V,\\{M_{n}(V)^{+}\\})$ be a matrix ordered space.

  1. (1)

$($[27], Proposition 1.8$).$

    1. (a)

If $V^{+}$ is proper, then $M_{n}(V)^{+}$ is proper for all $n\in\mathbb{N}$.

    2. (b)

If $V^{+}$ is generating, then $M_{n}(V)^{+}$ is generating for all
$n\in\mathbb{N}$.

  2. (2)

$($[27], Lemma 2.6$).$ If $e\in V^{+}$ is an order unit for $V_{sa}$. Then
$e^{n}$ is an order unit for $M_{n}(V)_{sa}$ for all $n\in\mathbb{N}$ ( where
$e^{n}:=e\oplus\cdots\oplus e\in M_{n}(V)$).

In fact, matrix ordered spaces are normed spaces that is clear from the
following result:

######  Theorem 2.3 ([9]).

Let $(V,e)$ be a matrix order unit space. Then

  1. (1)

The order unit determines a matrix norm on $V$ in the following way:

$\|v\|_{n}:=\inf\\{k>0:\begin{bmatrix}ke^{n}&v\\\ v^{*}&ke^{n}\end{bmatrix}\in
M_{2n}(V)^{+}\\}\quad\textrm{for}~{}v\in M_{n}(V).$

  2. (2)

$\left\|\begin{bmatrix}u&0\\\
0&v\end{bmatrix}\right\|_{m+n}=\max\\{\|u\|_{m},\|v\|_{n}\\}$ for all $u\in
M_{m}(V)$ and $v\in M_{n}(V).$

  3. (3)

$\|\alpha v\beta\|_{n}\leq\|\alpha\|\|v\|_{n}\|\beta\|$ for all
$\alpha,\beta\in M_{n}.$

  4. (4)

$\|\alpha v\beta\|_{n}=\|v\|_{n}$ for all $\alpha,\beta\in M_{n}$ such that
$\alpha^{*}\alpha=I_{n}$ and $\beta\beta^{*}=I_{n}$ (i.e. for $\alpha$
isometry and $\beta$ co-isometry).

  5. (5)

$\|v\|_{n}=\|v^{*}\|_{n}=\left\|\begin{bmatrix}0&v\\\
v^{*}&0\end{bmatrix}\right\|_{2n}.$

In other words, $(V,\\{\|\cdot\|_{n}\\})$ is a $L^{\infty}$-matricially normed
space [31].

In [20, 21, 22, 23, 24, 25], some variants of orthoganity in ordered normed
spaces have been introduced and studied in terms of norm by Karn. We recall
some of them here:

Let $u,v\in M_{n}(V)^{+}$. We say that $u$ is $\infty$-_orthogonal_ , (we
write $u\perp_{\infty}v$), if $\|u+kv\|_{n}=\max\\{\|u\|,\|kv\|_{n}\\}$ for
all $k\in\mathbb{R}$ [23, Definition 3.6]. Recall that for $u,v\in
M_{n}(V)^{+}\setminus\\{0\\}$, we have $u\perp_{\infty}v$ if and only if
$\|\|u\|_{n}^{-1}u+\|v\|_{n}^{-1}v\|_{n}=1$ [21, Theorem 3.3]. For $u,v\in
M_{n}(V)^{+}\setminus\\{0\\}$, we say that $u\perp_{\infty}^{a}v$, if
$u_{1}\perp_{\infty}v_{1}$ whenever $0\leq u_{1}\leq u$ and $0\leq v_{1}\leq
v$. It was proved in [22, 23] that if $A$ is a C∗-algebra and if $a,b\in
A^{+}\setminus\\{0\\}$, then $a\perp_{\infty}^{a}b$ if and only if $ab=0$
(that is, $a$ is algebraically orthogonal to $b$).

###  2.2. Absolute Matrix Order Unit Spaces

In this subsection, we briefly recall absolutely matrix ordered spaces and
absolute matrix order unit spaces which are matricial versions are absolutely
ordered spaces and absolute order unit spaces respectively. The theory of
absolutely ordered spaces and absolute order unit spaces have been introduced
and studied by Karn in [23], and the theory of absolutely matrix ordered
spaces and absolute matrix order unit spaces have been introduced and studied
by Karn and the author in [24]. We begin with the definition of absolutely
matrix ordered spaces.

######  Definition 2.4 ([24], Definition 4.1).

Let $(V,\\{\ M_{n}(V)^{+}\\})$ be a matrix ordered space and assume that
$|\cdot|_{m,n}:M_{m,n}(V)\to M_{n}(V)^{+}$ for $m,n\in\mathbb{N}$. Let us
write $|\cdot|_{n,n}=|\cdot|_{n}$ for every $n\in\mathbb{N}$. Then
$\left(V,\\{M_{n}(V)^{+}\\},\\{|\cdot|_{m,n}\\}\right)$ is called an
_absolutely matrix ordered space_ , if it satisfies the following conditions:

  1. $1.$

For all $n\in\mathbb{N}$, $(M_{n}(V)_{sa},M_{n}(V)^{+},|\cdot|_{n})$ is an
absolutely ordered space;

  2. $2.$

For $v\in M_{m,n}(V),\alpha\in M_{r,m}$ and $\beta\in M_{n,s},$ we have

$|\alpha v\beta|_{r,s}\leq\|\alpha\|||v|_{m,n}\beta|_{n,s};$

  3. $3.$

For $v\in M_{m,n}(V)$ and $w\in M_{r,s}(V),$ we have

$|v\oplus w|_{m+r,n+s}=|v|_{m,n}\oplus|w|_{r,s}.$

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

Let $A$ be a $C^{*}$-algebra. Then, for each $n\in\mathbb{N},M_{n}(A)$ is a
$C^{*}$-algebra. If $M_{n}(A)^{+}$ denotes the set of all the positive
elements in $M_{n}(A),$ then $(A,\\{M_{n}(A)^{+}\\})$ is a matrix ordered
space. For $m,n\in\mathbb{N},$ we define $|\cdot|_{m,n}:M_{m,n}(A)\to
M_{n}(A)^{+}$ given by $|a|_{m,n}=(a^{*}a)^{\frac{1}{2}}$ for all $a\in
M_{m,n}(A).$ Then $(A,\\{M_{n}(A)^{+}\\},\\{|\cdot|_{m,n}\\})$ is an
absolutely matrix ordered space.

The following two results describes some basic properties of absolutely matrix
ordered spaces.

######  Proposition 2.6 ([24], Proposition 4.2).

Let $(V,\\{M_{n}(V)^{+}\\},\\{|\cdot|_{m,n}\\})$ be an absolutely matrix
ordered space.

  1. (1)

If $\alpha\in M_{r,m}$ is an isometry i.e. $\alpha^{*}\alpha=I_{m},$ then
$|\alpha v|_{r,n}=|v|_{m,n}$ for any $v\in M_{m,n}(V).$

  2. (2)

If $v\in M_{m,n}(V),$ then $\left|\begin{bmatrix}0_{m}&v\\\
v^{*}&0_{n}\end{bmatrix}\right|_{m+n}=|v^{*}|_{n,m}\oplus|v|_{m,n}.$

  3. (3)

$\begin{bmatrix}|v^{*}|_{n,m}&v\\\ v^{*}&|v|_{m,n}\end{bmatrix}\in
M_{m+n}(V)^{+}$ for any $v\in M_{m,n}(V).$

  4. (4)

$|v|_{m,n}=\left|\begin{bmatrix}v\\\ 0\end{bmatrix}\right|_{m+r,n}$ for any
$v\in M_{m,n}(V)$ and $r\in\mathbb{N}.$

  5. (5)

$|v|_{m,n}\oplus 0_{s}=\left|\begin{bmatrix}v&0\end{bmatrix}\right|_{m,n+s}$
for any $v\in M_{m,n}(V)$ and $s\in\mathbb{N}.$

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

Let $V$ be an absolutely matrix ordered space and let $n\in\mathbb{N}.$ Then
$|\alpha^{*}v\alpha|_{n}=\alpha^{*}|v|_{n}\alpha,$ if $v\in M_{n}(V)$ and
$\alpha\in M_{n}$ is a unitary.

Next, we recall the another variant of orthogonality for general elements in
absolutely matrix ordered spaces in terms of absolute value, introduced and
studied by Karn and the author in [25]. This orthogonaly for positive elements
in absolutely ordered spaces has been introduced and studied earlier by Karn
in [23].

######  Definition 2.8 ([25], Definitions 2.2 and 2.4).

Let $\left(V,\\{M_{n}(V)^{+}\\},\\{|\cdot|_{m,n}\\}\right)$ be an _absolutely
matrix ordered space_. Let $u,v\in M_{n}(V)^{+}$ for some $n\in\mathbb{N}.$ We
say that $u$ is _orthogonal_ to $v,$ (we write it, $u\perp v$), if
$|u-v|_{n}=u+v.$ Now, if $u,v\in M_{n}(V)_{sa},$ we say that $u$ is
_orthogonal_ to $v,$ (we still write it, $u\perp v$), if
$|u|_{n}\perp|v|_{n}.$ Further, if $u,v\in M_{m,n}(V)$ for some
$m,n\in\mathbb{N}$, we say that $u$ is _orthogonal_ to $v$, (we continue to
write, $u\perp v$), if $\begin{bmatrix}0&u\\\
u^{*}&0\end{bmatrix}\perp\begin{bmatrix}0&v\\\ v^{*}&0\end{bmatrix}$ in
$M_{m+n}(V)_{sa}$.

The next result associates orthogonality of general elements in absolutely
matrix ordered spaces to orthogonality of positive elements.

######  Remark 2.9 ([25], Remark 2.2(2)).

Let $V$ be an absolutely matrix ordered space and let $u,v\in M_{m,n}(V)$ for
some $m,n\in\mathbb{N}.$ Then $u\perp v$ if and only if
$|u|_{m,n}\perp|v|_{m,n}$ and $|u^{*}|_{n,m}\perp|v^{*}|_{n,m}.$

The following result tells that absolute values are additive on orthogonal
elements.

######  Proposition 2.10 ([25], Proposition 2.3(1)).

Let $V$ be an absolutely matrix ordered space and let $u,v\in M_{m,n}(V)$.
Then $u\perp v$ if and only if, $|u\pm v|_{m,n}=|u|_{m,n}+|v|_{m,n}$ and
$|u^{*}\pm v^{*}|_{n,m}=|u^{*}|_{n,m}+|v^{*}|_{n,m}.$

Next, we prove that orthogonality in absolutely matrix ordered spaces is
invariant under scalar multiplication.

######  Proposition 2.11.

Let $V$ be an absolutely matrix ordered space and let $u,v\in M_{m,n}(V)$ such
that $u\perp v.$ Then $\alpha u\perp\beta v$ for all
$\alpha,\beta\in\mathbb{C}.$

###### Proof.

Since $u\perp v,$ we have $\left|\begin{bmatrix}0&u\\\
u^{*}&0\end{bmatrix}\right|_{m+n}\perp\left|\begin{bmatrix}0&v\\\
v^{*}&0\end{bmatrix}\right|_{m+n}.$ Let $\alpha,\beta\in\mathbb{C}.$ If
$\alpha=0$ or $\beta=0,$ then $\alpha u\perp\beta v.$ Now, assume $\alpha\neq
0,\beta\neq 0.$ Thus $\alpha=|\alpha|e^{i\zeta_{1}}$ and
$\beta=|\beta|e^{i\zeta_{2}}$ for some $\zeta_{1},\zeta_{2}\in\mathbb{R}.$
Then $\begin{bmatrix}e^{i\zeta_{1}}I_{m}&0\\\
0&e^{-i\zeta_{1}}I_{n}\end{bmatrix}$ and
$\begin{bmatrix}e^{i\zeta_{2}}I_{m}&0\\\ 0&e^{-i\zeta_{2}}I_{n}\end{bmatrix}$
are unitary matrices such that $\begin{bmatrix}e^{i\zeta_{1}}I_{m}&0\\\
0&e^{-i\zeta_{1}}I_{n}\end{bmatrix}\begin{bmatrix}0&u\\\
u^{*}&0\end{bmatrix}=\begin{bmatrix}0&e^{i\zeta_{1}}u\\\
(e^{i\zeta_{1}}u)^{*}&0\end{bmatrix}$ and
$\begin{bmatrix}e^{i\zeta_{2}}I_{m}&0\\\
0&e^{-i\zeta_{2}}I_{n}\end{bmatrix}\begin{bmatrix}0&v\\\
v^{*}&0\end{bmatrix}=\begin{bmatrix}0&e^{i\zeta_{2}}v\\\
(e^{i\zeta_{2}}v)^{*}&0\end{bmatrix}.$ By Proposition 2.6(1), we get that
$\left|\begin{bmatrix}0&e^{i\zeta_{1}}u\\\
(e^{i\zeta_{1}}u)^{*}&0\end{bmatrix}\right|_{m+n}=\left|\begin{bmatrix}0&u\\\
u^{*}&0\end{bmatrix}\right|_{m+n}\perp\left|\begin{bmatrix}0&v\\\
v^{*}&0\end{bmatrix}\right|_{m+n}=\left|\begin{bmatrix}0&e^{i\zeta_{2}}v\\\
(e^{i\zeta_{2}}v)^{*}&0\end{bmatrix}\right|_{m+n}.$

Next, either $|\alpha|\leq|\beta|$ or $|\beta|\leq|\alpha|.$ Assume that
$|\alpha|\leq|\beta|.$ Then, by [23, Definition 3.4(3)], we have that
$|\beta|\left|\begin{bmatrix}0&e^{i\zeta_{1}}u\\\
(e^{i\zeta_{1}}u)^{*}&0\end{bmatrix}\right|_{m+n}\perp|\beta|\left|\begin{bmatrix}0&e^{i\zeta_{2}}v\\\
(e^{i\zeta_{2}}v)^{*}&0\end{bmatrix}\right|_{m+n}.$ Since
$|\alpha|\left|\begin{bmatrix}0&e^{i\zeta_{1}}u\\\
(e^{i\zeta_{1}}u)^{*}&0\end{bmatrix}\right|_{m+n}\leq|\beta|\left|\begin{bmatrix}0&e^{i\zeta_{1}}u\\\
(e^{i\zeta_{1}}u)^{*}&0\end{bmatrix}\right|_{m+n},$ again by [23, Definition
3.4(4)], we conclude that $\left|\begin{bmatrix}0&\alpha u\\\ (\alpha
u)^{*}&0\end{bmatrix}\right|_{m+n}=|\alpha|\left|\begin{bmatrix}0&e^{i\zeta_{1}}u\\\
(e^{i\zeta_{1}}u)^{*}&0\end{bmatrix}\right|_{m+n}\perp|\beta|\left|\begin{bmatrix}0&e^{i\zeta_{2}}v\\\
(e^{i\zeta_{2}}v)^{*}&0\end{bmatrix}\right|_{m+n}=\left|\begin{bmatrix}0&\beta
v\\\ (\beta v)^{*}&0\end{bmatrix}\right|_{m+n}$ so that $\alpha u\perp\beta
v.$ Similarly, the case $|\beta|\leq|\alpha|$ can be taken care. ∎

Finally, we bind up this subsection by recalling absolute matrix order unit
spaces.

######  Definition 2.12 ([24], Definition 4.3).

Let $(V,\\{M_{n}(V)^{+}\\},e)$ be a matrix order unit space such that

  1. (a)

$\left(V,\\{M_{n}(V)^{+}\\},\\{|\cdot|_{m,n}\\}\right)$ is an absolutely
matrix ordered space; and

  2. (b)

$\perp=\perp_{\infty}^{a}$ on $M_{n}(V)^{+}$ for all $n\in\mathbb{N}.$

Then $(V,\\{M_{n}(V)^{+}\\},\\{|\cdot|_{m,n}\\},e)$ is called an _absolute
matrix order unit space_.

######  Example 2.13 ([24], Example 4.4).

For any $C^{*}$-algebra $A,$ we have $\perp=\perp_{\infty}^{a}$ on
$M_{n}(A)^{+}$ for all $n\in\mathbb{N}.$ Therefore, every unital
$C^{*}$-algebra is an absolute matrix order unit space.

###  2.3. Order Projections in Absolute Matrix Order Unit Spaces

The notion of order projections has been introduced and studied in absolute
order unit spaces by Karn in [23]. It generalizes the notion of projections in
$C^{*}$-algebras. In the collaboration with Karn, author has generalized the
notion of order projections in absolute order unit spaces to order projections
in absolute matrix order unit spaces [25]. In [25], we have also introduced
and studied the notion of partial isometries in absolute matrix order unit
spaces which generalizes the notion of partial isometries in $C^{*}$-algebras.
For details, see [23, 25].

In this subsection, we recall the notions of order projections and partial
isometries in absolute matrix order unit spaces.

######  Definition 2.14 ([25], Definition 3.1).

Let $(V,e)$ be an absolute matrix order unit space and let $v\in M_{n}(V)$ for
some $n\in\mathbb{N}.$ Then $v$ is said to be _order projection_ , if
$v^{*}=v$ and $|2v-e^{n}|_{n}=e^{n}.$ Now, let $v\in M_{m,n}(V)$ for some
$m,n\in\mathbb{N}.$ Then $v$ is said to be _partial isometry_ , if $|v|_{m,n}$
and $|v^{*}|_{n,m}$ are order projections.

The set of all the order projections in $M_{n}(V)$ will be denoted by
$\mathcal{OP}_{n}(V)$ and the set of all partial isometries in $M_{m,n}(V)$
will be denoted by $\mathcal{PI}_{m,n}(V).$ For $m=n$, we write
$\mathcal{PI}_{m,n}(V)=\mathcal{PI}_{n}(V)$. For $n=1$, we shall write
$\mathcal{PI}(V)$ for $\mathcal{PI}_{1}(V)$ and $\mathcal{OP}(V)$ for
$\mathcal{OP}_{1}(V)$.

First, let’s recall the following property of order projections:

######  Proposition 2.15 ([25], Proposition 3.2).

Let $V$ be an absolute matrix order unit space and let $m,n\in\mathbb{N}.$
Then $p\in\mathcal{OP}_{m}(V),q\in\mathcal{OP}_{n}(V)$ if and only if $p\oplus
q\in\mathcal{OP}_{m+n}(V)$

The following result describes how othogonality helps partial isometries for
their sum to be again a partial isometry.

######  Proposition 2.16 ([25], Corollary 3.1).

Let $V$ be an absolute matrix order unit space and let $v_{1},\dots,v_{k}\in
M_{m,n}(V)$ be mutually orthogonal vectors for some $k,m,n\in\mathbb{N}$. Then
$v_{1},\dots,v_{k}\in\mathcal{PI}_{m,n}(V)$ if, and only if,
$\displaystyle\sum_{i=1}^{k}v_{i}\in\mathcal{PI}_{m,n}(V).$

Next, we recall a result that we need in the section 4. To prove this result,
we need to recall the following result:

######  Proposition 2.17 ([25], Proposition 3.3).

Let $V$ be an absolute matrix order unit space and let $\alpha\in M_{m,n}$
such that $\alpha^{*}\alpha=I_{n}.$ Then $\alpha u\alpha^{*}\perp\alpha
v\alpha^{*}$ for $u,v\in M_{n}(V)^{+}$ with $u\perp v.$ In particular, if
$p\in\mathcal{OP}_{n}(V),$ then $\alpha p\alpha^{*}\in\mathcal{OP}_{m}(V).$

######  Corollary 2.18.

Let $V$ be an absolue matrix order unit space. Also, let
$v\in\mathcal{PI}_{n}(V)$ and $\alpha\in M_{m,n}$ such that
$\alpha^{*}\alpha=I_{n}.$ Then $\alpha p\alpha^{*}\in\mathcal{PI}_{m}(V).$

###### Proof.

Since $\alpha^{*}\alpha=I_{n},$ we have $m\geq n$ and
$\alpha=\delta\begin{bmatrix}I_{n}\\\ 0_{m-n,n}\end{bmatrix}$ for some unitary
matrix $\delta$. Then $\alpha v\alpha^{*}=\delta(v\oplus 0_{m-n})\delta^{*}$
and $(\alpha v\alpha^{*})^{*}=\alpha v^{*}\alpha^{*}=\delta(v^{*}\oplus
0_{m-n})\delta^{*}$ and by Lemma 2.7, we get that $|\alpha
v\alpha^{*}|_{n}=\delta(|v|_{n}\oplus 0_{m-n})\delta^{*}$ and $|(\alpha
v\alpha^{*})|_{n}=\delta(|v^{*}|_{n}\oplus 0_{m-n})\delta^{*}.$ Next, we have
$|v|_{n}$ and $|v^{*}|_{n}\in\mathcal{OP}_{n}(V),$ thus by Proposition 2.15,
we have $|v^{*}|_{n}\oplus 0_{m-n}$ and $|v^{*}|_{n}\oplus
0_{m-n}\in\mathcal{OP}_{m}(V).$ Finally, by Proposition 2.17, we get that
$\alpha p\alpha^{*}\in\mathcal{PI}_{m}(V).$ ∎

In [26], Karn and the author have described the direct limit of absolute
matrix spaces. We denote direct limit of an absolute matrix order unit space
$V$ by $M_{\infty}(V).$ The direct limit of an absolute matrix order unit
space $V$ is identified with $\displaystyle\bigcup_{n=1}^{\infty}M_{n}(V)$
(see [26, Theorems 3.13 and 3.14]). Thus, we write:
$M_{\infty}(V)=\displaystyle\bigcup_{n=1}^{\infty}M_{n}(V)$. Further, under
this identification, the corresponding set of projections is identified with
$\mathcal{OP}_{\infty}(V)=\displaystyle\bigcup_{n=1}^{\infty}\mathcal{OP}_{n}(V)$.
For more information, we refer to [26].

In the next definition, a relation is defined on the set of all the order
projections.

######  Definition 2.19 ([25], Definition 4.1).

Let $V$ be an absolute matrix order unit space. Then we define a relation
$\sim$ on $\mathcal{OP}_{\infty}(V)$ by the following way:

Given $p\in\mathcal{OP}_{m}(V)$ and $q\in\mathcal{OP}_{n}(V)$, we say that $p$
is partial isometric equivalent to $q$, (we write, $p\sim q$), if there exists
$v\in\mathcal{PI}_{m,n}(V)$ such that $p=|v^{*}|_{n,m}$ and $q=|v|_{m,n}$.

We recall the following properties of this relation.

######  Proposition 2.20 ([25], Propositions 4.1 and 4.2).

Let $V$ be an absolute matrix order unit space and let
$p,q,r,p^{\prime},q^{\prime}\in\mathcal{OP}_{\infty}(V)$. Then

  1. (1)

If $m,n\in\mathbb{N}$ and let $p\in\mathcal{OP}_{m}(V)$, then $p\sim p\oplus
0_{n}$ and $p\sim 0_{n}\oplus p$;

  2. (2)

If $p\sim q$ and $p^{\prime}\sim q^{\prime}$ with $p\perp p^{\prime}$ and
$q\perp q^{\prime}$, then $p+p^{\prime}\sim q+q^{\prime}$;

  3. (3)

If $p\sim p^{\prime}$ and $q\sim q^{\prime},$ then $p\oplus q\sim
p^{\prime}\oplus q^{\prime};$

  4. (4)

$p\oplus q\sim q\oplus p;$

  5. (5)

If $p,q\in\mathcal{OP}_{n}(V)$ for some $n\in\mathbb{N}$ such that $p\perp q,$
then $p+q\sim p\oplus q;$

  6. (6)

$(p\oplus q)\oplus r=p\oplus(q\oplus r);$

  7. (7)

$\sim$ is an equivalence relation on $\mathcal{OP}_{\infty}(V)$, if the
following condition holds: (T): If $u\in\mathcal{PI}_{m,n}(V)$ and
$v\in\mathcal{PI}_{l,n}(V)$ with $|u|_{m,n}=|v|_{l,n}$, then there exists a
$w\in\mathcal{PI}_{m,l}(V)$ such that $|w^{*}|_{l,m}=|u^{*}|_{n,m}$ and
$|w|_{m,l}=|v^{*}|_{n,l}$.

We extend $\sim$ to the following relation on $\mathcal{OP}_{\infty}(V)$:

######  Definition 2.21 ([26], Definition 4.4).

Let $V$ be an absolute matrix order unit space. For
$p,q\in\mathcal{OP}_{\infty}(V),$ we say that $p\approx q$, if there exists
$r\in\mathcal{OP}_{\infty}(V)$ such that $p\oplus r\sim q\oplus r.$

The newly defined relation $\approx$ enjoys the following properties:

######  Proposition 2.22 ([26], Proposition 4.5).

Let $V$ be an absolute matrix order unit space and let
$p,q\in\mathcal{OP}_{\infty}(V)$.

  1. (1)

$p\sim q$ implies $p\approx q$.

  2. (2)

If $V$ satisfies condition (T), then $\approx$ is an equivalence relation.

Next, we recall one of the main result in [26] around which whole theory of
[26] revolves.

######  Theorem 2.23 ([26], Theorem 4.8).

Let $(V,e)$ be an absolute matrix order unit space satisfying (T) and consider
$\mathcal{OP}_{\infty}(V)\times\mathcal{OP}_{\infty}(V)$. For all
$p_{1},p_{2},q_{1},q_{2}\in\mathcal{OP}_{\infty}(V),$ we define
$(p_{1},q_{1})\equiv(p_{2},q_{2})$ if and only if $p_{1}\oplus q_{2}\approx
p_{2}\oplus q_{1}.$ Then $\equiv$ is an equivalence relation on
$\mathcal{OP}_{\infty}(V)\times\mathcal{OP}_{\infty}(V).$ Consider

$K_{0}(V)=\\{[(p_{,}q)]:p,q\in\mathcal{OP}_{\infty}\\}$

where $[(p,q)]$ is the equivalence class of $(p,q)$ in
$(\mathcal{OP}_{\infty}(V)\times\mathcal{OP}_{\infty}(V),\equiv).$ For all
$p_{1},p_{2},q_{1},q_{2}\in\mathcal{OP}_{\infty}(V),$ we write

$[(p_{1},q_{1})]+[(p_{2},q_{2})]=[(p_{1}\oplus p_{2},q_{1}\oplus q_{2})].$

Then $(K_{0}(V),+)$ is an abelian group.

######  Remark 2.24 ([26], Remark 4.9).

The abelian group $K_{0}(V)$ is called the $K_{0}(V)$-group of the absolute
matrix order unit space $V.$ If $A$ is a $C^{*}$-algebra, then $K_{0}(A)$ is
Grothendieck group of $A$ for projections. Thus $K_{0}$-group for absolute
matrix order unit spaces is a generalization of $K_{0}$-group for
$C^{*}$-algebras.

For the definitions of group cone, proper cone, generating cone, ordered
abelian group, ditinguished order unit and ordered abelian group with
ditinguished order unit etc., see [26, Subsection 4.1] and for the definition
of finite order projections, see [25, Definition 5.1].

The next two results tells that $K_{0}(V)$ receives order structure and order
unit from an absolute order unit space $V$ in heredity so that it becomes an
ordered abelian group with a distinguished order unit.

######  Theorem 2.25 ([26], Theorem 4.10).

Let $V$ be an absolute matrix order unit space satisfying (T). Put
$K_{0}(V)^{+}=\\{[(p,0)]:p\in\mathcal{OP}_{\infty}(V)\\}.$ Then

  1. (a)

$K_{0}(V)^{+}$ is a group cone in $K_{0}(V)$.

  2. (b)

If $e^{n}$ is finite for all $n\in\mathbb{N}$, then $K_{0}(V)^{+}$ is proper.

  3. (c)

$K_{0}(V)^{+}$ is generating.

In other words, if $e^{n}$ is finite for all $n\in\mathbb{N}$, then
$(K_{0}(V),K_{0}(V)^{+})$ is an ordered abelian group.

######  Corollary 2.26 ([26], Corollary 4.11).

Let $(V,e)$ be an absolute matrix order unit space satisfying (T) and let
$e^{n}$ be finite for all $n\in\mathbb{N}$. Then $(K_{0}(V),K_{0}(V)^{+})$ is
an ordered abelian group with distinguished order unit $[(e,0)].$ In other
words, for each $g\in K_{0}(V),$ there exists $n\in\mathbb{N}$ such that
$-n[(e,0)]\leq g\leq n[(e,0)].$

We recall suitable morphisms on absolute matrix order unit spaces, called
completely $|\cdot|$-preserving maps. These maps are generalization of
$|\cdot|$-preserving maps on absolute order unit spaces to absolute matrix
order unit spaces [24]. The author with Karn has proved that under certain
conditions such maps are isometries on these spaces (see [24, Theorem 3.3 and
4.6]).

######  Definition 2.27 ([24], Definition 4.5).

Let $V$ and $W$ be absolute matrix order unit spaces and let $\phi:V\to W$ be
a $\ast$-linear map. We say that $\phi$ is completely $|\cdot|$-preserving, if
$\phi_{n}:M_{n}(V)\to M_{n}(W)$ is an $|\cdot|$-preserving map for each
$n\in\mathbb{N}.$ Further, $\phi$ is said to be unital completely
$|\cdot|$-preserving, if it is completely $|\cdot|$-preserving and it also
preserves order unit.

Let $V$ and $W$ be absolute matrix order unit spaces. We denote the zero map
between $V$ and $W$ by $0_{W,V}.$ Similarly, the identity map on $V$ is
denoted by $I_{V}.$ Furthermore, if $V$ and $W$ satisfy $(T),$ then we denote
the zero group homomorphism between $K_{0}(V)$ and $K_{0}(W)$ by
$0_{K_{0}(W),K_{0}(V)}$ and the identity map on $K_{0}(V)$ is denoted by
$I_{K_{0}(V)}.$

Finally, the following result describes the functoriality of $K_{0}.$

######  Theorem 2.28 ([26], Corollary 5.3 and Remark 5.4).

Let $U,V$ and $W$ be absolute matrix order unit spaces satisfying (T). Then

  1. (a)

$K_{0}(I_{V})=I_{K_{0}(V)};$

  2. (b)

If $\phi:U\to V$ and $\psi:V\to W$ be unital completely $|\cdot|$-preserving
maps, then $K_{0}(\psi\circ\phi)=K_{0}(\psi)\circ K_{0}(\phi);$

  3. (c)

$K_{0}(0_{W,V})=0_{K_{0}(W),K_{0}(V)}.$

In particular, $K_{0}$ is a functor from category of absolute matrix order
unit spaces with morphisms as unital completely $|\cdot|$-preserving maps to
category of abelian groups.

##  3\. $K_{1}$-group corresponding to an absolute matrix order unit space

From this section, we begin our work. The notion of unitary elements in an
absolute matrix order unit space has been introduced by Karn and the author in
[25]. In this section, we study basic properties of unitary elements in
absolute matrix order unit spaces. We also study path homotopy equivalence of
unitary elements in absolute matrix order unit spaces. By path homotopy
equivalence, we define and study some variants of equivalence of unitary
elements which will help us to describe the Grothendieck group $K_{1}(V)$ of
an absolute matrix order unit space $V$ for unitary elements. The
$K_{1}$-group for absolute matrix order unit spaces is a generalization of
$K_{1}$-group for $C^{*}$-algebras. Later, we study order structure and
functoriality of $K_{1}(V).$ Now, we begin with the following definition:

######  Definition 3.1 ([25], Definition 3.1(2)).

Let $(V,e)$ be an absolute matrix order unit space and let $u\in M_{n}(V)$ for
some $n\in\mathbb{N}.$ Then $u$ is said to be _unitary_ , if
$|u|_{n}=e^{n}=|u^{*}|_{n}.$ We denote the set of all the unitary elements in
$M_{n}(V)$ by $\mathcal{U}_{n}(V).$ Under the identification
$M_{\infty}(V)=\displaystyle\bigcup_{n=1}^{\infty}M_{n}(V),$ the corresponding
set of unitaries is identified with
$\mathcal{U}_{\infty}(V)=\displaystyle\bigcup_{n=1}^{\infty}\mathcal{U}_{n}(V).$

We recall some properties of unitary elements.

######  Proposition 3.2.

Let $V$ be an absolute matrix order unit space. Then

  1. (1)

If $m,n\in\mathbb{N}$ and $u\in\mathcal{U}_{m}(V),~{}v\in\mathcal{U}_{n}(V),$
then $u\oplus v=\begin{bmatrix}u&0\\\
0&v\end{bmatrix}\in\mathcal{U}_{m+n}(V).$ In particular, $\oplus$ defines a
binary operation on $\mathcal{U}_{\infty}(V).$

  2. (2)

If $v\in\mathcal{U}_{n}(V)$ and $\alpha\in\mathcal{U}_{n}(\mathbb{C}),$ then
$\alpha^{*}v\alpha\in\mathcal{U}_{n}(V).$

  3. (3)

For each $v\in\mathcal{U}_{n}(V),w=\begin{bmatrix}0&v\\\
v^{*}&0\end{bmatrix}\in\mathcal{U}_{2n}(V)$ with $w^{*}=w.$

###### Proof.

Let $u\in\mathcal{U}_{m}(V),v\in\mathcal{U}_{n}(V)$ and
$\alpha\in\mathcal{U}_{n}(\mathbb{C})$ for some $m,n\in\mathbb{N}.$ Then, by
Definition 2.4(3), we get that $|u\oplus v|_{m+n}=\left|\begin{bmatrix}u&0\\\
0&v\end{bmatrix}\right|_{m+n}=\begin{bmatrix}|u|_{m}&0\\\
0&|v|_{n}\end{bmatrix}=\begin{bmatrix}e^{m}&0\\\
0&e^{n}\end{bmatrix}=e^{m+n}.$ Similarly, we get that $|(u\oplus
v)^{*}|_{m+n}=e^{m+n}.$ Thus $u\oplus v\in\mathcal{U}_{m+n}(V).$

Next, by Lemma 2.7, we get that
$|\alpha^{*}v\alpha|_{n}=\alpha^{*}|v|_{n}\alpha=\alpha^{*}e^{n}\alpha=\alpha^{*}\alpha
e^{n}=I_{n}e^{n}=e^{n}.$ Similarly, we get that
$|(\alpha^{*}v\alpha)^{*}|_{n}=|\alpha^{*}v^{*}\alpha|_{n}=e^{n}.$ Thus
$\alpha^{*}v\alpha\in\mathcal{U}_{n}(V).$

Put $w=\begin{bmatrix}0&v\\\ v^{*}&0\end{bmatrix}.$ Then $w^{*}=w$ and by
Proposition 2.6(2), we have $|w|_{m+n}=|v^{*}|_{n}\oplus|v|_{n}=e^{n}\oplus
e^{n}=e^{2n}$ so that $w\in\mathcal{U}_{2n}(V).$ ∎

By Theorem 2.3, we know that absolute matrix order unit spaces are normed
spaces. Thus, we can talk about norm continuous functions in an absolute
matrix order unit space. Next, we study the path homotopy of unitary elements
in absolute matrix order unit spaces.

######  Definition 3.3.

Let $(V,e)$ be an absolute matrix order unit space. Let
$u,v\in\mathcal{U}_{n}(V)$ for some $n\in\mathbb{N}.$ We say that $u$ is path
homotopic to $v$ (we write it, $u\sim_{h}v$) if there exists a continuous
function $f:[0,1]\to\mathcal{U}_{n}(V)$ such that $f(0)=u$ and $f(1)=v.$

The path homotopy of unitaries is an equivalence realtion.

######  Proposition 3.4.

Let $(V,e)$ be an absolute matrix order unit space and let $n\in\mathbb{N}.$
Then $\sim_{h}$ is an equivalence relation on $\mathcal{U}_{n}(V).$

###### Proof.

Let $u,v$ and $w\in\mathcal{U}_{n}(V).$ Put $f(t)=u$ for all $t\in[0,1].$ Then
$f$ is a continuous function with $f(0)=u=f(1).$ Thus $u\sim_{h}u.$

If $f_{1},f_{2}:[0,1]\to\mathcal{U}_{n}(V)$ be continuous functions with
$f_{1}(0)=u,f_{1}(1)=v=f_{2}(0)$ and $f_{2}(1)=w,$ then the functions
$g_{1},g_{2}:[0,1]\to\mathcal{U}_{n}(V)$ defined by

$g_{1}(t)=f_{1}(1-t)$

and

$g_{2}(t)=\begin{cases}f_{1}(2t)&\text{at t $\in$ [0,$\frac{1}{2}$]}\\\
f_{2}(2t-1)&\text{elsewhere}\end{cases}$

are also continuous with $g_{1}(0)=v,g_{1}(1)=u=g_{2}(0)$ and $g_{2}(1)=w.$
Thus $u\sim_{h}v$ and $v\sim_{h}w$ implies $v\sim_{h}u$ and $u\sim_{h}w.$ ∎

In next two results, we study some properties of the path homotopy of unitary
elements in absolute matrix order unit spaces.

######  Proposition 3.5.

Let $V$ be an absolute matrix order unit space. Then

  1. (1)

If $u\sim_{h}u^{\prime}$ and $v\sim_{h}v^{\prime}$ in $\mathcal{U}_{m}(V)$ and
$\mathcal{U}_{n}(V)$ respectively, then $u\oplus v\sim_{h}u^{\prime}\oplus
v^{\prime}$ in $\mathcal{U}_{m+n}(V).$

  2. (2)

If $u\sim_{h}v$ and $\alpha\sim_{h}\beta$ in $~{}\mathcal{U}_{n}(V)$ and
$~{}\mathcal{U}_{n}(\mathbb{C})$ respectively, then
$\alpha^{*}u\alpha\sim_{h}\beta^{*}v\beta$ in $\mathcal{U}_{n}(V)$.

###### Proof.

Let $f_{1}:[0,1]\to\mathcal{U}_{m}(V)$ and $f_{2}:[0,1]\to\mathcal{U}_{n}(V)$
be continuous functions such that $f_{1}(0)=u,f_{1}(1)=u^{\prime},f_{2}(0)=v$
and $f_{2}(1)=v^{\prime}.$ Put $g(t)=f_{1}(t)\oplus f_{2}(t)$ for all
$t\in[0,1].$ Then $g(0)=u\oplus v,g(1)=u^{\prime}\oplus v^{\prime}$ and by
Proposition 3.2(1), we get that $g(t)\in\mathcal{U}_{m+n}(V)$ for all
$t\in[0,1]$. By Theorem 2.3(2), we have
$\|g(t_{1})-g(t_{2})\|_{m+n}=\max\\{\|f_{1}(t_{1})-f_{1}(t_{2})\|_{m},\|f_{2}(t_{1})-f_{2}(t_{2})\|_{n}\\}$
for all $t_{1},t_{2}\in[0,1].$ Since $f_{1}$ and $f_{2}$ are continuous, we
get that $g:[0,1]\to\mathcal{U}_{m+n}(V)$ is also a continuous function. Thus
$u\oplus v\sim_{h}u^{\prime}\oplus v^{\prime}$ in $\mathcal{U}_{m+n}(V).$

Next, let $f_{1}:[0,1]\to\mathcal{U}_{n}(\mathbb{C})$ and
$f_{2}:[0,1]\to\mathcal{U}_{n}(V)$ are continuous functions such that
$f_{1}(0)=\alpha,f_{1}(1)=\beta,f_{2}(0)=u$ and $f_{2}(1)=v.$ Define
$f_{1}^{*}:[0,1]\to\mathcal{U}_{n}(\mathbb{C})$ such that
$f_{1}^{*}(t)=f_{1}(t)^{*}.$ Since $\|\delta^{*}\|=\|\delta\|$ for all
$\delta\in M_{n}$ and $f_{1}$ is continuous, thus $f_{1}^{*}$ is also
continuous. Put $g(t)=f_{1}^{*}(t)f_{2}(t)f_{1}(t)$ for all $t\in[0,1].$ Then
$g(0)=\alpha^{*}u\alpha,g(1)=\beta^{*}v\beta$ and by Proposition 3.2(2), we
get that $g(t)\in\mathcal{U}_{n}(V)$ for all $t\in[0,1].$ Now $\|\alpha
v\beta\|_{n}\leq\|\alpha\|\|v\|_{n}\|\beta\|$ for all $v\in M_{n}(V),\alpha$
and $\beta\in M_{n}$, therefore $g:[0,1]\to\mathcal{U}_{n}(V)$ is also a
continuous function. Hence $\alpha^{*}u\alpha\sim_{h}\beta^{*}v\beta$ in
$\mathcal{U}_{n}(V)$. ∎

######  Lemma 3.6.

Let $V$ be an absolute matrix order unit space. Then $u\oplus v\sim_{h}v\oplus
u$ in $\mathcal{U}_{m+n}(V)$ for all $u\in\mathcal{U}_{m}(V)$ and
$v\in\mathcal{U}_{n}(V).$

###### Proof.

Let $u\in\mathcal{U}_{m}(V)$ and $v\in\mathcal{U}_{n}(V).$ Put
$\alpha=\begin{bmatrix}0_{m,n}&I_{m}\\\ I_{n}&0_{n,m}\end{bmatrix}.$ Then
$\alpha\in\mathcal{U}_{m+n}(\mathbb{C})$ so that
$\sigma(\alpha)\subsetneq\mathbb{T},$ where $\sigma(\alpha)$ and $\mathbb{T}$
denote the spectrum of $\alpha$ and the unit circle in $\mathbb{C}$
respectively. Next, by [30, Lemma 2.1.3(ii)], we have $I_{m+n}\sim_{h}z$ in
$\mathcal{U}_{m+n}(\mathbb{C}).$ Thus, by Proposition 3.5(2), we get that
$u\oplus v=I_{m+n}\cdot(u\oplus v)\cdot I_{m+n}\sim_{h}z^{*}\cdot(u\oplus
v)\cdot z=v\oplus u$ in $\mathcal{U}_{m+n}(V).$ ∎

Now, we define a new relation on unitaries by help of path homotopy.

######  Definition 3.7.

Let $V$ be an absolute matrix order unit space. Define the relation $\sim_{1}$
on $\mathcal{U}_{\infty}(V)$ by given $u\in\mathcal{U}_{m}(V)$ and
$v\in\mathcal{U}_{n}(V),~{}u\sim_{1}v$ if and only if there exists
$k\in\mathbb{N},~{}k>max\\{m,n\\}$ such that $u\oplus e^{k-m}\sim_{h}v\oplus
e^{k-n}$ in $\mathcal{U}_{k}(V).$

The following result describes that new relation is stronger one.

######  Corollary 3.8.

Let $(V,e)$ be an absolute matrix order unit space. Then

  1. (1)

$\sim_{h}$ implies $\sim_{1}$ in $\mathcal{U}_{m}(V)$ for any
$m\in\mathbb{N}.$

  2. (2)

$\sim_{1}$ is an equivalence relation on $\mathcal{U}_{\infty}(V).$

###### Proof.

Let $u,v\in\mathcal{U}_{m}(V)$ with $u\sim_{h}v.$ Since $e^{n}\sim_{h}e^{n}$
for all $n\in\mathbb{N},$ by Proposition 3.5(1), we get $u\oplus
e^{k-m}=u\oplus e^{n}\sim_{h}v\oplus e^{n}=v\oplus e^{k-m}$ for $k=m+n>m.$
Thus $u\sim_{1}v$ in $\mathcal{U}_{m}(V).$

Next, let $u\in\mathcal{U}_{l}(V),v\in\mathcal{U}_{m}(V)$ and
$w\in\mathcal{U}_{n}(V)$ with $u\sim_{1}v$ and $v\sim_{1}w.$ Then $u\sim_{1}u$
as $u\sim_{h}u$ and there exist $k_{1},k_{2}\in\mathbb{N}$ with
$k_{1}>\max\\{l,m\\}$ and $k_{2}>\max\\{m,n\\}$ such that $u\oplus
e^{k_{1}-l}\sim_{h}v\oplus e^{k_{1}-m}$ and $v\oplus
e^{k_{2}-m}\sim_{h}w\oplus e^{k_{2}-n}$ in $\mathcal{U}_{k_{1}}(V)$ and
$\mathcal{U}_{k_{2}}(V)$ respectively. Put $k=k_{1}+k_{2}$ so that
$k>\max\\{l,m,n\\}$ and by Proposition 3.5(1), we have

$u\oplus e^{k-l}=(u\oplus e^{k_{1}-l})\oplus e^{k_{2}}\sim_{h}(v\oplus
e^{k_{1}-m})\oplus e^{k_{2}}=v\oplus e^{k-m}$

and

$v\oplus e^{k-m}=(v\oplus e^{k_{2}-m})\oplus e^{k_{1}}\sim_{h}(w\oplus
e^{k_{2}-n})\oplus e^{k_{1}}=w\oplus e^{k-n}$

in $\mathcal{U}_{k}(V).$ Now, by Proposition 3.4, we get $v\oplus
e^{k-m}\sim_{h}u\oplus e^{k-l}$ and $u\oplus e^{k-l}\sim_{h}w\oplus e^{k-n}$
in $\mathcal{U}_{k}(V).$ Thus $v\sim_{1}u$ and $u\sim_{1}w$ in
$\mathcal{U}_{\infty}(V).$

∎

We study some properties of newly defined equivalence relation.

######  Proposition 3.9.

Let $(V,e)$ be an absolute matrix order unit space and let
$u,v,w,u^{\prime},v^{\prime}\in\mathcal{U}_{\infty}(V).$ Then

  1. (1)

$(u\oplus v)\oplus w=u\oplus(v\oplus w).$

  2. (2)

$u\sim_{1}u\oplus e^{n}$ for all $n\in\mathbb{N}.$

  3. (3)

$u\oplus v\sim_{1}v\oplus u.$

  4. (4)

If $u\sim_{1}u^{\prime}$ and $v\sim_{1}v^{\prime},$ then $u\oplus
v\sim_{1}u^{\prime}\oplus v^{\prime}.$

###### Proof.

Let $u\in\mathcal{U}_{m}(V).$ For fixed $n\in\mathbb{N},$ we can find
$k\in\mathbb{N}$ such that $k>m+n.$ Then by Proposition 3.4, we get $u\oplus
e^{k-m}\sim_{h}u\oplus e^{k-m}=(u\oplus e^{n})\oplus e^{k-m-n}$ so that
$u\sim_{1}u\oplus e^{n}.$

By Lemma 3.6, we have $u\oplus v\sim_{h}v\oplus u.$ Now, by Corollary 3.8(1),
we get $u\oplus v\sim_{1}v\oplus u.$

Next, let $u^{\prime}\in\mathcal{U}_{l}(V),v\in\mathcal{U}_{m^{\prime}}(V)$
and $v^{\prime}\in\mathcal{U}_{l^{\prime}}(V)$ such that $u\sim_{1}u^{\prime}$
and $v\sim_{1}v^{\prime}.$ Then there exist
$k_{1},k_{2}\in\mathbb{N},k_{1}>\max\\{m,l\\},k_{2}>\max\\{m^{\prime},l^{\prime}\\}$
with $u\oplus e^{k_{1}-m}\sim_{h}u^{\prime}\oplus e^{k1-l}$ and $v\oplus
e^{k_{2}-m^{\prime}}\sim_{h}v^{\prime}\oplus e^{k_{2}-l^{\prime}}.$ Put
$k=k_{1}+k_{2}>\\{m+m^{\prime},l+l^{\prime}\\}.$ Using part (1), Lemma 3.6 and
Propositions 3.5(1) and 3.4 respectively, we get

$\displaystyle(u\oplus v)\oplus e^{k-m-m^{\prime}}$ $\displaystyle\sim_{h}$
$\displaystyle(u\oplus e^{k_{1}-m})\oplus(v\oplus e^{k_{2}-m^{\prime}})$
$\displaystyle\sim_{h}$ $\displaystyle(u^{\prime}\oplus
e^{k_{1}-l})\oplus(v^{\prime}\oplus e^{k_{2}-l^{\prime}})$
$\displaystyle\sim_{h}$ $\displaystyle(u^{\prime}\oplus v^{\prime})\oplus
e^{k-l-l^{\prime}}$

so that $u\oplus v\sim_{1}u^{\prime}\oplus v^{\prime}.$ ∎

Next, we again define a new relation on unitaries by $\sim_{1}.$

######  Definition 3.10.

Let $V$ be an absolute matrix order unit space. For
$u,v\in\mathcal{U}_{\infty}(V),$ we say that $u\approx_{1}v,$ if there exists
$w\in\mathcal{U}_{\infty}(V)$ such that $u\oplus w\sim_{1}v\oplus w.$

The following result shows that $\approx_{1}$ is stronger than $\sim_{1}.$

######  Proposition 3.11.

Let $V$ be an absolute matrix order unit space and let $n\in\mathbb{N}.$ Then

  1. (1)

$\sim_{1}$ implies $\approx_{1}.$

  2. (2)

$\approx_{1}$ is an equivalence relation on $\mathcal{U}_{\infty}(V).$

###### Proof.

By Proposition 3.4, we have $u\oplus w\sim_{h}u\oplus w$ for all
$u,w\in\mathcal{U}_{\infty}(V).$ Then by Corollary 3.8(1), we get $u\oplus
w\sim_{1}u\oplus w$ so that $u\approx_{1}u$ for all
$u\in\mathcal{U}_{\infty}(V).$

Let $u_{1},u_{2},u_{3}\in\mathcal{U}_{\infty}(V)$ such that
$u_{1}\approx_{1}u_{2}$ and $u_{2}\approx_{1}u_{3}.$ Then $u_{1}\oplus
w_{1}\sim_{1}u_{2}\oplus w_{1}$ and $u_{2}\oplus w_{2}\sim_{1}u_{3}\oplus
w_{2}$ for some $w_{1},w_{2}\in\mathcal{U}_{\infty}(V).$ By Corollary 3.8(2),
we have $u_{2}\oplus w_{1}\sim_{1}u_{1}\oplus w_{1}$ so that
$u_{2}\approx_{1}u_{1}.$ By Corollary 3.8 and by Proposition 3.9(4), we get
$(u_{1}\oplus w_{1})\oplus w_{2}\sim_{1}(u_{2}\oplus w_{1})\oplus w_{2}$ and
$(u_{2}\oplus w_{2})\oplus w_{1}\sim_{1}(u_{3}\oplus w_{2})\oplus w_{1}.$
Again applying Proposition 3.9(3), we have $w_{1}\oplus
w_{2}\sim_{1}w_{2}\oplus w_{1}.$ Then

$\displaystyle u_{1}\oplus(w_{1}\oplus w_{2})$ $\displaystyle=$
$\displaystyle(u_{1}\oplus w_{1})\oplus w_{2}\sim_{1}(u_{2}\oplus w_{1})\oplus
w_{2}$ $\displaystyle=$ $\displaystyle u_{2}\oplus(w_{1}\oplus
w_{2})\sim_{1}u_{2}\oplus(w_{2}\oplus w_{1})$ $\displaystyle=$
$\displaystyle(u_{2}\oplus w_{2})\oplus w_{1}\sim_{1}(u_{3}\oplus w_{2})\oplus
w_{1}$ $\displaystyle=$ $\displaystyle u_{3}\oplus(w_{2}\oplus
w_{1})\sim_{1}u_{3}\oplus(w_{1}\oplus w_{2}).$

Finally, by Corollary 3.8(2), we conclude $u_{1}\oplus(w_{1}\oplus
w_{2})\sim_{1}u_{3}\oplus(w_{1}\oplus w_{2}).$ Thus $u_{1}\approx_{1}u_{3}.$ ∎

The relation $\approx_{1}$ enjoys the following properties:

######  Corollary 3.12.

Let $(V,e)$ be an absolute matrix order unit space and let
$u,v,w,u^{\prime},v^{\prime}\in\mathcal{U}_{\infty}(V)$. Then

  1. $1.$

$u\approx_{1}u\oplus e^{n}$ for all $n\in\mathbb{N}.$

  2. $2.$

$u\oplus v\approx_{1}v\oplus u.$

  3. $3.$

If $u\approx_{1}u^{\prime}$ and $v\approx_{1}v^{\prime},$ then $u\oplus
v\approx_{1}u^{\prime}\oplus v^{\prime}.$

###### Proof.

By Propositions 3.9 and 3.11, it follows that $u\oplus v\approx_{1}v\oplus u$
and $u\approx_{1}u\oplus e^{n}$ for all $u,v\in\mathcal{U}_{\infty}(V)$ and
$n\in\mathbb{N}.$

Next, let $u,u^{\prime},v$ and $v^{\prime}\in\mathcal{U}_{\infty}(V)$ such
that $u\approx_{1}u^{\prime}$ and $v\approx_{1}v^{\prime}.$ Then $u\oplus
w_{1}\sim_{1}u^{\prime}\oplus w_{1}$ and $v\oplus
w_{2}\sim_{1}v^{\prime}\oplus w_{2}$ for some
$w_{1},w_{2}\in\mathcal{U}_{\infty}(V).$ By Proposition 3.9, we get

$\displaystyle(u\oplus v)\oplus(w_{1}\oplus w_{2})$ $\displaystyle=$
$\displaystyle u\oplus(v\oplus w_{1})\oplus w_{2}\sim_{1}u\oplus(w_{1}\oplus
v)\oplus w_{2}$ $\displaystyle=$ $\displaystyle(u\oplus w_{1})\oplus(v\oplus
w_{2})\sim_{1}(u^{\prime}\oplus w_{1})\oplus(v^{\prime}\oplus w_{2})$
$\displaystyle=$ $\displaystyle u^{\prime}\oplus(w_{1}\oplus v^{\prime})\oplus
w_{2}\sim_{1}u^{\prime}\oplus(v^{\prime}\oplus w_{1})\oplus w_{2}$
$\displaystyle=$ $\displaystyle(u^{\prime}\oplus v^{\prime})\oplus(w_{1}\oplus
w_{2}).$

Thus, by Corollary 3.8(2), we conclude that $(u\oplus v)\oplus(w_{1}\oplus
w_{2})\sim_{1}(u^{\prime}\oplus v^{\prime})\oplus(w_{1}\oplus w_{2})$ so that
$u\oplus v\approx_{1}u^{\prime}\oplus v^{\prime}.$ ∎

Now, we define a binary operatoin in the family of equivalence classes of
unitaries under $\approx_{1}.$

######  Proposition 3.13.

Let $(V,e)$ be an absolute matrix order unit space. For each
$u,v\in\mathcal{U}_{\infty}(V),$ let
$[u]=\\{w\in\mathcal{U}_{\infty}(V):w\approx_{1}u\\}$ and put
$[u]+[v]=[u\oplus v].$ Then, $+$ is a binary operation in the family of
equivalence classes $(\mathcal{U}_{\infty}(V)/\approx_{1},+)$. Also:

  1. (1)

$[u]+[e]=[u]$ for all $u\in\mathcal{U}_{\infty}(V);$

  2. (2)

$[u]+[v]=[v]+[u]$ for all $u,v\in\mathcal{U}_{\infty}(V);$

  3. (3)

$[u]+[w]=[v]+[w]$ for $u,v,w\in\mathcal{U}_{\infty}(V),$ then $[u]=[v].$

Thus, $(\mathcal{U}_{\infty}(V)\big{/}\approx_{1},+)$ is an abelian semi-group
satisfying the cancellation law.

###### Proof.

By Corollary 3.12(3), it follows that $+$ is well-defined on
$\mathcal{U}_{\infty}(V)\big{/}\approx_{1}.$ Note that (1) and (2) immediately
follow from 3.12(1) and (2) respectively. Next, we prove (3).

Let $u,v,w\in\mathcal{U}_{\infty}(V)$ such that $[u]+[w]=[v]+[w].$ Then
$u\oplus w\approx_{1}v\oplus w$ so that $u\oplus(w\oplus
x)\sim_{1}v\oplus(w\oplus x)$ for some $x\in\mathcal{U}_{\infty}(V).$ Thus,
$u\approx_{1}v$ so that $[u]=[v].$ ∎

Finally, the following result shows the importance of path homotopy of
unitaries to define a group.

######  Theorem 3.14.

Let $(V,e)$ be an absolute matrix order unit space and consider
$\mathcal{U}_{\infty}(V)\times\mathcal{U}_{\infty}(V).$ For all
$u_{1},u_{2},v_{1},v_{2}\in\mathcal{U}_{\infty}(V),$ we define
$(u_{1},v_{1})\equiv_{1}(u_{2},v_{2})$ if and only if $u_{1}\oplus
v_{2}\approx_{1}u_{2}\oplus v_{1}.$ Then, $\equiv_{1}$ is an equivalence
relation on $\mathcal{U}_{\infty}(V)\times\mathcal{U}_{\infty}(V).$ Consider:

$K_{1}(V)=\\{[(u,v)]:u,v\in\mathcal{U}_{\infty}(V)\\},$

where $[(u,v)]$ is the equivalence class of $(u,v)$ in
$(\mathcal{U}_{\infty}(V)\times\mathcal{U}_{\infty}(V),\equiv_{1}).$ For all
$u_{1},u_{2},v_{1},v_{2}\in\mathcal{U}_{\infty}(V),$ we write:

$[(u_{1},v_{1})]+[(u_{2},v_{2})]=[(u_{1}\oplus u_{2},v_{1}\oplus v_{2})].$

Then $(K_{1}(V),+)$ is an abelian group.

###### Proof.

It follows from Proposition 3.11(2) and Corollary 3.12 that the relation
$\equiv_{1}$ on $\mathcal{U}_{\infty}(V)\times\mathcal{U}_{\infty}(V)$ is
reflexive and symmetric. Let $(u_{1},v_{1})\equiv_{1}(u_{2},v_{2})$ and
$(u_{2},v_{2})\equiv_{1}(u_{3},v_{3})$ for some
$u_{1},u_{2},u_{3},v_{1},v_{2},v_{3}\in\mathcal{U}_{\infty}(V).$ Then
$u_{1}\oplus v_{2}\approx_{1}u_{2}\oplus v_{1}$ and $u_{2}\oplus
v_{3}\approx_{1}u_{3}\oplus v_{2}.$ By Proposition 3.11(2) and by Corollary
3.12(2) and (3), we get that $(u_{1}\oplus v_{3})\oplus(u_{2}\oplus
v_{2})\approx_{1}(u_{3}\oplus v_{1})\oplus(u_{2}\oplus v_{2})$ so that
$[u_{1}\oplus v_{3}]+[u_{2}\oplus v_{2}]=[u_{3}\oplus v_{1}]+[u_{2}\oplus
v_{2}].$ Then, by Proposition 3.13(3), we conclude that $[u_{1}\oplus
v_{3}]=[u_{3}\oplus v_{1}]$ so that $u_{1}\oplus v_{3}\approx_{1}u_{3}\oplus
v_{1}.$ Thus $(u_{1},v_{1})\equiv_{1}(u_{3},v_{3})$ so that $\equiv_{1}$ is
transitive. Hence, $\equiv_{1}$ is an equivalence relation on
$\mathcal{U}_{\infty}(V)\times\mathcal{U}_{\infty}(V).$

Now, we show that $+$ is well defined on $K_{1}(V).$ Let
$[(u_{1},v_{1})]=[(u_{1}^{\prime},v_{1}^{\prime})]$ and
$[(u_{2},v_{2})]=[(u_{2}^{\prime},v_{2}^{\prime})].$ Then
$(u_{1},v_{1})\equiv_{1}(u_{1}^{\prime},v_{1}^{\prime})$ and
$(u_{2},v_{2})\equiv_{1}(u_{2}^{\prime},v_{2}^{\prime})$ so that $u_{1}\oplus
v_{1}^{\prime}\approx_{1}u_{1}^{\prime}\oplus v_{1}$ and $u_{2}\oplus
v_{2}^{\prime}\approx_{1}u_{2}^{\prime}\oplus v_{2}.$ Again by Proposition
3.11(2) and by Corollary 3.12(2) and (3), we get that $(u_{1}\oplus
u_{2})\oplus(v_{1}^{\prime}\oplus
v_{2}^{\prime})\approx_{1}(u_{1}^{\prime}\oplus
u_{2}^{\prime})\oplus(v_{1}\oplus v_{2}).$ Thus
$[(u_{1},v_{1})]+[(u_{2},v_{2})]=[(u_{1}^{\prime},v_{1}^{\prime})]+[(u_{2}^{\prime},v_{2}^{\prime})]$
so that $+$ is well defined.

By Corollary 3.12(2) and (3), we have:

$(u_{1}\oplus u_{2})\oplus(v_{2}\oplus v_{1})\approx_{1}(u_{2}\oplus
u_{1})\oplus(v_{1}\oplus v_{2})$

for all $u_{1},u_{2},v_{1},v_{2}\in\mathcal{U}_{\infty}(V),$ so that $+$ is
commutative in $K_{1}(V).$

Let $u,v\in\mathcal{U}_{\infty}(V).$ By Corollary 3.12(1) and (3), we have
$(u\oplus e)\oplus(v\oplus e)\approx_{1}u\oplus v.$ Thus $[(e,e)]$ is an
identity element in $K_{1}(V).$

Associativity of $+$ on $K_{1}(V)$ follows from Proposition 3.9(1).

For any $u,v\in\mathcal{U}_{\infty}(V),$ we have $[(u\oplus v,v\oplus
u)]=[(e,e)].$ Hence $[(v,u)]$ is an inverse element of $[(u,v)]$ in
$K_{1}(V).$ ∎

######  Remark 3.15.

The abelian group $K_{1}(V)$ is called the $K_{1}(V)$-group of the absolute
matrix order unit space $V.$ If $A$ is a $C^{*}$-algebra, then $K_{1}(A)$ is
Grothendieck group of $A$ for unitary elements. Thus $K_{1}$-group for
absolute matrix order unit spaces is a generalization of $K_{1}$-group for
$C^{*}$-algebras.

###  3.1. Order structure in $K_{1}$

In this subsection, we prove that $K_{1}(V)$ bears order structure on it. We
start with the following discussion:

Let $A$ be a unital $C^{*}$-algebra with unity $1_{A}.$ Put
$K_{1}(A)^{+}=\\{[(v,1_{A})]:v\in\mathcal{U}_{\infty}(A)\\}.$ Then
$K_{1}(A)^{+}$ is a group cone and generating. But $K_{1}(A)^{+}$ is not
proper. By Whitehead lemma $($see [30, Lemma 2.1.5]$)$, we have $v\oplus
v^{*}\sim_{h}1_{A}^{2n}$ for any $v\in\mathcal{U}_{n}(V),n\in\mathbb{N}.$ Then
$[(v,1_{A})]=[(v,1_{A}^{n})]=[(1_{A}^{n},v^{*})]=[(1_{A},v^{*})]$ so that
$\pm[(v,1_{A})]\in K_{1}(A)^{+}.$ Thus $K_{1}(A)^{+}$ is not proper.

Under certain conditions, we prove that $(K_{1}(V),K_{1}(V)^{+})$ is an
ordered abelian group with a distinguished order unit. For that, we need to
prove the following result:

######  Theorem 3.16.

Let $V$ be an absolute matrix order unit space. Put
$K_{1}(V)^{+}=\\{[(v,e)]:v\in\mathcal{U}_{\infty}(V)\\}.$ Then

  1. (1)

$K_{1}(V)^{+}$ is a group cone in $K_{1}(V)$.

  2. (2)

$K_{1}(V)^{+}$ is generating.

  3. (3)

If $v\oplus v^{*}\sim_{h}e^{2n}$ in $\mathcal{U}_{2n}(V)$ for all
$n\in\mathbb{N}$ and $v\in\mathcal{U}_{n}(V),$ then for each $g\in K_{1}(V),$
there exists $m\in\mathbb{N}$ such that $-m[(e,e)]\leq g\leq m[(e,e)].$

###### Proof.

It is routine to verify (1) and (2). Next, we prove (3). Assume that $v\oplus
v^{*}\sim_{h}e^{2n}$ in $\mathcal{U}_{2n}(V)$ for all $n\in\mathbb{N}$ and
$v\in\mathcal{U}_{n}(V).$ First, let $v\in\mathcal{U}_{n}(V)$ for some
$n\in\mathbb{N}.$ By part (1), we have

$\displaystyle[(v,e)]$ $\displaystyle=$ $\displaystyle[(v,e^{n})]$
$\displaystyle\leq$ $\displaystyle[(v,e^{n})]+[(v^{*},e^{n})]$
$\displaystyle=$ $\displaystyle[(v\oplus v^{*},e^{n}\oplus e^{n})]$
$\displaystyle=$ $\displaystyle[(e^{2n},e^{2n})]$ $\displaystyle=$
$\displaystyle 2n[(e,e)].$

Next, let $g\in K_{1}(V).$ Then by part (2), we have $g=[(u,e)]-[(v,e)]$ for
some $u,v\in\mathcal{U}_{\infty}(V).$ Without loss of generality, we may
assume that $u,v\in\mathcal{U}_{n}(V)$ for some $n\in\mathbb{N}.$ Since
$-[(v,e)]\leq g\leq[(u,e)],$ we get that $-2n[(e,e)]\leq g\leq 2n[(e,e)].$ Put
$m=2n.$ Thus $-m[(e,e)]\leq g\leq m[(e,e)].$ ∎

######  Corollary 3.17.

Let $(V,e)$ be an absolute matrix order unit space. Then by Theorem 3.16, we
conclude that $(K_{1}(V),K_{1}(V)^{+})$ is an ordered abelian group. Moreover,
if $v\oplus v^{*}\sim_{h}e^{2n}$ in $\mathcal{U}_{2n}(V)$ for all
$n\in\mathbb{N}$ and $v\in\mathcal{U}_{n}(V),$ then $(K_{1}(V),K_{1}(V)^{+})$
is an ordered abelian group with distinguished order unit $[(e,e)].$

###  3.2. Functoriality of $K_{1}$

Let $V$ be an absolute matrix order unit space. Then $v\longmapsto[(v,e)]$
defines a map $\Upsilon_{V}:\mathcal{U}_{\infty}(V)\to K_{1}(V)$. If $V$ and
$W$ are complex vector spaces and if $\phi:V\to W$ be a linear map, we denote
the corresponding map from $M_{\infty}(V)$ to $M_{\infty}(W)$ again by $\phi$.
In this sense, $\phi_{\mid_{M_{n}(V)}}=\phi_{n}$ for all $n\in\mathbb{N}$. In
the next result, we describe the functorial nature of $K_{1}$. For this, first
we prove the following commutative property of $K_{1}(V)$:

######  Theorem 3.18.

Let $(V,e_{V})$ and $(W,e_{W})$ be absolute matrix order unit spaces and let
$\phi:V\to W$ be a unital completely $|\cdot|$-preserving map. Then there
exists a unique group homomorphism $K_{1}(\phi):K_{1}(V)\to K_{1}(W)$ such
that the following diagram commutes:

${\mathcal{U}_{\infty}(V)}$${\mathcal{U}_{\infty}(W)}$${K_{1}(V)}$${K_{1}(W)}$$\scriptstyle{\phi}$$\scriptstyle{\Upsilon_{V}}$$\scriptstyle{\Upsilon_{W}}$$\scriptstyle{K_{1}(\phi)}$

###### Proof.

Let $n\in\mathbb{N}$ and $v\in\mathcal{U}_{n}(V).$ Since
$|v|_{n}=e^{n}=|v^{*}|_{n}$ and $\phi$ is unital completely
$|\cdot|$-preserving map, we get that
$|\phi(v)|_{n}=\phi(|v|_{n})=\phi(e_{V}^{n})=\phi(e_{V})^{n}=e_{W}^{n}$ and
$|\phi(v)^{*}|_{n}=|\phi(v^{*})|_{n}=\phi(|v^{*}|_{n})=\phi(e_{v}^{n})=e_{W}^{n}$
so that $\phi(v)\in\mathcal{U}_{n}(W).$ Thus
$\phi(\mathcal{U}_{\infty}(V))\subset\mathcal{U}_{\infty}(W).$

By [24, Theorems 3.7 and 3.8], we get that $\phi$ is a contraction on
${M_{n}(V)}_{sa}$ for each $n\in\mathbb{N}.$ By Theorem 2.3(5), we conclude
that

$\displaystyle\|\phi(v)\|_{n}$ $\displaystyle=$
$\displaystyle\left\|\begin{bmatrix}0&\phi(v)\\\
\phi(v)^{*}&0\end{bmatrix}\right\|_{2n}$ $\displaystyle=$
$\displaystyle\left\|\begin{bmatrix}0&\phi(v)\\\
\phi(v^{*})&0\end{bmatrix}\right\|_{2n}$ $\displaystyle=$
$\displaystyle\left\|\phi\left(\begin{bmatrix}0&v\\\
v^{*}&0\end{bmatrix}\right)\right\|_{2n}$ $\displaystyle\leq$
$\displaystyle\left\|\begin{bmatrix}0&v\\\ v^{*}&0\end{bmatrix}\right\|_{2n}$
$\displaystyle=$ $\displaystyle\|v\|_{n}$

so that $\phi_{n}$ is a contraction on $M_{n}(V)$. Then $\phi$ is completely
contraction.

Now, let $w\in\mathcal{U}_{n}(V)$ such that $v\sim_{h}w$. As $\phi_{n}$ is
continuous, we get $\phi(v)\sim_{h}\phi(w)$ in $\mathcal{U}_{n}(W)$.

Next, put $K_{1}(\phi)([(v,w)])=[(\phi(v),\phi(w))]$ for each $[(v,w)]\in
K_{1}(V)$. We show that $K_{1}(\phi)$ is well-defined. Let
$[(v_{1},w_{1})]=[(v_{2},w_{2})]$ for some
$v_{1},v_{2},w_{1},w_{2}\in\mathcal{U}_{\infty}(V).$ Then there exists
$u\in\mathcal{U}_{\infty}(V)$ such that $v_{1}\oplus w_{2}\oplus
u\sim_{h}v_{2}\oplus w_{1}\oplus u.$ Thus
$\phi(v_{1})\oplus\phi(w_{2})\oplus\phi(u)\sim_{h}\phi(v_{2})\oplus\phi(w_{1})\oplus\phi(u)$
so that $[(\phi(v_{1}),\phi(w_{1}))]=[(\phi(v_{2}),\phi(w_{2}))].$ Hence
$K_{1}(\phi)$ is well-defined. For all $[(v_{1},w_{1})],[(v_{2},w_{2})]\in
K_{1}(V)$, we have that

$\displaystyle K_{1}(\phi)([(v_{1},w_{1})]+[(v_{2},w_{2})])$ $\displaystyle=$
$\displaystyle K_{1}(\phi)([(v_{1}\oplus v_{2},w_{1}\oplus w_{2})])$
$\displaystyle=$ $\displaystyle[(\phi(v_{1}\oplus v_{2}),\phi(w_{1}\oplus
w_{2}))]$ $\displaystyle=$
$\displaystyle[(\phi(v_{1})\oplus\phi(v_{2}),\phi(w_{1})\oplus\phi(w_{2}))]$
$\displaystyle=$
$\displaystyle[(\phi(v_{1}),\phi(w_{1}))]+[(\phi(v_{2}),\phi(w_{2}))]$
$\displaystyle=$ $\displaystyle
K_{1}(\phi)([(v_{1},w_{1})])+K_{1}(\phi)([(v_{2},w_{2})])$

so that $K_{1}(\phi)$ is a group homomorphism. By construction $K_{1}$
satisfies the diagram.

Uniqueness of $K_{1}(\phi)$:- Let $\mathcal{H}:K_{1}(V)\to K_{1}(W)$ be a
group homomorphism satisfying the same diagram. Then
$K_{1}(\phi)(\Upsilon_{V}(v))=\Upsilon_{W}(\phi(v))=\mathcal{H}(\Upsilon_{V}(v))$
for all $v\in\mathcal{U}_{\infty}(V)$. Thus we get that

$\displaystyle K_{1}(\phi)([(v,w)])$ $\displaystyle=$ $\displaystyle
K_{1}(\phi)([(v,e)]-[(w,e)])$ $\displaystyle=$ $\displaystyle
K_{1}(\phi)(\Upsilon_{V}(v)-\Upsilon_{V}(w))$ $\displaystyle=$ $\displaystyle
K_{1}(\phi)(\Upsilon_{V}(v))-K_{1}(\phi)(\Upsilon_{V}(w))$ $\displaystyle=$
$\displaystyle\mathcal{H}(\Upsilon_{V}(v))-\mathcal{H}(\Upsilon_{V}(w))$
$\displaystyle=$ $\displaystyle\mathcal{H}(\Upsilon_{V}(v)-\Upsilon_{V}(w))$
$\displaystyle=$ $\displaystyle\mathcal{H}([(v,w)])$

for all $[(v,w)]\in K_{1}(V)$. Hence $K_{1}(\phi)=\mathcal{H}$. ∎

Let $V$ and $W$ be absolute matrix order unit spaces. We denote the identity
maps on $K_{1}(V)$ by $I_{K_{1}(V)}.$

######  Corollary 3.19.

Let $U,V$ and $W$ be absolute matrix order unit spaces, and $\phi:U\to V$ and
$\psi:V\to W$ be unital completely $|\cdot|$-preserving maps. Then

  1. (a)

$K_{1}(I_{V})=I_{K_{1}(V)};$

  2. (b)

$K_{1}(\psi\circ\phi)=K_{1}(\psi)\circ K_{1}(\phi),$

###### Proof.

  1. (a)

Let $v,w\in\mathcal{U}_{\infty}(V)$. Then

$\displaystyle K_{1}(I_{V})([(v,w)])$ $\displaystyle=$
$\displaystyle[(I_{V}(v),I_{V}(w))]$ $\displaystyle=$ $\displaystyle[(v,w)]$

so that by Theorem 3.18, $K_{1}(I_{V})=I_{K_{1}(V)}$.

  2. (b)

For any $[(u,v)]\in K_{1}(U)$, we get that

$\displaystyle K_{1}(\psi\circ\phi)([(u,v)])$ $\displaystyle=$
$\displaystyle[(\psi\circ\phi(u),\psi\circ\phi(v))]$ $\displaystyle=$
$\displaystyle[(\psi(\phi(u)),\psi(\phi(v)))]$ $\displaystyle=$ $\displaystyle
K_{1}(\psi)[(\phi(u),\phi(v))]$ $\displaystyle=$ $\displaystyle
K_{1}(\psi)(K_{1}(\phi)([(u,v)]))$ $\displaystyle=$ $\displaystyle
K_{1}(\psi)\circ K_{1}(\phi)([(u,v)]).$

Thus, again by Theorem 3.18, we conclude that
$K_{1}(\psi\circ\phi)=K_{1}(\psi)\circ K_{1}(\phi)$.

∎

######  Remark 3.20.

It follows from Corollary 3.19 that $K_{1}$ is a functor from category of
absolute matrix order unit spaces with morphisms as unital completely
$|\cdot|$-preserving maps to category of abelian groups.

In the end of the subsection, we note the following isomorphic property of
$K_{1}.$

######  Corollary 3.21.

Let $V$ and $W$ be isomorphic absolute matrix order unit spaces (isomorphic in
the sense that there exists a unital, bijective completely
$|\cdot|$-preserving map between $V$ and $W$). Then $K_{1}(V)$ and $K_{1}(W)$
are group isomorphic.

###### Proof.

Let $\phi:V\to W$ be unital completely $|\cdot|$-preserving map. Then
$\phi^{-1}$ is also unital completely $|\cdot|$-preserving map. Since
$\phi^{-1}\circ\phi=I_{V}$ and $\phi\circ\phi^{-1}=I_{W}$, by Corollary
3.19(a) and (b), we get that $K_{1}(\phi^{-1})\circ K_{1}(\phi)=I_{K_{1}(V)}$
and $K_{1}(\phi)\circ K_{1}(\phi^{-1})=I_{K_{1}(W)}$. Thus
$K_{1}(\phi):K_{1}(V)\to K_{1}(W)$ is a surjective group isomorphism and
$K_{1}(\phi)^{-1}=K_{1}(\phi^{-1})$. Hence $K_{1}(V)$ and $K_{1}(W)$ are group
isomorphic. ∎

##  4\. $K$-group corresponding to an absolute matrix order unit space

This section is very similar to the previous one except few results.
Nevertheless, for the sake of completeness and accuracy of results, let’s have
a quick look at all the results with proofs as we need some results of this
section for the next section, where we derive a relation among
$K_{0}(V),K_{1}(V)$ and $K(V)$ for an absolute matrix order unit space $V$.
Note that $K(V)$ is described in this section. Let’s start the section.

The notion of partial unitary elements in an absolute matrix order unit space
has been introduced by Karn and the author in [25]. In this section, we study
basic properties of partial unitary elements in absolute matrix order unit
spaces. We also study path homotopy equivalence of partial unitary elements in
absolute matrix order unit spaces. By path homotopy equivalence, we define and
study some variants of equivalence of partial unitary elements which will help
us to describe the Grothendieck group $K(V)$ of an absolute matrix order unit
space $V$ for partial unitary elements. The $K$-group for absolute matrix
order unit spaces is a generalization of $K$-group for $C^{*}$-algebras.
Later, we study order structure and functoriality of $K(V).$ Now, we begin
with the following definition:

######  Definition 4.1 ([25], Definition 3.1(5)).

Let $V$ be an absolute matrix order unit space and let $u\in M_{n}(V)$ for
some $n\in\mathbb{N}.$ Then $u$ is said to be _partial unitary_ , if
$|u|_{n}=|u^{*}|_{n}$ is an order projection. We denote the set of all the
partial unitary elements in $M_{n}(V)$ by $\mathcal{PU}_{n}(V).$ Under the
identification $M_{\infty}(V)=\displaystyle\bigcup_{n=1}^{\infty}M_{n}(V),$
the corresponding set of partial unitaries is identified with
$\mathcal{PU}_{\infty}(V)=\displaystyle\bigcup_{n=1}^{\infty}\mathcal{PU}_{n}(V).$

We recall some properties of partial unitary elements.

######  Proposition 4.2.

Let $V$ be an absolute matrix order unit space. Then

  1. (1)

If $m,n\in\mathbb{N}$ and
$u\in\mathcal{PU}_{m}(V),~{}v\in\mathcal{PU}_{n}(V),$ then $u\oplus
v=\begin{bmatrix}u&0\\\ 0&v\end{bmatrix}\in\mathcal{PU}_{m+n}(V).$ In
particular, $\oplus$ defines a binary operation on $\mathcal{PU}_{\infty}(V).$

  2. (2)

If $v\in\mathcal{PU}_{n}(V)$ and $\alpha\in\mathcal{U}_{n}(\mathbb{C}),$ then
$\alpha^{*}v\alpha\in\mathcal{PU}_{n}(V).$

  3. (3)

For each $v\in\mathcal{PI}_{m,n}(V),w=\begin{bmatrix}0&v\\\
v^{*}&0\end{bmatrix}\in\mathcal{PI}_{m+n}(V)$ with $w^{*}=w$ so that
$w\in\mathcal{PU}_{m+n}(V)$

###### Proof.

Let $u\in\mathcal{PU}_{m}(V)$ and $v\in\mathcal{PU}_{n}(V).$ Then
$|u|_{m}=|u^{*}|_{m}\in\mathcal{OP}_{m}(V)$ and
$|v|_{n}=|v^{*}|_{n}\in\mathcal{OP}_{n}(V).$ By Proposition 2.17, we get that
$|u\oplus
v|_{m+n}=|u|_{m}\oplus|v|_{n}=|u^{*}|_{m}\oplus|v^{*}|_{n}=|u^{*}\oplus
v^{*}|_{m+n}=|(u\oplus v)^{*}|_{m+n}\in\mathcal{OP}_{m+n}(V).$ Thus $u\oplus
v\in\mathcal{PU}_{m+n}(V).$

Next, by Lemma 2.7 and by Corollary 2.18, we get
$|\alpha^{*}v\alpha|_{n}=\alpha^{*}|v|_{n}\alpha=\alpha^{*}|v^{*}|_{n}\alpha=|(\alpha^{*}v\alpha)^{*}|_{n}$
and $\alpha^{*}v\alpha\in\mathcal{PI}_{n}(V)$. Thus
$\alpha^{*}v\alpha\in\mathcal{PU}_{n}(V).$

Let $v\in\mathcal{PI}_{m,n}(V).$ Thus $|v^{*}|_{n,m}\in\mathcal{OP}_{m}(V)$
and $|v|_{m,n}\in\mathcal{OP}_{n}(V).$ Put $w=\begin{bmatrix}0&v\\\
v^{*}&0\end{bmatrix}.$ Then $w^{*}=w$ and by Proposition 2.17, we have
$|w|_{2n}=|v^{*}|_{n}\oplus|v|_{n}\in\mathcal{OP}_{m+n}(V)$ so that
$w\in\mathcal{PI}_{m+n}(V).$ Hence $w\in\mathcal{PU}_{m+n}(V).$ ∎

Next, we study the path homotopy of partial unitary elements in absolute
matrix order unit spaces.

######  Definition 4.3.

Let $V$ be an absolute matrix order unit space. Let
$u,v\in\mathcal{PU}_{n}(V)$ for some $n\in\mathbb{N}.$ We say that $u$ is path
homotopic to $v$ (we continue to write it, $u\sim_{h}v$) if there exists a
continuous function $f:[0,1]\to\mathcal{PU}_{n}(V)$ such that $f(0)=u$ and
$f(1)=v.$

Following the proof mentioned in Proposition 3.4, we see that the path
homotopy of partial unitaries is also an equivalence realtion.

######  Proposition 4.4.

Let $V$ be an absolute matrix order unit space and let $n\in\mathbb{N}.$ Then
$\sim_{h}$ is an equivalence relation on $\mathcal{PU}_{n}(V).$

In next two results, we study some properties of the path homotopy of partial
unitary elements in absolute matrix order unit spaces.

######  Proposition 4.5.

Let $V$ be an absolute matrix order unit space. Then

  1. (1)

If $u\sim_{h}u^{\prime}$ and $v\sim_{h}v^{\prime}$ in $\mathcal{PU}_{m}(V)$
and $\mathcal{PU}_{n}(V)$ respectively, then $u\oplus
v\sim_{h}u^{\prime}\oplus v^{\prime}$ in $\mathcal{PU}_{m+n}(V).$

  2. (2)

If $u\sim_{h}v$ and $\alpha\sim_{h}\beta$ in $~{}\mathcal{PU}_{n}(V)$ and
$~{}\mathcal{U}_{n}(\mathbb{C})$ respectively, then
$\alpha^{*}u\alpha\sim_{h}\beta^{*}v\beta$ in $\mathcal{PU}_{n}(V)$.

###### Proof.

Let $f_{1}:[0,1]\to\mathcal{PU}_{m}(V)$ and
$f_{2}:[0,1]\to\mathcal{PU}_{n}(V)$ be continuous functions such that
$f_{1}(0)=u,f_{1}(1)=u^{\prime},f_{2}(0)=v$ and $f_{2}(1)=v^{\prime}.$ Define
$g:[0,1]\to\mathcal{PU}_{m+n}(V)$ such that $g(t)=f_{1}(t)\oplus f_{2}(t)$ for
all $t\in[0,1].$ Then $g(0)=u\oplus v,g(1)=u^{\prime}\oplus v^{\prime}$ and
$g$ is also continuous. Thus $u\oplus v\sim_{h}u^{\prime}\oplus v^{\prime}$ in
$\mathcal{PU}_{m+n}(V).$

Next, let $f_{1}:[0,1]\to\mathcal{U}_{n}(\mathbb{C})$ and
$f_{2}:[0,1]\to\mathcal{PU}_{n}(V)$ are continuous functions such that
$f_{1}(0)=\alpha,f_{1}(1)=\beta,f_{2}(0)=u$ and $f_{2}(1)=v.$ Define
$g:[0,1]\to\mathcal{PU}_{n}(V)$ such that $g(t)=f_{1}^{*}(t)f_{2}(t)f_{1}(t)$
for all $t\in[0,1].$ Then $g(0)=\alpha^{*}u\alpha,g(1)=\beta^{*}v\beta$ and
$g$ is continuous. Hence $\alpha^{*}u\alpha\sim_{h}\beta^{*}v\beta$ in
$\mathcal{PU}_{n}(V).$ ∎

######  Lemma 4.6.

Let $V$ be an absolute matrix order unit space. Then $u\oplus v\sim_{h}v\oplus
u$ in $\mathcal{PU}_{m+n}(V)$ for all $u\in\mathcal{PU}_{m}(V)$ and
$v\in\mathcal{PU}_{n}(V).$

###### Proof.

Again, by $I_{m+n}\sim_{h}z$ in $\mathcal{U}_{m+n}(\mathbb{C}),$ we get that
$u\oplus v=I_{m+n}\cdot(u\oplus v)\cdot I_{m+n}\sim_{h}z^{*}\cdot(u\oplus
v)\cdot z=v\oplus u$ in $\mathcal{PU}_{m+n}(V).$ ∎

Now, we define a new relation on partial unitaries by help of path homotopy.

######  Definition 4.7.

Let $V$ be an absolute matrix order unit space. Define the relation $\sim_{K}$
on $\mathcal{PU}_{\infty}(V)$ by given $u\in\mathcal{PU}_{m}(V)$ and
$v\in\mathcal{PU}_{n}(V),~{}u\sim_{K}v$ if and only if there exists
$k\in\mathbb{N},~{}k>max\\{m,n\\}$ such that $u\oplus 0_{k-m}\sim_{h}v\oplus
0_{k-n}$ in $\mathcal{PU}_{k}(V).$

The following result describes that new relation is stronger one.

######  Corollary 4.8.

Let $V$ be an absolute matrix order unit space. Then

  1. (1)

$\sim_{h}$ implies $\sim_{K}$ in $\mathcal{PU}_{m}(V)$ for any
$m\in\mathbb{N}.$

  2. (2)

$\sim_{K}$ is an equivalence relation on $\mathcal{PU}_{\infty}(V).$

###### Proof.

Let $m\in\mathbb{N}$ and $u,v\in\mathcal{PU}_{m}(V)$ with $u\sim_{h}v.$ Since
$0_{n}\sim_{h}0_{n}$ for all $n\in\mathbb{N},$ by Proposition 4.5(1), we get
$u\oplus 0_{k-m}=u\oplus 0_{n}\sim_{h}v\oplus 0_{n}=v\oplus 0_{k-m}$ for
$k=m+n>m.$ Thus $u\sim_{K}v$ in $\mathcal{PU}_{m}(V).$

Next, let $u\in\mathcal{PU}_{l}(V),v\in\mathcal{PU}_{m}(V)$ and
$w\in\mathcal{PU}_{n}(V)$ with $u\sim_{K}v$ and $v\sim_{K}w.$ Then
$u\sim_{K}u$ as $u\sim_{h}u$ and there exist $k_{1},k_{2}\in\mathbb{N}$ with
$k_{1}>\max\\{l,m\\}$ and $k_{2}>\max\\{m,n\\}$ such that $u\oplus
0_{k_{1}-l}\sim_{h}v\oplus 0_{k_{1}-m}$ and $v\oplus
0_{k_{2}-m}\sim_{h}w\oplus 0_{k_{2}-n}$ in $\mathcal{PU}_{k_{1}}(V)$ and
$\mathcal{PU}_{k_{2}}(V)$ respectively. Put $k=k_{1}+k_{2}$ so that
$k>\max\\{l,m,n\\}$ and by Proposition 4.5(1), we have

$u\oplus 0_{k-l}=(u\oplus 0_{k_{1}-l})\oplus 0_{k_{2}}\sim_{h}(v\oplus
0_{k_{1}-m})\oplus 0_{k_{2}}=v\oplus 0_{k-m}$

and

$v\oplus 0_{k-m}=(v\oplus 0_{k_{2}-m})\oplus 0_{k_{1}}\sim_{h}(w\oplus
0_{k_{2}-n})\oplus 0_{k_{1}}=w\oplus 0_{k-n}$

in $\mathcal{PU}_{k}(V).$ Now, by Proposition 4.4, we get $v\oplus
0_{k-m}\sim_{h}u\oplus 0_{k-l}$ and $u\oplus 0_{k-l}\sim_{h}w\oplus 0_{k-n}$
in $\mathcal{PU}_{k}(V).$ Thus $v\sim_{K}u$ and $u\sim_{K}w$ in
$\mathcal{PU}_{\infty}(V).$

∎

We study some properties of newly defined equivalence relation.

######  Proposition 4.9.

Let $V$ be an absolute matrix order unit space and let
$u,v,w,u^{\prime},v^{\prime}\in\mathcal{PU}_{\infty}(V).$ Then

  1. (1)

$(u\oplus v)\oplus w=u\oplus(v\oplus w).$

  2. (2)

$u\sim_{K}u\oplus 0_{n}$ for all $n\in\mathbb{N}.$

  3. (3)

$u\oplus v\sim_{K}v\oplus u.$

  4. (4)

If $u\sim_{K}u^{\prime}$ and $v\sim_{K}v^{\prime},$ then $u\oplus
v\sim_{K}u^{\prime}\oplus v^{\prime}.$

###### Proof.

Let $u\in\mathcal{PU}_{m}(V).$ For fixed $n\in\mathbb{N},$ we can find
$k\in\mathbb{N}$ such that $k>m+n.$ Then by Proposition 4.4, we get $u\oplus
0_{k-m}\sim_{h}u\oplus 0_{k-m}=(u\oplus 0_{n})\oplus 0_{k-m-n}$ so that
$u\sim_{K}u\oplus 0_{n}.$

By Lemma 4.6, we have $u\oplus v\sim_{h}v\oplus u.$ Now, by Corollary 4.8(1),
we get $u\oplus v\sim_{K}v\oplus u.$

Next, let $u^{\prime}\in\mathcal{PU}_{l}(V),v\in\mathcal{PU}_{m^{\prime}}(V)$
and $v^{\prime}\in\mathcal{PU}_{l^{\prime}}(V)$ such that
$u\sim_{K}u^{\prime}$ and $v\sim_{K}v^{\prime}.$ Then there exist
$k_{1},k_{2}\in\mathbb{N},k_{1}>\max\\{m,l\\},k_{2}>\max\\{m^{\prime},l^{\prime}\\}$
with $u\oplus 0_{k_{1}-m}\sim_{h}u^{\prime}\oplus 0_{k1-l}$ and $v\oplus
0_{k_{2}-m^{\prime}}\sim_{h}v^{\prime}\oplus 0_{k_{2}-l^{\prime}}.$ Put
$k=k_{1}+k_{2}>\\{m+m^{\prime},l+l^{\prime}\\}.$ Using part (1), Lemma 4.6 and
Propositions 4.5(1) and 4.4 respectively, we get

$\displaystyle(u\oplus v)\oplus 0_{k-m-m^{\prime}}$ $\displaystyle\sim_{h}$
$\displaystyle(u\oplus 0_{k_{1}-m})\oplus(v\oplus 0_{k_{2}-m^{\prime}})$
$\displaystyle\sim_{h}$ $\displaystyle(u^{\prime}\oplus
0_{k_{1}-l})\oplus(v^{\prime}\oplus 0_{k_{2}-l^{\prime}})$
$\displaystyle\sim_{h}$ $\displaystyle(u^{\prime}\oplus v^{\prime})\oplus
0_{k-l-l^{\prime}}$

so that $u\oplus v\sim_{K}u^{\prime}\oplus v^{\prime}.$ ∎

Next, we again define a new relation on partial unitaries by $\sim_{K}.$

######  Definition 4.10.

Let $V$ be an absolute matrix order unit space. For
$u,v\in\mathcal{PU}_{\infty}(V),$ we say that $u\approx_{K}v,$ if there exists
$w\in\mathcal{PU}_{\infty}(V)$ such that $u\oplus w\sim_{K}v\oplus w.$

The following result shows that $\approx_{K}$ is stronger than $\sim_{K}.$

######  Proposition 4.11.

Let $V$ be an absolute matrix order unit space and let $n\in\mathbb{N}.$ Then

  1. (1)

$\sim_{K}$ implies $\approx_{K}$ in $\mathcal{PU}_{\infty}(V).$

  2. (2)

$\approx_{K}$ is an equivalence relation on $\mathcal{PU}_{\infty}(V).$

###### Proof.

By Proposition 4.4, we have $u\oplus w\sim_{h}u\oplus w$ for all
$u,w\in\mathcal{PU}_{\infty}(V).$ Then by Corollary 4.8(1), we get $u\oplus
w\sim_{K}u\oplus w$ so that $u\approx_{K}u$ for all
$u\in\mathcal{PU}_{\infty}(V).$

Let $u_{1},u_{2},u_{3}\in\mathcal{PU}_{\infty}(V)$ such that
$u_{1}\approx_{K}u_{2}$ and $u_{2}\approx_{K}u_{3}.$ Then $u_{1}\oplus
w_{1}\sim_{K}u_{2}\oplus w_{1}$ and $u_{2}\oplus w_{2}\sim_{K}u_{3}\oplus
w_{2}$ for some $w_{1},w_{2}\in\mathcal{PU}_{\infty}(V).$ By Corollary 4.8(2),
we have $u_{2}\oplus w_{1}\sim_{K}u_{1}\oplus w_{1}$ so that
$u_{2}\approx_{K}u_{1}.$ By Corollary 4.8 and by Proposition 4.9(4), we get
$(u_{1}\oplus w_{1})\oplus w_{2}\sim_{K}(u_{2}\oplus w_{1})\oplus w_{2}$ and
$(u_{2}\oplus w_{2})\oplus w_{1}\sim_{K}(u_{3}\oplus w_{2})\oplus w_{1}.$
Again applying Proposition 4.9(3), we have $w_{1}\oplus
w_{2}\sim_{K}w_{2}\oplus w_{1}.$ Then

$\displaystyle u_{1}\oplus(w_{1}\oplus w_{2})$ $\displaystyle=$
$\displaystyle(u_{1}\oplus w_{1})\oplus w_{2}\sim_{K}(u_{2}\oplus w_{1})\oplus
w_{2}$ $\displaystyle=$ $\displaystyle u_{2}\oplus(w_{1}\oplus
w_{2})\sim_{K}u_{2}\oplus(w_{2}\oplus w_{1})$ $\displaystyle=$
$\displaystyle(u_{2}\oplus w_{2})\oplus w_{1}\sim_{K}(u_{3}\oplus w_{2})\oplus
w_{1}$ $\displaystyle=$ $\displaystyle u_{3}\oplus(w_{2}\oplus
w_{1})\sim_{K}u_{3}\oplus(w_{1}\oplus w_{2}).$

Finally, by Corollary 4.8(2), we conclude $u_{1}\oplus(w_{1}\oplus
w_{2})\sim_{K}u_{3}\oplus(w_{1}\oplus w_{2}).$ Thus $u_{1}\approx_{K}u_{3}.$ ∎

The relation $\approx_{K}$ enjoys the following properties:

######  Corollary 4.12.

Let $V$ be an absolute matrix order unit space and let
$u,v,w,u^{\prime},v^{\prime}\in\mathcal{U}_{\infty}(V)$. Then

  1. $1.$

$u\approx_{K}u\oplus 0_{n}$ for all $n\in\mathbb{N}.$

  2. $2.$

$u\oplus v\approx_{K}v\oplus u.$

  3. $3.$

If $u\approx_{K}u^{\prime}$ and $v\approx_{K}v^{\prime},$ then $u\oplus
v\approx_{K}u^{\prime}\oplus v^{\prime}.$

###### Proof.

By Propositions 4.9 and 4.11, it follows that $u\oplus v\approx_{K}v\oplus u$
and $u\approx_{K}u\oplus 0_{n}$ for all $u,v\in\mathcal{U}_{\infty}(V)$ and
$n\in\mathbb{N}.$

Next, let $u,u^{\prime},v$ and $v^{\prime}\in\mathcal{PU}_{\infty}(V)$ such
that $u\approx_{K}u^{\prime}$ and $v\approx_{K}v^{\prime}.$ Then $u\oplus
w_{1}\sim_{K}u^{\prime}\oplus w_{1}$ and $v\oplus
w_{2}\sim_{K}v^{\prime}\oplus w_{2}$ for some
$w_{1},w_{2}\in\mathcal{PU}_{\infty}(V).$ By Proposition 4.9, we get

$\displaystyle(u\oplus v)\oplus(w_{1}\oplus w_{2})$ $\displaystyle=$
$\displaystyle u\oplus(v\oplus w_{1})\oplus w_{2}\sim_{K}u\oplus(w_{1}\oplus
v)\oplus w_{2}$ $\displaystyle=$ $\displaystyle(u\oplus w_{1})\oplus(v\oplus
w_{2})\sim_{K}(u^{\prime}\oplus w_{1})\oplus(v^{\prime}\oplus w_{2})$
$\displaystyle=$ $\displaystyle u^{\prime}\oplus(w_{1}\oplus v^{\prime})\oplus
w_{2}\sim_{K}u^{\prime}\oplus(v^{\prime}\oplus w_{1})\oplus w_{2}$
$\displaystyle=$ $\displaystyle(u^{\prime}\oplus v^{\prime})\oplus(w_{1}\oplus
w_{2}).$

Thus, by Corollary 4.8(2), we conclude that $(u\oplus v)\oplus(w_{1}\oplus
w_{2})\sim_{K}(u^{\prime}\oplus v^{\prime})\oplus(w_{1}\oplus w_{2})$ so that
$u\oplus v\approx_{K}u^{\prime}\oplus v^{\prime}.$ ∎

Now, we define a binary operatoin in the family of equivalence classes of
partial unitaries under $\approx_{K}.$

######  Proposition 4.13.

Let $V$ be an absolute matrix order unit space. For each
$u,v\in\mathcal{PU}_{\infty}(V),$ let
$[u]=\\{w\in\mathcal{PU}_{\infty}(V):w\approx_{K}u\\}$ and put
$[u]+[v]=[u\oplus v].$ Then, $+$ is a binary operation in the family of
equivalence classes $(\mathcal{PU}_{\infty}(V)/\approx_{K},+)$. Also:

  1. (1)

$[u]+[0]=[u]$ for all $u\in\mathcal{PU}_{\infty}(V);$

  2. (2)

$[u]+[v]=[v]+[u]$ for all $u,v\in\mathcal{PU}_{\infty}(V);$

  3. (3)

$[u]+[w]=[v]+[w]$ for $u,v,w\in\mathcal{PU}_{\infty}(V),$ then $[u]=[v].$

Thus, $(\mathcal{PU}_{\infty}(V)\big{/}\approx_{K},+)$ is an abelian semi-
group satisfying the cancellation law.

###### Proof.

By Corollary 4.12(3), it follows that $+$ is well-defined on
$\mathcal{PU}_{\infty}(V)\big{/}\approx_{K}.$ Note that (1) and (2)
immediately follow from 4.12(1) and (2) respectively. Next, we prove (3).

Let $u,v,w\in\mathcal{PU}_{\infty}(V)$ such that $[u]+[w]=[v]+[w].$ Then
$u\oplus w\approx_{K}v\oplus w$ so that $u\oplus(w\oplus
x)\sim_{K}v\oplus(w\oplus x)$ for some $x\in\mathcal{PU}_{\infty}(V).$ Thus,
$u\approx_{K}v$ so that $[u]=[v].$ ∎

Finally, the following result shows the importance of path homotopy of partial
unitaries to define a group.

######  Theorem 4.14.

Let $V$ be an absolute matrix order unit space and consider
$\mathcal{PU}_{\infty}(V)\times\mathcal{PU}_{\infty}(V).$ For all
$u_{1},u_{2},v_{1},v_{2}\in\mathcal{PU}_{\infty}(V),$ we define
$(u_{1},v_{1})\equiv_{K}(u_{2},v_{2})$ if and only if $u_{1}\oplus
v_{2}\approx_{K}u_{2}\oplus v_{1}.$ Then, $\equiv_{K}$ is an equivalence
relation on $\mathcal{PU}_{\infty}(V)\times\mathcal{PU}_{\infty}(V).$
Consider:

$K(V)=\\{[(u,v)]:u,v\in\mathcal{PU}_{\infty}(V)\\},$

where $[(u,v)]$ is the equivalence class of $(u,v)$ in
$(\mathcal{PU}_{\infty}(V)\times\mathcal{PU}_{\infty}(V),\equiv_{K}).$ For all
$u_{1},u_{2},v_{1},v_{2}\in\mathcal{PU}_{\infty}(V),$ we write:

$[(u_{1},v_{1})]+[(u_{2},v_{2})]=[(u_{1}\oplus u_{2},v_{1}\oplus v_{2})].$

Then $(K(V),+)$ is an abelian group.

###### Proof.

It follows from Proposition 4.11(2) and Corollary 4.12 that the relation
$\equiv_{K}$ on $\mathcal{PU}_{\infty}(V)\times\mathcal{PU}_{\infty}(V)$ is
reflexive and symmetric. Let $(u_{1},v_{1})\equiv_{K}(u_{2},v_{2})$ and
$(u_{2},v_{2})\equiv_{K}(u_{3},v_{3})$ for some
$u_{1},u_{2},u_{3},v_{1},v_{2},v_{3}\in\mathcal{PU}_{\infty}(V).$ Then
$u_{1}\oplus v_{2}\approx_{K}u_{2}\oplus v_{1}$ and $u_{2}\oplus
v_{3}\approx_{K}u_{3}\oplus v_{2}.$ By Proposition 4.11(2) and by Corollary
4.12(2) and (3), we get that $(u_{1}\oplus v_{3})\oplus(u_{2}\oplus
v_{2})\approx_{K}(u_{3}\oplus v_{1})\oplus(u_{2}\oplus v_{2})$ so that
$[u_{1}\oplus v_{3}]+[u_{2}\oplus v_{2}]=[u_{3}\oplus v_{1}]+[u_{2}\oplus
v_{2}].$ Then, by Proposition 4.13(3), we conclude that $[u_{1}\oplus
v_{3}]=[u_{3}\oplus v_{1}]$ so that $u_{1}\oplus v_{3}\approx_{K}u_{3}\oplus
v_{1}.$ Thus $(u_{1},v_{1})\equiv_{K}(u_{3},v_{3})$ so that $\equiv_{K}$ is
transitive. Hence, $\equiv_{K}$ is an equivalence relation on
$\mathcal{PU}_{\infty}(V)\times\mathcal{PU}_{\infty}(V).$

Now, we show that $+$ is well defined on $K(V).$ Let
$[(u_{1},v_{1})]=[(u_{1}^{\prime},v_{1}^{\prime})]$ and
$[(u_{2},v_{2})]=[(u_{2}^{\prime},v_{2}^{\prime})].$ Then
$(u_{1},v_{1})\equiv_{K}(u_{1}^{\prime},v_{1}^{\prime})$ and
$(u_{2},v_{2})\equiv_{K}(u_{2}^{\prime},v_{2}^{\prime})$ so that $u_{1}\oplus
v_{1}^{\prime}\approx_{K}u_{1}^{\prime}\oplus v_{1}$ and $u_{2}\oplus
v_{2}^{\prime}\approx_{K}u_{2}^{\prime}\oplus v_{2}.$ Again by Proposition
4.11(2) and by Corollary 4.12(2) and (3), we get that $(u_{1}\oplus
u_{2})\oplus(v_{1}^{\prime}\oplus
v_{2}^{\prime})\approx_{K}(u_{1}^{\prime}\oplus
u_{2}^{\prime})\oplus(v_{1}\oplus v_{2}).$ Thus
$[(u_{1},v_{1})]+[(u_{2},v_{2})]=[(u_{1}^{\prime},v_{1}^{\prime})]+[(u_{2}^{\prime},v_{2}^{\prime})]$
so that $+$ is well defined.

By Corollary 4.12(2) and (3), we have:

$(u_{1}\oplus u_{2})\oplus(v_{2}\oplus v_{1})\approx_{K}(u_{2}\oplus
u_{1})\oplus(v_{1}\oplus v_{2})$

for all $u_{1},u_{2},v_{1},v_{2}\in\mathcal{PU}_{\infty}(V),$ so that $+$ is
commutative in $K(V).$

Let $u,v\in\mathcal{PU}_{\infty}(V).$ By Corollary 4.12(1) and (3), we have
$(u\oplus 0)\oplus(v\oplus 0)\approx_{K}u\oplus v.$ Thus $[(0,0)]$ is an
identity element in $K(V).$

Associativity of $+$ on $K(V)$ follows from Proposition 4.9(1).

For any $u,v\in\mathcal{PU}_{\infty}(V),$ we have $[(u\oplus v,v\oplus
u)]=[(0,0)].$ Hence $[(v,u)]$ is an inverse element of $[(u,v)]$ in $K(V).$ ∎

######  Remark 4.15.

The abelian group $K(V)$ is called the $K(V)$-group of the absolute matrix
order unit space $V.$ If $A$ is a $C^{*}$-algebra, then $K(A)$ is Grothendieck
group of $A$ for partial unitary elements. Thus $K$-group for absolute matrix
order unit spaces is a generalization of $K$-group for $C^{*}$-algebras.

###  4.1. Order structure in $K$

In this subsection, we prove that $K(V)$ also bears order structure on it. We
start with the following discussion:

Let $A$ be a unital $C^{*}$-algebra. Then

  1. (1)

$p\sim_{h}q$ in $\mathcal{OP}(A)$ implies $p\sim q.$

  2. (2)

$|\cdot|$ is a continuous function.

For proof of part (1), see [30, Proposition 2.2.7]. Proof of part (2) can also
be found somewhere in literature. However, for the sake of completeness, we
provide its proof. For proof of part (2), we use functional calculus of self-
adjoint elements in a C∗-algebra [18]. It is sufficient to prove that
$\sqrt{a_{n}}\longrightarrow\sqrt{a}$ whenever $a_{n},a\in A^{+}$ such that
$a_{n}\longrightarrow a.$ Let $a_{n},a\in A^{+}$ such that
$a_{n}\longrightarrow a.$ Then there exists $M>0$ such that
$\|a_{n}\|_{n},\|a\|\leq M$ so that $\sigma(a_{n}),\sigma(a)\subset[0,M]$ for
all $n\in\mathbb{N}.$ Put $f(t)=\sqrt{t}.$ By Weierstrass Approximation
theorem, there exists a sequence of polynomials $\\{p_{m}(t)\\}$ converging
uniformly to $f$ on $[0,M].$ Note that
$\|p_{m}(a)-f(a)\|,\|p_{m}(a_{n})-f(a_{n})\|\leq\|p_{m}-f\|_{\infty}=\sup\\{|p_{m}(t)-f(t)|:t\in[0,M]\\}.$
Let $\epsilon>0.$ Choose $m_{0}\in\mathbb{N}$ such that
$\|p_{m_{0}}-f\|_{\infty}<\frac{\epsilon}{3}.$ Since $m_{0}$ is fixed and
$a_{n}\longrightarrow a,$ we can also choose $n_{0}\in\mathbb{N}$ such that
$\|p_{m_{0}}(a_{n})-p_{m_{0}}(a)\|<\frac{\epsilon}{3}$ for all $n\geq n_{0}.$
Then

$\displaystyle\|\sqrt{a_{n}}-\sqrt{a}\|$ $\displaystyle=$
$\displaystyle\|f(a_{n})-f(a)\|$ $\displaystyle\leq$
$\displaystyle\|f(a_{n})-p_{m_{0}}(a_{n})\|+\|p_{m_{0}}(a_{n})-p_{m_{0}}(a)\|+\|p_{m_{0}}(a)-f(a)\|$
$\displaystyle<$
$\displaystyle\frac{\epsilon}{3}+\frac{\epsilon}{3}+\frac{\epsilon}{3}$
$\displaystyle=$ $\displaystyle\epsilon$

for all $n\geq n_{0}.$ Thus $\sqrt{a_{n}}\longrightarrow\sqrt{a}.$

Next, we prove that $(K(V),K(V)^{+})$ is also an ordered abelian group. For
that, we need to prove the following result:

######  Theorem 4.16.

Let $V$ be an absolute matrix order unit space. Put
$K(V)^{+}=\\{[(v,0)]:v\in\mathcal{PU}_{\infty}(V)\\}.$ Then

  1. (1)

$K(V)^{+}$ is a group cone in $K(V)$.

  2. (2)

$K(V)^{+}$ is generating.

Moreover, if $V$ satisfies:

  1. (a)

$p\sim_{h}q$ in $\mathcal{OP}_{n}(V)$ implies $p\sim q;$

  2. (b)

$e^{n}$ is finite; and

  3. (c)

$|\cdot|_{n}$ is continuous

for all $n\in\mathbb{N}$ and $p,q\in\mathcal{OP}_{n}(V).$ Then we also have:

  1. (3)

$K(V)^{+}$ is proper.

###### Proof.

It is routine to verify (1) and (2). Next, we prove (3). Assume that $V$
satisfies $(a),(b)$ and $(c).$ Let $g\in K(V)^{+}\cap-K(V)^{+}.$ There exist
$u\in\mathcal{PU}_{m}(V)$ and $v\in\mathcal{PU}_{n}(V)$ such that
$g=[(u,0)]=[(0,v)]$. Then $(u,0)\equiv(0,v)$ so that $u\oplus
v\approx_{K}0_{m}\oplus 0_{n}$. Thus $u\oplus v\oplus w\sim_{h}0_{m}\oplus
0_{n}\oplus w$ for some $w\in\mathcal{OP}_{l}(V).$ By continuity of
$|\cdot|_{m+n+l},$ it follows that
$|u|_{m}\oplus|v|_{n}\oplus|w|_{l}\sim_{h}0_{m}\oplus 0_{n}\oplus|w|_{l}.$
Next, by assumption, we have $|u|_{m}\oplus|v|_{n}\oplus|w|_{l}\sim
0_{m}\oplus 0_{n}\oplus|w|_{l}.$ Since $e^{m+n+l}$ is finte, and
$|u|_{m}\oplus|v|_{n}\oplus|w|_{l}$ and $0_{m}\oplus
0_{n}\oplus|w|_{l}\in\mathcal{OP}_{m+n+l}(V)$ such that $0_{m}\oplus
0_{n}\oplus|w|_{l}\leq|u|_{m}\oplus|v|_{n}\oplus|w|_{l},$ by [25, Corollary
5.1], we conclude that $|u|_{m}\oplus|v|_{n}\oplus|w|_{l}=0_{m}\oplus
0_{n}\oplus|w|_{l}$. Then $|u|_{m}\oplus|v|_{n}=0_{m}\oplus 0_{n}$ so that
$|u|_{m}=0_{m}$ and $|v|_{n}=0_{n}.$ By [25, Proposition 2.4], we have $u=0$
and $v=0.$ Thus $g=0.$ ∎

######  Corollary 4.17.

Let $V$ be an absolute matrix order unit space. Then by Theorem 4.16, we
conclude that $(K(V),K(V)^{+})$ is an ordered abelian group.

###  4.2. Functoriality of $K$

Let $V$ be an absolute matrix order unit space. Then $v\longmapsto[(v,0)]$
defines a map $\Omega_{V}:\mathcal{PU}_{\infty}(V)\to K(V)$. In the next
result, we describe the functorial nature of $K$. For this, first we prove the
following commutative property of $K(V)$:

######  Theorem 4.18.

Let $V$ and $W$ be absolute matrix order unit spaces and let $\phi:V\to W$ be
a unital completely $|\cdot|$-preserving map. Then there exists a unique group
homomorphism $K(\phi):K(V)\to K(W)$ such that the following diagram commutes:

${\mathcal{PU}_{\infty}(V)}$${\mathcal{PU}_{\infty}(W)}$${K(V)}$${K(W)}$$\scriptstyle{\phi}$$\scriptstyle{\Omega_{V}}$$\scriptstyle{\Omega_{W}}$$\scriptstyle{K(\phi)}$

###### Proof.

By [24, Theorem 3.7(1)], we get that
$\phi(\mathcal{OP}_{\infty}(V))\subset\mathcal{OP}_{\infty}(W)$ so that
$\phi(\mathcal{PU}_{\infty}(V))\subset\mathcal{PU}_{\infty}(W)$. Let
$v,w\in\mathcal{PU}_{\infty}(V)$ such that $v\sim_{h}w$. Then
$\phi(v)\sim_{h}\phi(w).$

Next, put $K(\phi)([(v,w)])=[(\phi(v),\phi(w))]$ for each $[(v,w)]\in K(V)$.
We show that $K(\phi)$ is well-defined. Let $[(v_{1},w_{1})]=[(v_{2},w_{2})]$
for some $v_{1},v_{2},w_{1},w_{2}\in\mathcal{PU}_{\infty}(V).$ Then there
exists $u\in\mathcal{PU}_{\infty}(V)$ such that $v_{1}\oplus w_{2}\oplus
u\sim_{h}v_{2}\oplus w_{1}\oplus u.$ Thus
$\phi(v_{1})\oplus\phi(w_{2})\oplus\phi(u)\sim_{h}\phi(v_{2})\oplus\phi(w_{1})\oplus\phi(u)$
so that $[(\phi(v_{1}),\phi(w_{1}))]=[(\phi(v_{2}),\phi(w_{2}))].$ Hence
$K(\phi)$ is well-defined. For all $[(v_{1},w_{1})],[(v_{2},w_{2})]\in K(V)$,
we have that

$\displaystyle K(\phi)([(v_{1},w_{1})]+[(v_{2},w_{2})])$ $\displaystyle=$
$\displaystyle K(\phi)([(v_{1}\oplus v_{2},w_{1}\oplus w_{2})])$
$\displaystyle=$ $\displaystyle[(\phi(v_{1}\oplus v_{2}),\phi(w_{1}\oplus
w_{2}))]$ $\displaystyle=$
$\displaystyle[(\phi(v_{1})\oplus\phi(v_{2}),\phi(w_{1})\oplus\phi(w_{2}))]$
$\displaystyle=$
$\displaystyle[(\phi(v_{1}),\phi(w_{1}))]+[(\phi(v_{2}),\phi(w_{2}))]$
$\displaystyle=$ $\displaystyle
K(\phi)([(v_{1},w_{1})])+K(\phi)([(v_{2},w_{2})])$

so that $K(\phi)$ is a group homomorphism. By construction $K$ satisfies the
diagram.

Uniqueness of $K(\phi)$:- Let $\mathcal{H}:K(V)\to K(W)$ be a group
homomorphism satisfying the same diagram. Then
$K(\phi)(\Omega_{V}(v))=\Omega_{V}(\phi(v))=\mathcal{H}(\Omega_{V}(v))$ for
all $v\in\mathcal{PU}_{\infty}(V)$. Thus we get that

$\displaystyle K(\phi)([(v,w)])$ $\displaystyle=$ $\displaystyle
K(\phi)([(v,0)]-[(w,0)])$ $\displaystyle=$ $\displaystyle
K(\phi)(\Omega_{V}(v)-\Omega_{V}(w))$ $\displaystyle=$ $\displaystyle
K(\phi)(\Omega_{V}(v))-K(\phi)(\Omega_{V}(w))$ $\displaystyle=$
$\displaystyle\mathcal{H}(\Omega_{V}(v))-\mathcal{H}(\Omega_{V}(w))$
$\displaystyle=$ $\displaystyle\mathcal{H}(\Omega_{V}(v)-\Omega_{V}(w))$
$\displaystyle=$ $\displaystyle\mathcal{H}([(v,w)])$

for all $[(v,w)]\in K(V)$. Hence $K(\phi)=\mathcal{H}$. ∎

Let $V$ and $W$ be absolute matrix order unit spaces. We denote the zero group
homomorphism between $K(V)$ and $K(W)$ by $0_{K(W),K(V)}.$ The identity map on
$K(V)$ is denoted by $I_{K(V)}$.

######  Corollary 4.19.

Let $U,V$ and $W$ be absolute matrix order unit spaces and let $\phi:U\to V$
and $\psi:V\to W$ be unital completely $|\cdot|$-preserving maps. Then

  1. (a)

$K(I_{V})=I_{K(V)};$

  2. (b)

$K(\psi\circ\phi)=K(\psi)\circ K(\phi);$

  3. (c)

$K(0_{W,V})=0_{K(W),K(V)}.$

###### Proof.

  1. (a)

Let $v,w\in\mathcal{PU}_{\infty}(V)$. Then

$\displaystyle K(I_{V})([(v,w)])$ $\displaystyle=$
$\displaystyle[(I_{V}(v),I_{V}(w))]$ $\displaystyle=$ $\displaystyle[(v,w)]$

so that by Theorem 4.18, $K(I_{V})=I_{K(V)}$.

  2. (b)

For any $[(v,w)]\in K(U)$, we get that

$\displaystyle K(\psi\circ\phi)([(v,w)])$ $\displaystyle=$
$\displaystyle[(\psi\circ\phi(v),\psi\circ\phi(w))]$ $\displaystyle=$
$\displaystyle[(\psi(\phi(v)),\psi(\phi(w)))]$ $\displaystyle=$ $\displaystyle
K(\psi)[(\phi(v),\phi(w))]$ $\displaystyle=$ $\displaystyle
K(\psi)(K(\phi)([(v,w)]))$ $\displaystyle=$ $\displaystyle K(\psi)\circ
K(\phi)([(v,w)]).$

Thus by Theorem 4.18, we conclude that $K(\psi\circ\phi)=K(\psi)\circ
K(\phi)$.

  3. (c)

$K(0_{W,V})([(v,w)])=[(0_{W,V}(v),0_{W,V}(w))]=[(0,0)]$ for all $[(v,w)]\in
K(V)$. Thus again using 4.18, we get that $K(0_{W,V})=0_{K(W),K(V)}$.

∎

######  Remark 4.20.

It follows from Corollary 4.19 that $K$ is a functor from category of absolute
matrix order unit spaces with morphisms as unital completely
$|\cdot|$-preserving maps to category of abelian groups.

In the end of the subsection, we note the following isomorphic property of
$K.$

######  Corollary 4.21.

Let $V$ and $W$ be isomorphic absolute matrix order unit spaces (isomorphic in
the sense that there exists a unital, bijective completely
$|\cdot|$-preserving map between $V$ and $W$). Then $K(V)$ and $K(W)$ are
group isomorphic.

###### Proof.

Let $\phi:V\to W$ be unital completely $|\cdot|$-preserving map. Then
$\phi^{-1}$ is also unital completely $|\cdot|$-preserving map. Since
$\phi^{-1}\circ\phi=I_{V}$ and $\phi\circ\phi^{-1}=I_{W}$, by Corollary
4.19(a) and (b), we get that $K(\phi^{-1})\circ K(\phi)=I_{K(V)}$ and
$K(\phi)\circ K(\phi^{-1})=I_{K(W)}$. Thus $K(\phi):K(V)\to K(W)$ is a
surjective group isomorphism and $K(\phi)^{-1}=K(\phi^{-1})$. Hence $K(V)$ and
$K(W)$ are group isomorphic. ∎

##  5\. Relation among $K_{0},K_{1}$ and $K$

In this section, we derive a relation among $K_{0},K_{1}$ and $K.$ Therefore,
we start this section with the following result:

######  Proposition 5.1.

Let $(V,e)$ be an absolute matrix order unit space and let
$v_{i}\in\mathcal{PI}_{n}(V)$ for $i=1,2,3,...,m$ such that $v_{i}\perp vj$
for all $i\neq j.$ If
$\displaystyle\sum_{i=1}^{m}|v_{i}|=e^{n}=\displaystyle\sum_{i=1}^{m}|v_{i}^{*}|,$
then $\displaystyle\sum_{i=1}^{m}v_{i}\in\mathcal{U}_{n}(V).$

###### Proof.

Let $v_{i}\in\mathcal{PI}_{n}(V)$ for $i=1,2,3,...,m$ such that $v_{i}\perp
v_{j}$ for all $i\neq j.$ Assume that
$\displaystyle\sum_{i=1}^{m}|v_{i}|=e^{n}=\displaystyle\sum_{i=1}^{m}|v_{i}^{*}|.$
We prove the above result in the following two cases:

Case 1: $m=2$. Let $v_{1},v_{2}\in\mathcal{PI}_{n}(V)$ such that $v_{1}\perp
v_{2}$ and $|v_{1}|_{n}+|v_{2}|_{n}=e^{n}=|v_{1}^{*}|_{n}+|v_{2}^{*}|_{n}.$ By
Proposition 2.10, we have $|v_{1}+v_{2}|_{n}=|v_{1}|_{n}+|v_{2}|_{n}$ and
$|v_{1}^{*}+v_{2}^{*}|_{n}=|v_{1}^{*}|_{n}+|v_{2}^{*}|_{n}.$ Thus
$|v_{1}+v_{2}|_{n}=e^{n}$ and $|(v_{1}+v_{2})^{*}|_{n}=e^{n}$ so that
$v_{1}+v_{2}\in\mathcal{U}_{n}(V).$

Case 2: $m>2$. Put $w_{m-1}=\displaystyle\sum_{i=1}^{m-1}v_{i}.$ Thus, by
Proposition 2.16, we have $w_{m-1}\in\mathcal{PI}_{n}(V).$ Next, by [23,
Definition 3.4(5)] and by Remark 2.9, we also note that
$|w_{m-1}|_{n}=\displaystyle\sum_{i=1}^{m-1}\left|v_{i}\right|_{n}\perp|v_{m}|_{n}$
and
$|w_{m-1}^{*}|_{n}=\displaystyle\sum_{i=1}^{m-1}\left|v_{i}^{*}\right|_{n}\perp|v_{m}^{*}|_{n}$
so that $w_{m-1}\perp v_{m}.$ Then, again by Proposition 2.10, we get
$|w_{m-1}|_{n}+|v_{m}|_{n}=\displaystyle\sum_{i=1}^{m}|v_{i}|=e^{n}$ and
$|w_{m-1}^{*}|_{n}+|v_{m}^{*}|_{n}=\displaystyle\sum_{i=1}^{m}|v_{i}^{*}|=e^{n}.$
Finally, by case (1), we conclude that
$\displaystyle\sum_{i=1}^{m}v_{i}=w_{m-1}+v_{m}\in\mathcal{U}_{n}(V).$ ∎

The following result describes that every partial unitary is associated to a
unitary element.

######  Lemma 5.2.

Let $(V,e)$ be an absolute matrix order unit space and let
$v\in\mathcal{PU}_{n}(V)$ for some $n\in\mathbb{N}.$ Then $v\perp
e^{n}-|v|_{n}$ and $v\pm(e^{n}-|v|_{n})\in\mathcal{U}_{n}(V).$

###### Proof.

Let $v\in\mathcal{PU}_{n}(V).$ Then
$|v|_{n}=|v^{*}|_{n}\in\mathcal{OP}_{n}(V)$ so that $|v|=|v^{*}|_{n}\perp
e^{n}-|v|_{n}.$ Thus, by Remark 2.9, we get that $v\perp\pm(e^{n}-|v|_{n}).$
Next, $v\pm(e^{n}-|v|_{n})\in\mathcal{U}_{n}(V)$ follows from the Proposition
5.1. ∎

Next result provides an alternative characterization of partial unitaries.

######  Proposition 5.3.

Let $V$ be an absolute matrix order unit space and let
$v\in\mathcal{PI}_{n}(V)$ for some $n\in\mathbb{N}.$ Then
$v\in\mathcal{PU}_{n}(V)$ if and only if $|v^{*}|_{n}\perp e^{n}-|v|_{n}$ and
$v+e^{n}-|v|_{n}\in\mathcal{U}_{n}(V).$

###### Proof.

Let $n\in\mathbb{N}$ and $v\in\mathcal{PI}_{n}(V).$ Then $|v|_{n}$ and
$|v^{*}|_{n}\in\mathcal{OP}_{n}(V).$ First, assume that
$v\in\mathcal{PU}_{n}(V),$ then $|v|_{n}=|v^{*}|_{n}.$ Thus, by Lemma 5.2, we
conclude that $|v^{*}|_{n}\perp e^{n}-|v|_{n}$ and
$v+e^{n}-|v|_{n}\in\mathcal{U}_{n}(V).$ Conversely, assume that
$|v^{*}|_{n}\perp e^{n}-|v|_{n}$ and $v+e^{n}-|v|_{n}\in\mathcal{U}_{n}(V).$
Then
$e^{n}=|(v+e^{n}-|v)^{*}|_{n}=|v^{*}+(e^{n}-|v|_{n})|_{n}=|v^{*}|_{n}+e^{n}-|v|_{n}$
so that $|v|_{n}=|v^{*}|_{n}.$ Hence $v\in\mathcal{PU}_{n}(V).$ ∎

The result stated below tells that converse part of the Lemma 5.2 also holds.

######  Corollary 5.4.

Let $(V,e)$ be an absolute matrix order unit space and let $n\in\mathbb{N}.$
Then $v\in\mathcal{PU}_{n}(V)$ if and only if $v\perp e^{n}-|v|_{n}$ and
$v\pm(e^{n}-|v|_{n})\in\mathcal{U}_{n}(V).$

###### Proof.

First, assume that $v\perp e^{n}-|v|_{n}$ and
$v\pm(e^{n}-|v|_{n})\in\mathcal{U}_{n}(V).$ Since $v\perp e^{n}-|v|_{n},$ we
have $|v^{*}|_{n}\perp e^{n}-|v|_{n}.$ By Proposition 5.3, we get that
$v\in\mathcal{PU}_{n}(V).$ Next, converse part immediately follows from the
Lemma 5.2. ∎

Next result tells that every partial unitary is arithmetic mean of two unitary
elements.

######  Proposition 5.5.

Let $(V,e)$ be an absolute matrix order unit space and let
$v\in\mathcal{PU}_{n}(V)$ for some $n\in\mathbb{N}.$ Then, there exist
$v_{1},v_{2}\in\mathcal{U}_{n}(V)$ such that $v=\frac{1}{2}(v_{1}+v_{2})$ and
$v_{1}-v_{2}\perp v_{1}+v_{2}.$

###### Proof.

Let $n\in\mathbb{N}$ and $v\in\mathcal{PU}_{n}(V).$ Since
$|v|_{n}=|v^{*}|_{n}\in\mathcal{OP}_{n}(V),$ we get that $v$ and
$e^{n}-|v|_{n}\in\mathcal{PI}_{n}(V)$ with $|v|_{n}\perp e^{n}-|v|_{n}.$ Put
$v_{1}=v-e^{n}+|v|_{n}$ and $v_{2}=v+e^{n}-|v|_{n}.$ Then, by Proposition 5.1,
we conclude that $v_{1},v_{2}\in\mathcal{U}_{n}(V).$ Note that
$v=\frac{1}{2}(v_{1}+v_{2}),|v_{1}-v_{2}|_{n}=2(e^{n}-|v|_{n}),|v_{1}+v_{2}|_{n}=2|v|_{n}.$
By Proposition 2.11, we also note that $v_{1}-v_{2}\perp v_{1}+v_{2}.$ ∎

In the following result, we prove that for any $n\in\mathbb{N},\sim_{h}$ in
$\mathcal{PU}_{n}(V)$ is preserved by $|\cdot|_{n}$ in $\mathcal{OP}_{n}(V)$
and $\mathcal{U}_{n}(V)$ in terms of corresponding elements under the
continuity of $|\cdot|_{n}.$

######  Proposition 5.6.

Let $(V,e)$ be an absolute matrix order unit space and let $n\in\mathbb{N}.$
If $u\sim_{h}v$ in $\mathcal{PU}_{n}(V)$ and $|\cdot|_{n}$ is continuous, then
$|u|_{n}\sim_{h}|v|_{n}$ and $u\pm(e^{n}-|u|_{n})\sim_{h}v\pm(e^{n}-|v|_{n})$
in $\mathcal{OP}_{n}(V)$ and $\mathcal{U}_{n}(V)$ respectively.

###### Proof.

Assume that $u\sim_{h}v$ in $\mathcal{PU}_{n}(V)$ and $|\cdot|_{n}$ is
continuous. Then there exists a continuous function
$f:[0,1]\to\mathcal{PU}_{n}(V)$ such that $f(0)=u$ and $f(1)=v.$ By Definition
4.1 and by Lemma 5.2, we have that $|f(t)|\in\mathcal{OP}_{n}(V)$ and
$f(t)\pm(e^{n}-|f(t)|_{n})\in\mathcal{U}_{n}(V).$ Put $g_{1}(t)=|f(t)|_{n}$
and $g_{2}^{\pm}(t)=f(t)\pm(e^{n}-|f(t)|_{n}).$ Since $f$ and $|\cdot|_{n}$
are continuous, we get that $g_{1}:[0,1]\to\mathcal{OP}_{n}(V)$ and
$g_{2}^{\pm}:[0,1]\to\mathcal{U}_{n}(V)$ are continuous functions such that
$g_{1}(0)=|u|_{n},g_{1}(1)=|v|_{n},g_{2}^{\pm}(0)=u\pm(e^{n}-|u|_{n})$ and
$g_{2}^{\pm}(1)=v\pm(e^{n}-|v|_{n}).$ Thus $|u|_{n}\sim_{h}|v|_{n}$ and
$u\pm(e^{n}-|u|_{n})\sim_{h}v\pm(e^{n}-|v|_{n})$ in $\mathcal{OP}_{n}(V)$ and
$\mathcal{U}_{n}(V)$ respectively. ∎

Let $(G_{1},*_{1})$ and $(G_{2},*_{2})$ be two groups. We write: $G_{1}\oplus
G_{2}=\\{(g_{1},g_{2}):g_{1}\in G_{1},g_{2}\in G_{2}\\}$ and
$(g_{1},g_{2})*(h_{1},h_{2})=(g_{1}*_{1}h_{1},g_{2}*_{2}h_{2}).$ Then
$(G_{1}\oplus G_{2},*)$ is also a group and it is called the direct sum of
$G_{1}$ and $G_{2}.$

Finally, we provides a realtion among $K_{0},K_{1}$ and $K$ by the following
result:

######  Theorem 5.7.

Let $(V,e)$ be an absolute matrix order unit space satisfying $(T)$ such that
$|\cdot|_{n}$ is continuous for all $n\in\mathbb{N}$. If $p\sim_{h}q$ implies
$p\sim q$ for all $p,q\in\mathcal{OP}_{n}(V)$ and $n\in\mathbb{N},$ then there
exists a surjective group homomorphism $\theta:K(V)\to K_{0}(V)\oplus
K_{1}(V).$ In this case, $K(V)\big{/}\ker(\theta)\cong K_{0}(V)\oplus
K_{1}(V).$

###### Proof.

Put $\eta([(u,0)])=[(|u|_{m},0)]\in K_{0}(V)$ for all
$u\in\mathcal{PU}_{m}(V),m\in\mathbb{N}.$ Let $u,v\in\mathcal{PU}_{\infty}(V)$
such that $[(u,0)]=[(v,0)].$ Without loss of generality, we may assume that
$u,v\in\mathcal{PU}_{m}(V)$ for some $m\in\mathbb{N}.$ Therefore $u\oplus
w\oplus 0_{l}\sim_{h}v\oplus w\oplus 0_{l}$ in $\mathcal{PU}_{m+n+l}(V)$ for
some $w\in\mathcal{PU}_{n}(V)$ and $n,l\in\mathbb{N}.$ Since $|\cdot|_{m+n+l}$
is continuous, by Proposition 5.6, we have that $|u|_{m}\oplus|w|_{n}\oplus
0_{l}\sim_{h}|v|_{m}\oplus|w|_{n}\oplus 0_{l}$ in $\mathcal{OP}_{m+n+l}(V).$
By assumption, we get that $|u|_{m}\oplus|w|_{n}\oplus
0_{l}\sim|v|_{m}\oplus|w|_{n}\oplus 0_{l}$ so that
$[(|u|_{m},0)]+[(0,|w|_{n})]=[(|v|_{m},0)]+[(0,|w|_{n})].$ By cancellation law
in $K_{0}(V),$ we get that $[(|u|_{m},0)]=[(|v|_{m},0)].$ Thus, $\eta$ is
well-defined. Next, we have $[(u,v)]=[(u,0)]-[(v,0)]$ for all
$u,v\in\mathcal{PU}_{\infty}(V).$ Define
$\eta([(u,v)])=\eta([(u,0)])-\eta([(v,0)])$ for all
$u,v\in\mathcal{PU}_{\infty}(V).$ Then, $\eta$ extends to $K(V).$ It is
routine to verify that $\eta:K(V)\to K_{0}(V)$ is a group homomorphism. By
Lemma 5.2, it follows that $u+e^{m}-|u|_{m}\in\mathcal{U}_{m}(V).$ Also, put
$\mu([(u,0)])=[(u+e^{m}-|u|_{m},e)]\in K_{1}(V)$ for all
$u\in\mathcal{PU}_{m}(V).$ Again, by Proposition 5.6, we note that
$(u+e^{m}-|u|_{m})\oplus(w+e^{n}-|w|_{n})\oplus e^{l}=(u\oplus w\oplus
0_{l})+e^{m+n+l}-(|u|_{m}\oplus|w|_{n}\oplus 0_{l})\sim_{h}(v\oplus w\oplus
0_{l})+e^{m+n+l}-(|v|_{m}\oplus|w|_{n}\oplus
0_{l})=(v+e^{m}-|w|_{m})\oplus(v+e^{n}-|w|_{n})\oplus e^{l}.$ We can choose
$l>\max\\{m,n\\}$ so that
$(u+e^{m}-|u|_{m})\oplus(w+e^{n}-|w|_{n})\sim_{1}(v+e^{m}-|w|_{m})\oplus(v+e^{n}-|w|_{n})$
and consequently $(u+e^{m}-|u|_{m})\approx_{1}(v+e^{m}-|w|_{m}).$ Then
$[(u+e^{m}-|u|_{m},e)]=[(v+e^{m}-|w|_{m},e)].$ Thus, $\mu$ is also well-
defined. Define $\mu([(u,v)])=\mu([(u,0)])-\mu([(v,0)])$ for all
$u,v\in\mathcal{PU}_{\infty}(V).$ It is also routine to verify that
$\mu:K(V)\to K_{1}(V)$ is a group homomorphism.

Now, define a new group homomorphism $\theta:K(V)\to K_{0}(V)\oplus K_{1}(V)$
by $\theta([(u,0)])=(\eta([(u,0)]),\mu([(u,0)]))$ for all
$u\in\mathcal{PU}_{\infty}(V).$ We claim that $\theta$ is surjective. Let
$p\in\mathcal{OP}_{m}(V)$ and $v\in\mathcal{U}_{n}(V)$ for some
$m,n\in\mathbb{N}.$ Without loss of generality, we can assume that $m=n.$ Put
$p^{\prime}=e^{n}-p.$ Then
$\theta([(p^{\prime},0)])=([(p^{\prime},0)],[(e^{n},e)])$ and
$\theta([(v,0)])=([(e^{n},0)],[(v,e)]).$ By Proposition 2.20(5), we get that
$e^{n}\sim p\oplus p^{\prime}$ and hence
$[(e^{n},0)]=[(p,0)]+[(p^{\prime},0)].$ Thus, we have

$\displaystyle\theta([(v,0)]-[(p^{\prime},0)])$ $\displaystyle=$
$\displaystyle\theta([(v,0)])-\theta([(p^{\prime},0)])$ $\displaystyle=$
$\displaystyle([(e^{n},0)],[(v,e)])-([(p^{\prime},0)],[(e^{n},e)])$
$\displaystyle=$
$\displaystyle([(e^{n},0)]-[(p^{\prime},0)],[(v,e)]-[(e^{n},e)])$
$\displaystyle=$ $\displaystyle([(p,0)],[(v\oplus e,e\oplus e^{n})])$
$\displaystyle=$ $\displaystyle([(p,0)],[(v,e)])$

so that $\theta$ is surjective. Finally, by first isomorphism theorem of
groups, we conclude that $K(V)\big{/}\ker(\theta)\cong K_{0}(V)\oplus
K_{1}(V).$ ∎

######  Remark 5.8.

If $\theta$ is injective, then $K(V)\cong K_{0}(V)\oplus K_{1}(V).$

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