# Multicomplex Ideals, Modules and Hilbert Spaces

Derek Courchesne1 and Sébastien Tremblay2

(May 8, 2024)

###### Abstract

In this article we study some algebraic aspects of multicomplex numbers
$\mathbb{M}_{n}$. For $n\geq 2$ a canonical representation is defined in terms
of the multiplication of $n-1$ idempotent elements. This representation
facilitates computations in this algebra and makes it possible to introduce a
generalized conjugacy $\Lambda_{n}$, i.e. a composition of the $n$
multicomplex conjugates $\Lambda_{n}:=\dagger_{1}\cdots\dagger_{n}$, as well
as a multicomplex norm. The multicomplex algebra equipped with this norm and
$\Lambda_{n}$ as the involution forms a C∗-algebra. The ideals of the ring of
multicomplex numbers are then studied in details. Free
$\mathbb{M}_{n}$-modules and their linear operators are considered. Finally,
we develop Hilbert spaces on the multicomplex algebra.

Keywords. Multicomplex numbers, Commutative ring, C∗-algebra, Multicomplex
ideals, Free modules, Hilbert spaces

##  1 Introduction

For over two decades, bicomplex numbers
$\mathbb{B}\mathbb{C}\simeq\mathbb{M}_{2}$ and its subring of hyperbolic
numbers – also known as perplex, duplex or pseudo-complex numbers – have
garnered significant attention in mathematical research [3, 6, 9, 11, 12, 13,
18] and found applications in various domains of physics, including quantum
mechanics [10, 14, 19, 20, 24, 25] and general relativity [1, 8, 22] as well
as in engineering [17, 26]. The pioneering work of Corrado Segre in 1892
introduced an infinite set of algebras, termed bicomplex $\mathbb{M}_{2}$,
tricomplex $\mathbb{M}_{3}$, and so forth, extending the realm of complex
numbers $\mathbb{C}\simeq\mathbb{M}_{1}$. About a century later Price wrote
his book [16] on the subject and more recently Luna-Elizarrarás et al.
published a comprehensive study [12] of the analysis and geometry of bicomplex
numbers.

The multicomplex algebra $\mathbb{M}_{n}$ characterized by a finite number $n$
of commutative imaginary units
$\mathrm{i}_{1},\mathrm{,}\ldots,\mathrm{i}_{n}$ is defined as

$\mathbb{M}_{n}=\\{\eta_{1}+\mathrm{i}_{n}\eta_{2}\mid\eta_{1},\eta_{2}\in\mathbb{M}_{n-1}\\},\qquad\mathbb{M}_{0}:=\mathbb{R}.$

With the exeption of a few articles related to differential equations [15,
23], this algebra largely unexplored. This can be attributed primarily to the
daunting computational challenges associated with multicomplex calculations,
as exemplified by the representation of multicomplex numbers adopted by Price
in chapter 5 of his book [16]. As a result, it seems that some gap in
literature persists concerning the general case of muticomplex algebra
$\mathbb{M}_{n}$.

Although the seminal books [12, 16] and other articles (see for instance [13,
18]) extensively cover bicomplex numbers $n=2$ and that Chapter 5 of Price’s
book is devoted to multicomplex, all this work is dedicated mainly from the
point of view of mathematical analysis. Since the multicomplex algebra is a
commutative unitary ring with zero divisors, with the exception of the recent
article [9] in the specific bicomplex case, the abstract algebraic properties
of this ring have never been studied in details to our knowledge.

In light of these considerations, this work endeavors to address this lacuna
by embarking on a study of multicomplex algebra. Our objectives are threefold:
firstly, to investigate the abstract algebraic structure of multicomplex
numbers thereby extending the existing discourse beyond bicomplex numbers.
Hence, the ideals and the quotient rings of multicomplex will be studied as
well as the $\mathbb{M}_{n}$-modules. Secondly, to introduce a natural
representation expressed in terms of the multiplication of idempotent elements
and the $n$ multicomplex conjugates facilitating computationally operations on
multicomplex numbers and allowing the introduction of a useful multicomplex
norm. And finally, by introducing the multicomplex Hilbert spaces, to furnish
readers with the requisite mathematical foundations essential for the
advancement and application of multicomplex algebra in diverse realms
encompassing fundamental mathematics, physics or applied sciences.

##  2 Multicomplex numbers algebra

###  2.1 Definition

The $n$th multicomplex numbers space $\mathbb{M}_{n}$ is defined by the space
obtained after a total of $n\in\mathbb{N}$ successive complexifications of the
reals, each time introducing a new imaginary unit $\mathrm{i}_{k}$ such that
$\mathrm{i}^{2}_{k}=-1$:

$\mathbb{M}_{n}:=\\{\eta_{1}+\mathrm{i}_{n}\eta_{2}\mid\eta_{1},\eta_{2}\in\mathbb{M}_{n-1}\\},\qquad\mathbb{M}_{0}:=\mathbb{R}.$
(1)

Each element of the set $\\{\mathrm{i}_{1},\ldots,\mathrm{i}_{n}\\}$ is called
a principal unit, and any multiplication of distinct elements from this set is
called a composite unit (e.g. $\mathrm{i}_{2}\mathrm{i}_{4}\mathrm{i}_{9}$).
To get the whole structure of the multicomplex numbers we need to add the
following properties:

  1. 1.

$(\mathrm{i}_{j}\mathrm{i}_{k})\mathrm{i}_{l}=\mathrm{i}_{j}(\mathrm{i}_{k}\mathrm{i}_{l})$,
$\forall j,k,l=1,\ldots,n$ (associativity of units)

  2. 2.

$\mathrm{i}_{j}\mathrm{i}_{k}=\mathrm{i}_{k}\mathrm{i}_{j}$, $\forall
j,k=1,\ldots,n$. (commutativity of units)

It is well known from this construction that we get a commutative, unitary
algebra over both $\mathbb{R}$ and $\mathbb{C}$ (considering
$\mathrm{i}=\mathrm{i}_{1}$) with zero divisors when the set $\mathbb{M}_{n}$
is equipped with term by term addition and multiplication defined in the usual
way [16]. Also, important subspaces trivially isomorphic to the complex
numbers are given by $\mathbb{C}(\mathrm{i}_{k})=\\{a+b\mathrm{i}_{k}\mid
a,b\in\mathbb{R}\\}$.

The standard representation of a multicomplex number $\eta\in\mathbb{M}_{n}$
for the first values of $n$ is given by the following expressions

$\begin{array}[]{lrcl}\eta\in\mathbb{M}_{0}\simeq\mathbb{R}:&\eta&=&x_{0},\\\\[8.61108pt]
\eta\in\mathbb{M}_{1}\simeq\mathbb{C}:&\eta&=&x_{0}+x_{1}\mathrm{i}_{1},\\\\[8.61108pt]
\eta\in\mathbb{M}_{2}:&\eta&=&x_{0}+x_{1}\mathrm{i}_{1}+x_{2}\mathrm{i}_{2}+x_{12}\mathrm{i}_{1}\mathrm{i}_{2},\\\\[8.61108pt]
\eta\in\mathbb{M}_{3}:&\eta&=&x_{0}+x_{1}\mathrm{i}_{1}+x_{2}\mathrm{i}_{2}+x_{3}\mathrm{i}_{3}+x_{12}\mathrm{i}_{1}\mathrm{i}_{2}+x_{13}\mathrm{i}_{1}\mathrm{i}_{3}\\\\[8.61108pt]
&&&+\ x_{23}\mathrm{i}_{2}\mathrm{i}_{3}+\
x_{123}\mathrm{i}_{1}\mathrm{i}_{2}\mathrm{i}_{3},\\\ &\vdots\end{array}$ (2)

where the coefficients $x_{k}$ are real and $\mathbb{M}_{2},\mathbb{M}_{3}$
are the bicomplex and the tricomplex algebras, respectively [16]. One can
write the general expression of $\eta\in\mathbb{M}_{n}$ in terms of the power
set $\mathcal{P}\big{(}\\{1,\ldots,n\\}\big{)}$ :

$\eta=\sum_{\mathcal{A}\in\mathcal{P}_{n}}x_{\mathcal{A}}\mathrm{i}_{\mathcal{A}},\qquad\mathcal{P}_{n}:=\mathcal{P}\big{(}\\{1,\ldots,n\\}\big{)},\qquad
x_{\mathcal{A}}\in\mathbb{R},$

where the empty set in $\mathcal{P}_{n}$ is associated with the index zero and
$\mathrm{i}_{0}:=1$, the singleton $\\{k\\}\in\mathcal{P}_{n}$ is associated
with the index $k$, the set $\\{k,l\\}\in\mathcal{P}_{n}$ is associated with
the indices $kl$ such that $\mathrm{i}_{kl}:=\mathrm{i}_{k}\mathrm{i}_{l}$,
etc.

###  2.2 Conjugation and composition

A complex-like conjugation $\dagger_{k}$ is defined by

$\dagger_{0}:=\mathrm{id}\qquad\text{and}\qquad\dagger_{k}:\mathrm{i}_{k}\rightarrow-\mathrm{i}_{k},\hskip
8.53581ptk=1,\ldots,n$ (3)

where $\dagger_{0}$ is the identity map. We combine these conjugations with
the composition operation denoted $\circ$ and defined as follow :

$\eta^{\dagger_{j}\circ\dagger_{k}}:=(\eta^{\dagger_{j}})^{\dagger_{k}},\qquad\forall\eta\in\mathbb{M}_{n}\quad\text{
and }\quad 0\leq j,k\leq n.$ (4)

The composition is associative and commutative and the set $\ddagger$ of all
conjugates with the composition operation is a commutative group $(\ddagger,\
\circ)$ of order $2^{n}$, where each element is its own inverse and the
generators are $\dagger_{0},\dagger_{1},\ldots,\dagger_{n}$. This group is
isomorphic to $(\mathbb{Z}_{2}^{n},\ +_{2})$, see [4]. It is easy to show that
any composition of conjugates is distributive over addition and multiplication
for any multicomplex number. Hence, for $1\leq j_{1},j_{2},\ldots,j_{s}\leq n$
the conjugate $\dagger_{j_{1}}\dagger_{j_{2}}\cdots\dagger_{j_{s}}$ applied to
a multicomplex number $\eta\in\mathbb{M}_{n}$ changes the sign of the
principal units
$\mathrm{i}_{j_{1}},\mathrm{i}_{j_{2}},\ldots\mathrm{i}_{j_{s}}$ in $\eta$.
For instance, taking the composition of conjugates
$\eta^{\dagger_{1}\dagger_{3}}$ from an element $\eta\in\mathbb{M}_{3}$ in
(2), we obtain

$\eta^{\dagger_{1}\dagger_{3}}=x_{0}-x_{1}\mathrm{i}_{1}+x_{2}\mathrm{i}_{2}-x_{3}\mathrm{i}_{3}-x_{12}\mathrm{i}_{1}\mathrm{i}_{2}+x_{13}\mathrm{i}_{1}\mathrm{i}_{3}-x_{23}\mathrm{i}_{2}\mathrm{i}_{3}+x_{123}\mathrm{i}_{1}\mathrm{i}_{2}\mathrm{i}_{3}.$

###  2.3 Standard idempotent representation

When $n\geq 2$, it is possible to take advantage of the presence of zero
divisors and idempotent elements in the multicomplex algebra to get a basis of
$\mathbb{M}_{n}$ over $\mathbb{M}_{n-1}$ such that addition and multiplication
between multicomplex numbers are done componentwise. Take

$\gamma_{n}:=\frac{1}{2}(1+\mathrm{i}_{n-1}\mathrm{i}_{n})\hskip
8.53581pt\text{ and }\hskip
8.53581pt\gamma_{n}^{\prime}:=\frac{1}{2}(1-\mathrm{i}_{n-1}\mathrm{i}_{n}).$
(5)

Both $\gamma_{n}$ and $\gamma_{n}^{\prime}$ are zero divisors and idempotent
elements of $\mathbb{M}_{n}$, i.e.

$\gamma_{n}\cdot\gamma_{n}^{\prime}=0,\qquad\gamma_{n}^{2}=\gamma_{n},\qquad(\gamma_{n}^{\prime})^{2}=\gamma_{n}^{\prime}.$
(6)

Moreover, we also have the following relations for the multiplicative unit $1$
and the unit $\mathrm{i}_{n}$ expressed in terms of $\gamma_{n}$ and
$\gamma_{n}^{\prime}$:

$\displaystyle 1$ $\displaystyle=\gamma_{n}+\gamma_{n}^{\prime},$ (7)
$\displaystyle\mathrm{i}_{n}$
$\displaystyle=-\mathrm{i}_{n-1}(\gamma_{n}-\gamma_{n}^{\prime}).$

Now let $\eta\in\mathbb{M}_{n}$ be a multicomplex number with components
$\eta_{1},\eta_{2}\in\mathbb{M}_{n-1}$. Then by (7):

$\displaystyle\eta$
$\displaystyle=\eta_{1}+\eta_{2}\mathrm{i}_{n}=\eta_{1}(\gamma_{n}+\gamma_{n}^{\prime})-\eta_{2}\mathrm{i}_{n-1}(\gamma_{n}-\gamma_{n}^{\prime})$
$\displaystyle=(\eta_{1}-\eta_{2}\mathrm{i}_{n-1})\gamma_{n}+(\eta_{1}+\eta_{2}\mathrm{i}_{n-1})\gamma_{n}^{\prime}.$

Hence $\\{\gamma_{n},\ \gamma_{n}^{\prime}\\}$ forms a basis of
$\mathbb{M}_{n}$ over $\mathbb{M}_{n-1}$. Let $\eta,\zeta\in\mathbb{M}_{n}$
with respective components $\eta_{1},\eta_{2}$ and $\zeta_{1},\zeta_{2}$ in
$\mathbb{M}_{n-1}$ relative to that new basis. Then from (6) addition and
multiplication in $\mathbb{M}_{n}$ are done componentwise:

$\displaystyle\eta+\zeta$
$\displaystyle=(\eta_{1}\gamma_{n}+\eta_{2}\gamma_{n}^{\prime})+(\zeta_{1}\gamma_{n}+\zeta_{2}\gamma_{n}^{\prime})=(\eta_{1}+\zeta_{1})\gamma_{n}+(\eta_{2}+\zeta_{2})\gamma_{n}^{\prime},$
(8) $\displaystyle\eta\cdot\zeta$
$\displaystyle=(\eta_{1}\gamma_{n}+\eta_{2}\gamma_{n}^{\prime})\cdot(\zeta_{1}\gamma_{n}+\zeta_{2}\gamma_{n}^{\prime})=\eta_{1}\zeta_{1}\gamma_{n}+\eta_{2}\zeta_{2}\gamma_{n}^{\prime}.$

This representation of multicomplex numbers is called the standard idempotent
representation.

###  2.4 Canonical idempotent representation

Let us introduce the element $\Gamma_{n}\in\mathbb{M}_{n}$ defined as the
product of all consecutive idempotent numbers, i.e.

$\Gamma_{n}:=\gamma_{2}\gamma_{3}\cdots\gamma_{n},\qquad n\geq 2.$ (9)

By definition, we have the recursive relation
$\Gamma_{n+1}=\Gamma_{n}\gamma_{n+1}$. We define $\Gamma_{n}^{\ddagger}$ as
the set of all compositions of conjugates for the element $\Gamma_{n}$, i.e.

$\Gamma_{n}^{\ddagger}:=\big{\\{}\Gamma_{n}^{\dagger_{j_{1}}\dagger_{j_{2}}\cdots\dagger_{j_{k}}}\
|\ 0\leq j_{1}<j_{2}<\cdots<j_{k}\leq n,\quad k=1,\ldots,n\big{\\}}.$

######  Proposition 1.

For $n\geq 2$ the set $\Gamma_{n}^{\ddagger}$ has $2^{n-1}$ distinct elements.
If these elements are represented by $\varepsilon_{k}$ for
$k=1,\ldots,2^{n-1}$ we have

$\Gamma_{n+1}^{\ddagger}=\\{\varepsilon_{k}\gamma_{n+1}\mid
k=1,\ldots,2^{n-1}\\}\cup\\{\varepsilon_{k}\gamma_{n+1}^{\prime}\mid
k=1,\ldots,2^{n-1}\\}.$

###### Proof.

We proceed by induction over $n$. In the case $n=2$, we have
$\Gamma_{2}=\gamma_{2}$, and the values obtained when applying each
composition of conjugates are

$\begin{array}[]{c c c
c}\Gamma_{2}^{\dagger_{0}}=\gamma_{2},&\Gamma_{2}^{\dagger_{1}}=\gamma_{2}^{\prime},&\Gamma_{2}^{\dagger_{2}}=\gamma_{2}^{\prime},&\Gamma_{2}^{\dagger_{1}\dagger_{2}}=\gamma_{2},\end{array}$

such that $\Gamma_{2}^{\ddagger}$ contains two distinct elements
$\\{\varepsilon_{1}=\gamma_{2},\varepsilon_{2}=\gamma_{2}^{\prime}\\}$.
Moreover,

$\displaystyle\Gamma_{3}^{\ddagger}$
$\displaystyle:=\\{\Gamma_{3},\Gamma_{3}^{\dagger_{1}},\Gamma_{3}^{\dagger_{2}},\Gamma_{3}^{\dagger_{3}},\Gamma_{3}^{\dagger_{1}\dagger_{2}},\Gamma_{3}^{\dagger_{1}\dagger_{3}},\Gamma_{3}^{\dagger_{2}\dagger_{3}},\Gamma_{3}^{\dagger_{1}\dagger_{2}\dagger_{3}}\\}$
$\displaystyle=\\{\gamma_{2}\gamma_{3},(\gamma_{2}\gamma_{3})^{\dagger_{1}},(\gamma_{2}\gamma_{3})^{\dagger_{2}},(\gamma_{2}\gamma_{3})^{\dagger_{3}},(\gamma_{2}\gamma_{3})^{\dagger_{1}\dagger_{2}},(\gamma_{2}\gamma_{3})^{\dagger_{1}\dagger_{3}},(\gamma_{2}\gamma_{3})^{\dagger_{2}\dagger_{3}},(\gamma_{2}\gamma_{3})^{\dagger_{1}\dagger_{2}\dagger_{3}}\\}$
$\displaystyle=\\{\gamma_{2}\gamma_{3},\gamma_{2}^{\prime}\gamma_{3},\gamma_{2}^{\prime}\gamma_{3}^{\prime},\gamma_{2}\gamma_{3}^{\prime},\gamma_{2}\gamma_{3}^{\prime},\gamma_{2}^{\prime}\gamma_{3}^{\prime},\gamma_{2}^{\prime}\gamma_{3},\gamma_{2}\gamma_{3}\\}$
$\displaystyle=\\{\varepsilon_{1}\gamma_{3},\varepsilon_{2}\gamma_{3}\\}\cup\\{\varepsilon_{1}\gamma_{3}^{\prime},\varepsilon_{2}\gamma_{3}^{\prime}\\}.$

Assuming the statement to be true in the case $n\geq 2$, we set
$\Gamma_{n}^{\ddagger}=\\{\varepsilon_{k}\\}_{k=1}^{2^{n-1}}$. Since
$\Gamma_{n+1}=\Gamma_{n}\gamma_{n+1}$ and any composition of conjugates is
distributive over the product, then every elements of
$\Gamma_{n+1}^{\ddagger}$ can be written as

$\displaystyle\Gamma_{n+1}^{\dagger_{j_{1}}\cdots\dagger_{j_{s}}}$
$\displaystyle=(\Gamma_{n}\gamma_{n+1})^{\dagger_{j_{1}}\cdots\dagger_{j_{s}}}\quad\text{for
}\quad 0\leq j_{1}<j_{2}<\cdots<j_{s}\leq n+1$
$\displaystyle=(\Gamma_{n})^{\dagger_{j_{1}}\cdots\dagger_{j_{s}}}(\gamma_{n+1})^{\dagger_{j_{1}}\cdots\dagger_{j_{s}}}$
$\displaystyle=\varepsilon_{k}\gamma_{n+1}\quad\mathrm{or}\quad\varepsilon_{k}\gamma_{n+1}^{\prime}\quad\text{for
}\quad 1\leq k\leq 2^{n-1}.$

We verify easily that the pair of multicomplex numbers
$\varepsilon_{k}\gamma_{n+1}$ and $\varepsilon_{l}\gamma_{n+1}$ as well as
$\varepsilon_{k}\gamma_{n+1}^{\prime}$ and
$\varepsilon_{l}\gamma_{n+1}^{\prime}$ are distinct for $k,l=1,\ldots,2^{n-1}$
and $k\neq l$. Moreover, $\varepsilon_{k}\gamma_{n+1}$ and
$\varepsilon_{l}\gamma_{n+1}^{\prime}$ are also distinct for
$k,l=1,\ldots,2^{n-1}$. Therefore, we can conclude that
$\Gamma_{n+1}^{\ddagger}$ has $2\cdot 2^{n-1}$ distinct elements. ∎

######  Remark 2.

We can rename the elements of $\Gamma_{n+1}^{\ddagger}$ in Proposition 1 such
that this set can be rewritten in the form
$\\{\tilde{\varepsilon}_{k}\\}_{k=1}^{2^{n}}$. Indeed, for
$k=1,\ldots,2^{n-1}$ we set

$\tilde{\varepsilon}_{k}:=\varepsilon_{k}\gamma_{n+1}\qquad\text{and}\qquad\tilde{\varepsilon}_{2^{n-1}+k}:=\varepsilon_{k}\gamma_{n+1}^{\prime},$

where $\\{\varepsilon_{k}\\}_{k=1}^{2^{n-1}}$ is the set
$\Gamma_{n}^{\ddagger}$.

######  Proposition 3.

For $n\geq 2$ the elements $\\{\varepsilon_{k}\\}_{k=1}^{2^{n-1}}$ of
$\Gamma_{n}^{\ddagger}$ have the following properties:

$(i)\ \varepsilon_{k}\varepsilon_{l}=\delta_{kl}\varepsilon_{l}\qquad(ii)\
\varepsilon_{k}^{\Lambda_{n}}=\varepsilon_{k}\qquad(iii)\
\sum_{j=1}^{2^{n-1}}\varepsilon_{j}=1,$

for $1\leq k,l\leq 2^{n-1}$, where the symbol $\delta_{kl}$ is the usual
Kronecker delta and $\Lambda_{n}:=\dagger_{1}\dagger_{2}\cdots\dagger_{n}$ is
the composition of all single conjugates in $\mathbb{M}_{n}$.

###### Proof.

We proceed by induction over $n$. For $n=2$ we have from the proof of
Proposition 1 that $\varepsilon_{1}=\gamma_{2}$ and
$\varepsilon_{2}=\gamma_{2}^{\prime}$, hence
$\varepsilon_{k}\varepsilon_{l}=\delta_{kl}\varepsilon_{l}$ for $k,l=1,2$.
Moreover,
$\varepsilon_{1}^{\Lambda_{2}}=\gamma_{2}^{\dagger_{1}\dagger_{2}}=\gamma_{2}=\varepsilon_{1}$,
$\varepsilon_{2}^{\Lambda_{2}}=\varepsilon_{2}$ and
$\varepsilon_{1}+\varepsilon_{2}=\gamma_{2}+\gamma_{2}^{\prime}=1$.

Suppose now that all three properties of the proposition are satisfied for
$n\geq 2$. Then from Proposition 1 and Remark 2 we have
$\Gamma_{n+1}^{\ddagger}=\\{\tilde{\varepsilon}_{k}\\}_{k=1}^{2^{n}}$ such
that for $1\leq k,l\leq 2^{n-1}$ we obtain

$\displaystyle\tilde{\varepsilon}_{k}\cdot\tilde{\varepsilon}_{l}$
$\displaystyle=\varepsilon_{k}\gamma_{n+1}\cdot\varepsilon_{l}\gamma_{n+1}=\varepsilon_{k}\varepsilon_{l}\gamma_{n+1}=\delta_{kl}\varepsilon_{l}\gamma_{n+1}=\delta_{kl}\tilde{\varepsilon}_{l},$
$\displaystyle\tilde{\varepsilon}_{2^{n-1}+k}\cdot\tilde{\varepsilon}_{2^{n-1}+l}$
$\displaystyle=\varepsilon_{k}\gamma_{n+1}^{\prime}\cdot\varepsilon_{l}\gamma_{n+1}^{\prime}=\varepsilon_{k}\varepsilon_{l}\gamma_{n+1}^{\prime}=\delta_{kl}\varepsilon_{l}\gamma_{n+1}^{\prime}=\delta_{kl}\tilde{\varepsilon}_{2^{n-1}+l},$
$\displaystyle\tilde{\varepsilon}_{k}\cdot\tilde{\varepsilon}_{2^{n-1}+l}$
$\displaystyle=\varepsilon_{k}\gamma_{n+1}\cdot\varepsilon_{l}\gamma_{n+1}^{\prime}=0,$

which proves ($i$). Moreover, we have

$\tilde{\varepsilon}_{k}^{\Lambda_{n+1}}=(\varepsilon_{k}\gamma_{n+1})^{\Lambda_{n+1}}=\varepsilon_{k}^{\Lambda_{n}}\gamma_{n+1}^{\Lambda_{n+1}}=\varepsilon_{k}\gamma_{n+1}^{\Lambda_{n+1}}=\varepsilon_{k}(\gamma_{n+1})^{\dagger_{n}\dagger_{n+1}}=\varepsilon_{k}\gamma_{n+1}=\tilde{\varepsilon}_{k}$

and
$\tilde{\varepsilon}_{2^{n-1}+k}^{\Lambda_{n+1}}=\tilde{\varepsilon}_{2^{n-1}+k}$
by a similar calculation, which demonstrate ($ii$). Finally, we have

$\displaystyle\sum_{j=1}^{2^{n}}\tilde{\varepsilon}_{j}$
$\displaystyle=\sum_{k=1}^{2^{n-1}}\tilde{\varepsilon}_{k}+\sum_{k=1}^{2^{n-1}}\tilde{\varepsilon}_{2^{n-1}+k}=\sum_{k=1}^{2^{n-1}}\varepsilon_{k}\gamma_{n+1}+\sum_{k=1}^{2^{n-1}}\varepsilon_{k}\gamma_{n+1}^{\prime}$
$\displaystyle=(\gamma_{n+1}+\gamma_{n+1}^{\prime})\sum_{k=1}^{2^{n-1}}\varepsilon_{k}=1$

which shows ($iii$). ∎

These properties assure us that the distinct elements
$\\{\varepsilon_{k}\\}_{k=1}^{2^{n-1}}$ of $\Gamma_{n}^{\dagger}$ are linearly
independent over $\mathbb{C}$. Let
$\alpha_{1},\ldots,\alpha_{2^{n-1}}\in\mathbb{C}$, then
$\sum_{k=1}^{2^{n-1}}\alpha_{k}\varepsilon_{k}=0$ iff
$\alpha_{j}\varepsilon_{j}=0$ for all $j$, i.e. $\alpha_{j}=0$.

######  Proposition 4.

The set $\Gamma_{n}^{\ddagger}$ is a basis of $\mathbb{M}_{n}$ over
$\mathbb{C}$.

###### Proof.

We proceed by induction over $n$. In the case $n=2$, we know the statement is
true since $\Gamma_{2}^{\ddagger}=\\{\gamma_{2},\gamma_{2}^{\prime}\\}$ is the
basis for the standard idempotent representation of $\mathbb{M}_{2}$ over
$\mathbb{M}_{1}\simeq\mathbb{C}$. Suppose the statement is true for $n\geq 2$,
i.e. any number $\zeta\in\mathbb{M}_{n}$ can be written in the form
$\zeta=\sum_{k=1}^{2^{n-1}}z_{k}{\varepsilon_{k}},\ z_{k}\in\mathbb{C}.$ Here
$\\{\varepsilon_{k}\\}_{k=1}^{2^{n-1}}$ are the distinct elements of
$\Gamma_{n}^{\ddagger}$. Let $\eta\in\mathbb{M}_{n+1}$. From the standard
idempotent representation we have

$\eta=\zeta_{1}\gamma_{n+1}+\zeta_{2}\gamma_{n+1}^{\prime},\qquad\zeta_{1},\zeta_{2}\in\mathbb{M}_{n}.$
(10)

The induction hypothesis implies that
$\zeta_{1}=\sum_{k=1}^{2^{n-1}}\alpha_{k}\varepsilon_{k}$ and
$\zeta_{2}=\sum_{k=1}^{2^{n-1}}\beta_{k}\varepsilon_{k}$ for
$\alpha_{k},\beta_{k}\in\mathbb{C}$. By substituing in (10),

$\eta=\sum_{k=1}^{2^{n-1}}\alpha_{k}\varepsilon_{k}\gamma_{n+1}+\sum_{k=1}^{2^{n-1}}\beta_{k}\varepsilon_{k}\gamma_{n+1}^{\prime}=\sum_{k=1}^{2^{n-1}}\alpha_{k}\tilde{\varepsilon}_{k}+\sum_{k=1}^{2^{n-1}}\beta_{k}\tilde{\varepsilon}_{2^{n-1}+k},$

where the change of variables of Remark 2 is used. Hence, the set
$\Gamma_{n+1}^{\ddagger}=\\{\tilde{\varepsilon}_{k}\\}_{k=1}^{2^{n}}$ is a
basis of $\mathbb{M}_{n+1}$ over $\mathbb{C}$. ∎

######  Theorem 5 (Canonical idempotent representation).

For $n\geq 2$, any multicomplex element $\eta\in\mathbb{M}_{n}$ can be
represented by

$\eta=\sum_{k=1}^{2^{n-1}}z_{k}\varepsilon_{k},\qquad
z_{k}\in\mathbb{C}(\mathrm{i}_{i})$

and the $2^{n-1}$ idempotent elements $\varepsilon_{k}$ satisfy Proposition 3.

######  Remark 6.

To simplify the notation for the rest of the paper we will consider element
$\mathrm{i}_{1}$ as the usual imaginary complex number $\mathrm{i}$.
Therefore, we consider $\mathbb{C}(\mathrm{i}_{i})\simeq\mathbb{C}$ and
$\eta=\sum_{k=1}^{2^{n-1}}z_{k}\varepsilon_{k}$ where $z_{k}\in\mathbb{C}$ in
the last theorem. Moreover for any
$z\in\mathbb{C}(\mathrm{i}_{i})\simeq\mathbb{C}$ we have
$z^{\dagger_{1}}=\overline{z}$, where the bar represents the usual complex
conjugation.

###  2.5 Projections

For any multicomplex number
$\eta=\sum_{k=1}^{2^{n-1}}z_{k}\varepsilon_{k}\in\mathbb{M}_{n}$ written in
the canonical idempotent representation, we introduce the $j$th multicomplex
projection as the function $P_{j}:\mathbb{M}_{n}\rightarrow\mathbb{C}$ such
that $P_{j}(\eta)=z_{j}$. In what follows, the caret notation $\hat{\jmath}$
for indices will be used in relation with the $j$th projection in the
following manner :

$\eta_{\hat{\jmath}}:=P_{j}(\eta)\hskip 8.53581pt\text{ and }\hskip
8.53581pt\eta=\sum_{k=1}^{2^{n-1}}\eta_{\hat{k}}\varepsilon_{k}.$ (11)

In particular, for any $z\in\mathbb{C}$ we have from Proposition 3 (iii) that

$z=\sum_{k=1}^{2^{n-1}}z\varepsilon_{k}\quad\Rightarrow\quad
P_{j}(z)=z\quad\text{for}\quad 1\leq j\leq 2^{n-1},$

i.e. the $j$th projection is the identity map when applied on complex
elements. The projection operator is a linear operator for addition and
multiplication in $\mathbb{M}_{n}$, i.e. for all $\eta,\zeta\in\mathbb{M}_{n}$

$P_{j}(\eta+\zeta)=P_{j}(\eta)+P_{j}(\zeta)\quad\text{and}\quad
P_{j}(\eta\cdot\zeta)=P_{j}(\eta)\cdot P_{j}(\zeta).$

A multicomplex number $\eta$ is a zero divisor if and only if at least one of
its projections vanishes. Indeed, for two non zero elements
$\eta,\zeta\in\mathbb{M}_{n}$ such that $\eta\zeta=0$,
$\eta=\sum_{k=1}^{2^{n-1}}\eta_{\hat{k}}\varepsilon_{k}$ and
$\zeta=\sum_{k=1}^{2^{n-1}}\zeta_{\hat{k}}\varepsilon_{k}$ we have

$\eta\zeta=\sum_{k=1}^{2^{n-1}}\eta_{\hat{k}}\zeta_{\hat{k}}\varepsilon_{k}=0\quad\Rightarrow\quad\eta_{\hat{k}}=0\text{
or }\zeta_{\hat{k}}=0\quad\text{for}\quad 1\leq k\leq 2^{n-1}.$

Conversely, we have

$\eta_{\hat{\jmath}}=0\quad\Leftrightarrow\quad\varepsilon_{j}\eta=\varepsilon_{j}\eta_{\hat{\jmath}}=0$
(12)

which holds even if $\eta\neq 0$. The set of zero divisors including zero,
denoted by $\mathbb{M}_{n}^{-1}$, is then represented by

$\mathbb{M}_{n}^{-1}=\big{\\{}\eta=\sum_{k=1}^{2^{n-1}}\eta_{\hat{k}}\varepsilon_{k}\
|\ \eta_{\hat{\jmath}}=0\text{ for at least one }j\big{\\}}.$

For any
$\eta=\sum_{k=1}^{2^{n-1}}\eta_{\hat{k}}\varepsilon_{k}\in\mathbb{M}_{n}\backslash\mathbb{M}_{n}^{-1}$
it’s now easy to obtain its inverse:
$\eta^{-1}=\sum_{k=1}^{2^{n-1}}\eta^{-1}_{\hat{k}}\varepsilon_{k}$.

###  2.6 Multiperplex subalgebra

An important subalgebra called the multiperplex numbers and denoted
$\mathbb{D}_{n}$ is the one made up by multicomplex numbers invariant under
the $\Lambda_{n}$ composition of conjugates :

$\mathbb{D}_{n}=\\{\zeta\in\mathbb{M}_{n}\mid\zeta^{\Lambda}=\zeta\\}.$

######  Remark 7.

For the rest of this article we will omit index $n$ in $\Lambda$ except where
necessary for clarity.

Since all $\varepsilon_{k}$ from the canonical representation are invariant
under $\Lambda$ (see Proposition 3), $\Gamma_{n}^{\ddagger}$ is a basis of
$\mathbb{D}_{n}$. Expanding the equality $\zeta^{\Lambda}=\zeta$ we can see
that a multicomplex number is in the subalgebra $\mathbb{D}_{n}$ if and only
if all its components in the canonical representation are real :

$\zeta^{\Lambda}=\zeta\quad\Leftrightarrow\quad\sum_{k=1}^{2^{n-1}}\overline{\zeta_{\hat{k}}}\varepsilon_{k}=\sum_{k=1}^{2^{n-1}}\zeta_{\hat{k}}\varepsilon_{k}\quad\Leftrightarrow\quad\overline{\zeta_{\hat{k}}}=\zeta_{\hat{k}},$
(13)

i.e. $\zeta_{\hat{k}}\in\mathbb{R}$ for $k=1,\ldots,2^{n-1}$.

As a result, we obtain a broader notion of real and imaginary parts by
separating each multicomplex element into two parts contained in
$\mathbb{D}_{n}$. Let $\eta\in\mathbb{M}_{n}$, then each component
$\eta_{\hat{k}}$ of $\eta$ in the canonical idempotent representation can be
written as $\eta_{\hat{k}}=x_{k}+\mathrm{i}y_{k}$ for
$x_{k},y_{k}\in\mathbb{R}$:

$\eta=\sum_{k=1}^{2^{n-1}}(x_{k}+\mathrm{i}y_{k})\varepsilon_{k}=\sum_{k=1}^{2^{n-1}}x_{k}\varepsilon_{k}+\mathrm{i}\sum_{k=1}^{2^{n-1}}y_{k}\varepsilon_{k}=d_{1}+\mathrm{i}d_{2},$
(14)

where $d_{1},d_{2}\in\mathbb{D}_{n}$. The set of multiperplex numbers is a
vector space over $\mathbb{R}$ and for any $\eta,\zeta\in\mathbb{D}_{n}$ we
can define the following vector partial order $\leq$ :

$\eta\leq\zeta\quad\Leftrightarrow\quad\eta_{\hat{\jmath}}\leq\zeta_{\hat{\jmath}}\qquad\text{for}\qquad
j=1,\ldots,2^{n-1}.$

This is also known as the product order (or componentwise order) from [5]. We
denote $\mathbb{D}_{n}^{+}$ the set of all positive multiperplex numbers i.e.

$\mathbb{D}_{n}^{+}=\\{\eta\in\mathbb{D}_{n}\ |\ \eta\geq 0\\}.$

Let $\eta\in\mathbb{M}_{n}$. A natural and unique way to define a multicomplex
norm from the $n$ composition of conjugates
$\Lambda=\dagger_{1}\cdots\dagger_{n}$ is

$\|\eta\|:=\sqrt{\eta^{\Lambda}\eta}.$

As the multiplication acts componentwise in the canonical basis, the square
root is distributed over the components such that

$\|\eta\|:=\sqrt{\eta^{\Lambda}\eta}=\sum_{k=1}^{2^{n-1}}\sqrt{|\eta_{\hat{k}}|^{2}}\varepsilon_{k}=\sum_{k=1}^{2^{n-1}}|\eta_{\hat{k}}|\varepsilon_{k}\in\mathbb{D}_{n}^{+}$

and the result is a positive multiperplex number since
$|\eta_{\hat{\jmath}}|\geq 0$ for all $j=1,\ldots,2^{n-1}$. For all
$\eta,\zeta\in\mathbb{M}_{n}$ we have

$\|\eta+\zeta\|=\sum_{k=1}^{2^{n-1}}|\eta_{\hat{k}}+\zeta_{\hat{k}}|\varepsilon_{k}\leq\sum_{k=1}^{2^{n-1}}|\eta_{\hat{k}}|\varepsilon_{k}+\sum_{k=1}^{2^{n-1}}|\zeta_{\hat{k}}|\varepsilon_{k}=\|\eta\|+\|\zeta\|$

since
$|\eta_{\hat{\jmath}}+\zeta_{\hat{\jmath}}|\leq|\eta_{\hat{\jmath}}|+|\zeta_{\hat{\jmath}}|$
for all $j$. The others norm properties (absolute homogeneity and
positiveness) can be verified in the same way directly from the complex case.

The multicomplex algebra $\mathbb{M}_{n}$ equipped with this norm and the
conjugate $\Lambda$ as the involution is a C∗-algebra, see [2]. This fact
comes once again from the properties of complex numbers combined to the
properties of $\Lambda$ as a composition of conjugates.

##  3 Multicomplex ideals

###  3.1 Multiperplex ideals

A multicomplex ring ideal is a subring $I$ of $\mathbb{M}_{n}$ such that the
set $\eta I:=\\{\eta\,\zeta\mid\zeta\in I\\}$ is contained in $I$ for all
$\eta\in\mathbb{M}_{n}$ (the multiperplex ring ideal is defined in the same
way on $\mathbb{D}_{n}$). Starting with the description of multiperplex
ideals, we will see later that we can get the multicomplex ideals from the
complexification operation on vector spaces, thus covering the study of both
at the same time.

From properties (i) and (iii) of Proposition 3, the basis elements
$\Gamma_{n}^{\ddagger}=\\{\varepsilon_{k}\\}_{k=1}^{2^{n-1}}$ of the canonical
idempotent representation are an orthogonal decomposition of the identity [7].
This decomposition allow us to write the multiperplex ring as the following
finite direct sum:

$\mathbb{D}_{n}=\bigoplus_{k=1}^{2^{n-1}}\mathbb{D}_{n}\varepsilon_{k}.$ (15)

Each term $\mathbb{D}_{n}\varepsilon_{j}$ of this sum is the principal ideal
generated by the corresponding single element $\varepsilon_{j}$. Since this
decomposition is also a basis of the space, for all $\eta\in\mathbb{D}_{n}$ we
have
$\eta\,\varepsilon_{j}=\big{(}\sum_{k=1}^{2^{n-1}}\eta_{\hat{k}}\varepsilon_{k}\big{)}\varepsilon_{j}=\eta_{\hat{\jmath}}\,\varepsilon_{j}$
and $\mathbb{D}_{n}\varepsilon_{j}=\mathbb{R}\varepsilon_{j}$, meaning that
all principal ideals generated by an element of the canonical idempotent basis
are of the form $\mathbb{R}\varepsilon_{j}$ and the space $\mathbb{D}_{n}$ is
a direct sum of these ideals. We will show that not only $\mathbb{D}_{n}$, but
all multiperplex ideals can be characterized in the same way.

######  Proposition 8.

Let $I\neq\\{0\\}$ be a proper multiperplex ideal of $\mathbb{D}_{n}$. Then
all non-zero elements of $I$ are zero divisors.

###### Proof.

Let $\eta\in I$ be a non-zero element. Suppose that $\eta$ is not a zero
divisor, then $\eta^{-1}\,\eta=1\in I$. Hence, the presence of identity in $I$
implies $I=\mathbb{D}_{n}$, a contradiction. ∎

######  Proposition 9.

The principal ideal $\mathbb{R}\varepsilon_{j}$ is minimal for
$j=1,\ldots,2^{n-1}$.

###### Proof.

Let $I\subset\mathbb{R}\varepsilon_{j}$ be a subideal. Either $I=\\{0\\}$ or
there exists a real number $x\neq 0$ such that $x\varepsilon_{j}\in I$. Then
for all $y\in\mathbb{R}$, $(y/x)\cdot x\varepsilon_{j}=y\varepsilon_{j}\in I$
and $I=\mathbb{R}\varepsilon_{j}$. ∎

######  Lemma 10.

A nontrivial multiperplex ideal contains at least one minimal ideal
$\mathbb{R}\varepsilon_{j}$.

###### Proof.

Let $I\neq\\{0\\}$ a multiperplex ideal. Then there exists a non-zero element
$\eta\in I$ with at least one non-vanishing projection $\eta_{\hat{\jmath}}$.
Then for all $x\in\mathbb{R}$

$\eta\,\frac{x\varepsilon_{j}}{\eta_{\hat{\jmath}}}=\frac{x\eta_{\hat{\jmath}}\varepsilon_{j}}{\eta_{\hat{\jmath}}}=x\varepsilon_{j}\in
I\quad\Rightarrow\quad\mathbb{R}\varepsilon_{j}\subseteq I.$

∎

######  Lemma 11.

For any subset of indices $J\subseteq\\{1,\ldots,2^{n-1}\\}$, the direct sum
$\displaystyle\bigoplus_{j\in J}\mathbb{R}\varepsilon_{j}$ is a multiperplex
principal ideal generated by the element $\displaystyle\sum_{j\in
J}\varepsilon_{j}$.

###### Proof.

We have directly that

$\displaystyle\mathbb{D}_{n}\Big{(}\sum_{j\in J}\varepsilon_{j}\Big{)}$
$\displaystyle=\Big{\\{}\eta\sum_{j\in
J}\varepsilon_{j}\mid\eta\in\mathbb{D}_{n}\Big{\\}}=\Big{\\{}\sum_{j\in
J}\eta_{\hat{\jmath}}\varepsilon_{j}\mid\eta_{\hat{\jmath}}\in\mathbb{R}\Big{\\}}=\bigoplus_{j\in
J}\mathbb{R}\varepsilon_{j}.$

∎

######  Theorem 12.

A multiperplex ideal $I\neq\\{0\\}$ is a principal ideal of the form
$\displaystyle\bigoplus_{j\in J}\mathbb{R}\varepsilon_{j}$, where
$J\subseteq\\{1,\ldots,2^{n-1}\\}$.

###### Proof.

Let $I$ be a nontrivial multiperplex ideal and $\bigoplus_{j\in
J}\mathbb{R}\varepsilon_{j}\subseteq I$ the largest principal ideal contained
in $I$ (from Lemma 10, $I$ contains at least one $\mathbb{R}\varepsilon_{k}$).
If $I\backslash\bigoplus_{j\in J}\mathbb{R}\varepsilon_{j}=\varnothing$ then
$I=\bigoplus_{j\in J}\mathbb{R}\varepsilon_{j}$ which is the desired result.
If $I\backslash\bigoplus_{j\in J}\mathbb{R}\varepsilon_{j}\neq\varnothing$,
then (since $0\in\bigoplus_{j\in J}\mathbb{R}\varepsilon_{j}$) there exists a
non-zero element $\eta\in I\backslash\bigoplus_{j\in
J}\mathbb{R}\varepsilon_{j}$ with a non vanishing projection $\eta_{\hat{k}}$.
The existence of this element in the ideal implies that
$\mathbb{R}\varepsilon_{k}\subseteq I$ and

$\Big{(}\bigoplus_{j\in
J}\mathbb{R}\varepsilon_{j}\Big{)}\oplus\mathbb{R}\varepsilon_{k}\subseteq I.$

Contradicting the fact that $\bigoplus_{j\in J}\mathbb{R}\varepsilon_{j}$ is
the largest direct sum contained in $I$. Thus $I=\bigoplus_{j\in
J}\mathbb{R}\varepsilon_{j}$ and is a principal ideal from Lemma 11. ∎

Having characterized the multiperplex ideals in terms of the minimal ideals,
we turn our attention to the orthogonal hyperplanes $H_{j}$ defined as

$H_{j}:=\\{\eta\in\mathbb{D}_{n}\mid\eta\,\varepsilon_{j}=0\\}.$ (16)

This set corresponds to all multiperplex numbers with a vanishing $j$th
projection. We show easily that $H_{j}$ is an ideal of the multiperplex
numbers since $\zeta\,H_{j}\subseteq H_{j}$ for all $\zeta\in\mathbb{D}_{n}$.

######  Proposition 13.

For $j=1,\ldots,2^{n-1}$ the orthogonal hyperplane $H_{j}$ is a maximal
multiperplex ideal.

###### Proof.

Suppose $H_{j}$ is not maximal, then there is a proper ideal $I$ such that
$H_{j}\subset I$. Let $\eta\in I\backslash H_{j}$, then
$\eta_{\hat{\jmath}}\neq 0$ (otherwise $\eta\in H_{j}$). This implies that
$\mathbb{R}\varepsilon_{j}\subset I$ and

$H_{j}\oplus\mathbb{R}\varepsilon_{j}=\mathbb{D}_{n}\subseteq I.$

Thus we conclude that all orthogonal hyperplanes are maximal ideals. ∎

######  Proposition 14.

For any proper subset of indices $\varnothing\neq
J\subset\\{1,\ldots,2^{n-1}\\}$, we have

$\bigoplus_{j\in J}\mathbb{R}\varepsilon_{j}=\bigcap_{k\in
J^{\complement}}H_{k}.$

###### Proof.

We have directly

$\begin{array}[]{rcl}\eta\in\displaystyle\bigoplus_{j\in
J}\mathbb{R}\varepsilon_{j}&\Leftrightarrow&\eta_{\hat{k}}=0,\ \forall k\in
J^{\complement}\quad\Leftrightarrow\quad\eta\in H_{k},\ \forall k\in
J^{\complement}\\\\[8.61108pt]
&\Leftrightarrow&\eta\in\displaystyle\bigcap_{k\in
J^{\complement}}H_{k}.\end{array}$

∎

From these two last propositions and Theorem 12, we see that all multiperplex
ideals are written in terms of a direct sum of minimal ideals
$\mathbb{R}\varepsilon_{j}$ as well as intersections of maximal ideals
$H_{j}$. Moreover, there are no other ideals distinct from the
$\mathbb{R}\varepsilon_{j}$ and $H_{j}$ that are also minimal or maximal and
the characterization of multiperplex is thus completed.

###  3.2 Complexification and realization

Let $\mathcal{I}(\mathbb{D}_{n})$ and $\mathcal{I}(\mathbb{M}_{n})$ be the
respective sets of multiperplex and multicomplex ideals. The complexification
is always applied on a vector space (or algebra) $V$ over $\mathbb{R}$ as the
extension of the scalar multiplication over $\mathbb{C}$.

$(V,+,\cdot,\mathbb{R})\xrightarrow[]{\text{Complexification}}(V\oplus\mathrm{i}V,+,\cdot,\mathbb{C})$
(17)

Both bases and dimension of $V$ are preserved by the complexification.

######  Proposition 15.

The complexification of a multiperplex ideal is a multicomplex ideal.

###### Proof.

Let $I_{D}\in\mathcal{I}(\mathbb{D}_{n})$. Considering the complexification
$I_{D}\oplus\mathrm{i}I_{D}$ of $I_{D}$, we suppose that
$d_{1}+\mathrm{i}d_{2}\in I_{D}\oplus\mathrm{i}I_{D}$. Then, from (14) any
$\eta\in\mathbb{M}_{n}$ can be written as
$\eta=d_{1}^{\prime}+\mathrm{i}d_{2}^{\prime}$ where
$d_{1}^{\prime},d_{2}^{\prime}\in\mathbb{D}_{n}$. We find

$\eta\,(d_{1}+\mathrm{i}d_{2})=(d_{1}^{\prime}+\mathrm{i}d_{2}^{\prime})\,(d_{1}+\mathrm{i}d_{2})=(d_{1}^{\prime}d_{1}-d_{2}^{\prime}d_{2})+\mathrm{i}(d_{1}^{\prime}d_{2}+d_{2}^{\prime}d_{1}).$

Since $I_{D}$ is an ideal, all products of the right-hand side of the equation
are in $I_{D}$ and $\eta\,(d_{1}+\mathrm{i}d_{2})\in
I_{D}\oplus\mathrm{i}I_{D}$. ∎

From this proposition, the complexification can be seen as a well-defined
function from $\mathcal{I}(\mathbb{D}_{n})$ to $\mathcal{I}(\mathbb{M}_{n})$.

Let us now define the set operator $\mathcal{R}$ by

$\mathcal{R}(A):=\\{\eta\in A\mid\eta^{\Lambda}=\eta\\},\qquad\forall
A\subseteq\mathbb{M}_{n}.$ (18)

We note that for any multicomplex ideal $I_{M}$ (or multicomplex subring) then
$\mathcal{R}(I_{M})$ is nonempy since $0\in I_{M}$ and $0^{\Lambda}=0$.

######  Proposition 16.

Let $I_{M}$ be a multicomplex ideal. Then $\mathcal{R}(I_{M})$ is a
multiperplex ideal such that $\mathcal{R}$ is a mapping from the set of
multicomplex ideals to the set of multiperplex ideals, i.e.
$\mathcal{R}:\mathcal{I}(\mathbb{M}_{n})\rightarrow\mathcal{I}(\mathbb{D}_{n})$.

###### Proof.

Let $I_{M}\in\mathcal{I}(\mathbb{M}_{n})$. Then for all
$\zeta\in\mathbb{D}_{n}$ and $\eta\in\mathcal{R}(I_{M})$, we have

$(\zeta\,\eta)^{\Lambda}=\zeta^{\Lambda}\,\eta^{\Lambda}=\zeta\,\eta$

such that $\zeta\mathcal{R}(I_{M})\subseteq\mathcal{R}(I_{M})$. ∎

######  Proposition 17.

Let $I_{M}$ be a multicomplex ideal. If $\eta\in I_{M}$, then
$\eta^{\Lambda}\in I_{M}$.

###### Proof.

Let $\eta\in I_{M}$ written in the canonical representation :

$\eta=\displaystyle\sum_{k=1}^{2^{n-1}}\eta_{\hat{k}}\varepsilon_{k}$

and take $\zeta\in\mathbb{M}_{n}$ such that

$P_{k}(\zeta)=\zeta_{\hat{k}}=\begin{cases}0&\text{ if
}\eta_{\hat{k}}=0,\\\\[8.61108pt]
\overline{\eta_{\hat{k}}}^{2}/|\eta_{\hat{k}}|^{2}&\text{ if
}\eta_{\hat{k}}\neq 0,\end{cases}$

for $k=1,\ldots,2^{n-1}$. Then for all $k$,

$P_{k}(\zeta\,\eta)=\zeta_{\hat{k}}\,\eta_{\hat{k}}=\begin{cases}0&\text{ if
}\eta_{\hat{k}}=0\\\\[8.61108pt] \overline{\eta_{\hat{k}}}&\text{ if
}\eta_{\hat{k}}\neq 0\\\ \end{cases}\quad\Rightarrow\quad
P_{k}(\zeta\,\eta)=\overline{\eta_{\hat{k}}}.$

Thus $\zeta\,\eta=\eta^{\Lambda}$ and since $I_{M}$ is an ideal,
$\eta^{\Lambda}\in I_{M}$. ∎

######  Lemma 18.

The mapping
$\mathcal{R}:\mathcal{I}(\mathbb{M}_{n})\rightarrow\mathcal{I}(\mathbb{D}_{n})$
is the inverse of the complexification from $\mathcal{I}(\mathbb{D}_{n})$ to
$\mathcal{I}(\mathbb{M}_{n})$, i.e. for all
$I_{D}\in\mathcal{I}(\mathbb{D}_{n})$ and
$I_{M}\in\mathcal{I}(\mathbb{M}_{n})$

$\mathcal{R}(I_{D}\oplus\mathrm{i}I_{D})=I_{D}\qquad\text{ and
}\qquad\mathcal{R}(I_{M})\oplus\mathrm{i}\mathcal{R}(I_{M})=I_{M}.$

###### Proof.

Let $\eta=d_{1}+\mathrm{i}d_{2}\in I_{D}\oplus\mathrm{i}I_{D}$, then

$\begin{array}[]{rcl}\eta\in\mathcal{R}(I_{D}\oplus\mathrm{i}I_{D})&\Leftrightarrow&\eta^{\Lambda}=\eta\
\Leftrightarrow\
(d_{1}+\mathrm{i}d_{2})^{\Lambda}=d_{1}+\mathrm{i}d_{2}\\\\[8.61108pt]
&\Leftrightarrow&d_{2}=0\text{ and }\eta\in I_{D}.\end{array}$

Now, let
$\zeta=d_{1}^{\prime}+\mathrm{i}d_{2}^{\prime}\in\mathcal{R}(I_{M})\oplus\mathrm{i}\mathcal{R}(I_{M})$.
Since $\mathcal{R}(I_{M})\subseteq I_{M}$ from the definition of
$\mathcal{R}$, $\zeta$ is in $I_{M}$. Conversely, if
$\zeta=d_{1}+\mathrm{i}d_{2}\in I_{M}$ then from Proposition 17,
$\zeta^{\Lambda}=d_{1}-\mathrm{i}d_{2}\in I_{M}$ and

$\frac{\zeta+\zeta^{\Lambda}}{2}=d_{1}\in
I_{M},\qquad\frac{\zeta-\zeta^{\Lambda}}{2\mathrm{i}}=d_{2}\in I_{M}.$

Since $d_{1},d_{2}$ are multiperplex elements in $I_{M}$, they are invariant
under $\Lambda$, thus $d_{1},d_{2}\in\mathcal{R}(I_{M})$ and
$\zeta\in\mathcal{R}(I_{M})\oplus\mathrm{i}\mathcal{R}(I_{M})$. ∎

From ring theory, the intersection of two ideals is an ideal and this
operation defines an algebraic structure on $\mathcal{I}(\mathbb{D}_{n})$ and
$\mathcal{I}(\mathbb{M}_{n})$. More specifically,
$(\mathcal{I}(\mathbb{D}_{n}),\ \cap)$ and $(\mathcal{I}(\mathbb{M}_{n}),\
\cap)$ are both monoïds with respective identity elements $\mathbb{D}_{n}$ and
$\mathbb{M}_{n}$.

######  Theorem 19.

The complexification is a one-to-one correspondence from
$\mathcal{I}(\mathbb{D}_{n})$ to $\mathcal{I}(\mathbb{M}_{n})$ preserving the
intersection, i.e. for any $I_{1},I_{2}\in\mathcal{I}(\mathbb{D}_{n})$,

$(I_{1}\cap I_{2})\oplus\mathrm{i}(I_{1}\cap
I_{2})=(I_{1}\oplus\mathrm{i}I_{1})\cap(I_{2}\oplus\mathrm{i}I_{2}).$

###### Proof.

The existence of the inverse from Lemma 18 is sufficient to conclude that the
complexification is bijective between $\mathcal{I}(\mathbb{D}_{n})$ and
$\mathcal{I}(\mathbb{M}_{n})$. If $I_{1}$ and $I_{2}$ are two multiperplex
ideals then $\eta=d_{1}+\mathrm{i}d_{2}\in(I_{1}\cap
I_{2})\oplus\mathrm{i}(I_{1}\cap I_{2})$ if and only if

$\begin{array}[]{rcl}d_{1},d_{2}\in I_{1}\cap
I_{2}&\Leftrightarrow&d_{1},d_{2}\in I_{1}\text{ and }d_{1},d_{2}\in
I_{2}\\\\[8.61108pt] &\Leftrightarrow&d_{1}+\mathrm{i}d_{2}\in
I_{1}\oplus\mathrm{i}I_{1}\text{ and }d_{1}+\mathrm{i}d_{2}\in
I_{2}\oplus\mathrm{i}I_{2}\\\\[8.61108pt]
&\Leftrightarrow&\eta\in(I_{1}\oplus\mathrm{i}I_{1})\cap(I_{2}\oplus\mathrm{i}I_{2}).\end{array}$

∎

Note that the inclusion between ideals is also conserved since $I_{1}\subseteq
I_{2}\Leftrightarrow I_{1}\cap I_{2}=I_{1}$. From Theorem 19, for all
$I_{1},I_{2}\in\mathcal{I}(\mathbb{D}_{n})$,

$I_{1}\subseteq I_{2}\quad\Leftrightarrow\quad
I_{1}\oplus\mathrm{i}I_{1}\subseteq I_{2}\oplus\mathrm{i}I_{2}.$ (19)

###  3.3 Translation to the multicomplex ring

From Theorem 19, we get the multicomplex minimal and maximal ideals from the
complexification of $\mathbb{R}\varepsilon_{j}$ and $H_{j}$.

$\displaystyle\mathbb{R}\varepsilon_{j}$
$\displaystyle\rightarrow\mathbb{R}\varepsilon_{j}\oplus\mathrm{i}\mathbb{R}\varepsilon_{j}=\mathbb{C}\varepsilon_{j},$
(20) $\displaystyle H_{j}$ $\displaystyle\rightarrow
H_{j}\oplus\mathrm{i}H_{j}.$

The complexification of $H_{j}$ is itself a multicomplex hyperplane denoted
$E_{j}$ where

$E_{j}=H_{j}\oplus\mathrm{i}H_{j}=\\{\eta\in\mathbb{M}_{n}\mid\eta\,\varepsilon_{j}=0\\}.$
(21)

Indeed, for all $\eta=d_{1}+\mathrm{i}d_{2}\in\mathbb{M}_{n}$,
$d_{1}\varepsilon_{j}=0$ and $d_{2}\varepsilon_{j}=0$ if and only if
$\eta\,\varepsilon_{j}=0$. In the same way that $\mathbb{R}\varepsilon_{j}$
and $H_{j}$ can be seen as the building blocks of multiperplex ideals, a
complete characterization of multicomplex ideals is done in terms of
$\mathbb{C}\varepsilon_{j}$ and $E_{j}$.

######  Corollary 20.

All multicomplex ideals are principal ideals of the form

$\bigoplus_{j\in J}\mathbb{C}\varepsilon_{j}=\bigcap_{k\in
J^{\complement}}E_{k}$

generated by the elements $\displaystyle\sum_{j\in J}\varepsilon_{j}$, where
$\varnothing\neq J\subseteq\\{1,\ldots,2^{n-1}\\}$.

Taking any multicomplex ideal $I_{J}$ with a given subset of indices $J$, the
corresponding quotient ring $\mathbb{M}_{n}/I_{J}$ is written as

$\mathbb{M}_{n}/I_{J}=\\{\eta+I_{J}\mid\eta\in\mathbb{M}_{n}\\}=\\{\zeta+I_{J}\mid\zeta\in
I_{J^{\complement}}\\}\simeq I_{J^{\complement}}.$

The last part comes from the canonical representation which let us separate
$\eta$ into two distinct sums : one contained in $I_{J}$ and the other one in
$I_{J^{\complement}}$.

##  4 Free $\mathbb{M}_{n}$-module

###  4.1 Bases and subspaces

From [21], any free module over a commutative ring has a well-defined rank. As
$\mathbb{M}_{n}$ is a commutative algebra, any two basis sets of a free
$\mathbb{M}_{n}$-module have the same (fixed) cardinality $m$.

Let $W$ be a free $\mathbb{M}_{n}$-module and $\\{|w_{l}\rangle\\}_{l=1}^{m}$
a basis made up of $m<\infty$ elements. This describes the finite-dimensional
case, and an element $|\psi\rangle$ of $W$ is written as a linear combination
of the basis elements :

$|\psi\rangle=\sum_{l=1}^{m}\eta_{l}|w_{l}\rangle,\qquad\eta_{l}\in\mathbb{M}_{n}.$
(22)

An important subset $V\subset W$ is the set of all elements with coefficients
restricted to the field of complex numbers

$V:=\Big{\\{}\sum_{l=1}^{m}z_{l}|w_{l}\rangle\mid
z_{l}\in\mathbb{C}\Big{\\}}.$ (23)

This space $V$ is a $m$-dimensional vector space over $\mathbb{C}$, see [19].
We can use the canonical idempotent representation of multicomplex numbers to
further develop the expression of any $|\psi\rangle$ in $W$. For each
$\eta_{l}\in\mathbb{M}_{n}$ we have :

$\eta_{l}=\sum_{k=1}^{2^{n-1}}z_{l,\hat{k}}\varepsilon_{k},\qquad
z_{l,\hat{k}}\in\mathbb{C}.$ (24)

Substituting in (22), we get

$|\psi\rangle=\sum_{l=1}^{m}\Big{(}\sum_{k=1}^{2^{n-1}}z_{l,\hat{k}}\varepsilon_{k}\Big{)}|w_{l}\rangle=\sum_{k=1}^{2^{n-1}}\varepsilon_{k}\sum_{l=1}^{m}z_{l,\hat{k}}|w_{l}\rangle$
(25)

and we set

$|\psi\rangle_{\hat{k}}:=\sum_{l=1}^{m}z_{l,\hat{k}}|w_{l}\rangle.$ (26)

Hence, for a given basis of $W$ any element $|\psi\rangle\in W$ can be written
uniquely as

$|\psi\rangle=\sum_{k=1}^{2^{n-1}}\varepsilon_{k}|\psi\rangle_{\hat{k}},\qquad|\psi\rangle_{\hat{k}}\in
V.$ (27)

From that representation of any element $|\psi\rangle\in W$, we can define the
ket projector $P_{j}:W\rightarrow V$ as

$P_{j}|\psi\rangle:=|\psi\rangle_{\hat{\jmath}},\qquad j=1,\ldots,2^{n-1}.$
(28)

######  Remark 21.

Without ambiguity, we use the same notation $P_{j}$ both for the multicomplex
projector $P_{j}:\mathbb{M}_{n}\rightarrow\mathbb{C}$ defined in (11) and the
module projector $P_{j}:W\rightarrow V$ defined in (28).

One can show that $P_{j}:W\rightarrow V$ is quasilinear, i.e. for any
$|\psi\rangle,|\phi\rangle\in W$ and $\alpha\in\mathbb{M}_{n}$:

$\displaystyle P_{j}(|\psi\rangle+|\phi\rangle)$
$\displaystyle=P_{j}|\psi\rangle+P_{j}|\phi\rangle=|\psi\rangle_{\hat{\jmath}}+|\phi\rangle_{\hat{\jmath}}$
(29) $\displaystyle P_{j}(\alpha|\psi\rangle)$
$\displaystyle=P_{j}(\alpha)P_{j}|\psi\rangle=\alpha_{\hat{\jmath}}|\psi\rangle_{\hat{\jmath}},$

where $P_{j}(\alpha)$ is the multicomplex projector.

######  Remark 22.

The definitions of $V$ in (23) and the module projector $P_{j}$ in (28) depend
on the choice of the basis $\\{|w_{l}\rangle\\}_{k=1}^{m}$ since each
$|w_{l}\rangle$ could be expanded in a new basis with multicomplex
coefficients. The elements of $V$ or any projected ket
$|\psi\rangle_{\hat{\jmath}}$ would not necessarily have complex coefficients
when written in this new basis.

We say that a ket $|\psi\rangle$ is in the null cone of $W$ if for at least
one $1\leq j\leq 2^{n-1}$, $|\psi\rangle_{\hat{\jmath}}=0$. Using that
framework, we now turn our attention to the properties of an arbitrary basis
of $W$.

######  Theorem 23.

No basis elements of a free $\mathbb{M}_{n}$-module can belong to the null
cone.

###### Proof.

Let $|w_{p}\rangle$ be an element of a basis of $W$. We can write

$|w_{p}\rangle=\sum_{k=1}^{2^{n-1}}\varepsilon_{k}|w_{p}\rangle_{\hat{k}}.$

If we suppose that $|w_{p}\rangle$ is in the null cone, then
$|w_{p}\rangle_{\hat{l}}=0$ for at least one $\hat{l}$. This implies

$\varepsilon_{l}|w_{p}\rangle=\varepsilon_{l}|w_{p}\rangle_{\hat{l}}=0,$

but this last equation contradicts linear independence of the basis. ∎

For a given basis $\\{w_{l}\\}_{l=1}^{m}$ of $W$, let us now define

$\varepsilon_{k}V:=\big{\\{}\varepsilon_{k}\sum_{l=1}^{m}z_{l}|w_{l}\rangle\mid
z_{l}\in\mathbb{C}\big{\\}},\qquad k=1,\ldots,2^{n-1}.$

Trivially we see that any $\varepsilon_{k}V$ is an $m$-dimensional vector
space over $\mathbb{C}$ isomorphic to $V$, where
$\\{\varepsilon_{k}|w_{l}\rangle\\}_{l=1}^{m}$ is a basis. From (25) we see
that the $2^{n-1}\cdot m$ elements $\varepsilon_{k}|w_{l}\rangle$ are linearly
independent over $\mathbb{C}$. Indeed, $|\psi\rangle=0$ if and only if
$z_{l,\hat{k}}=0$ for all $l,k$, which implies the uniqueness of that
representation. Moreover, each term in the summation over $k$ belongs to
$\varepsilon_{k}V$. These two statements lead us to the next theorem.

######  Theorem 24.

An $m$-dimensional $\mathbb{M}_{n}$-module $W$ is a $(2^{n-1}\,m)$-dimensional
vector space over $\mathbb{C}$ where

$W=\varepsilon_{1}V\oplus\varepsilon_{2}V\oplus\cdots\oplus\varepsilon_{2^{n-1}}V.$

######  Theorem 25.

Let $P_{j}:W\rightarrow V$ be a projector for a given $1\leq j\leq 2^{n-1}$
and $V$ the subspace defined with respect to the basis
$\\{|w_{l}\rangle\\}_{l=1}^{m}$ of $W$. If $\\{|s_{l}\rangle\\}_{l=1}^{m}$ is
another basis of $W$, then $\\{P_{j}|s_{l}\rangle\\}_{l=1}^{m}$ is a basis of
$V$.

###### Proof.

We first show that the kets
$P_{j}|s_{1}\rangle,P_{j}|s_{2}\rangle,\ldots,P_{j}|s_{m}\rangle$ are linearly
independent for any fixed value $j$. Let $\alpha_{l}\in\mathbb{C}$ for
$l=1,\ldots,m$ and

$\sum_{l=1}^{m}\alpha_{l}P_{j}|s_{l}\rangle=0.$

By defining $\beta_{l}:=\alpha_{l}\varepsilon_{j}$, we obtain
$P_{k}(\beta_{l})=\delta_{kj}\alpha_{l}$. From which we get

$\begin{array}[]{l}P_{k}\Big{(}\displaystyle\sum_{l=1}^{m}\beta_{l}|s_{l}\rangle\Big{)}=\displaystyle\sum_{l=1}^{m}P_{k}(\beta_{l})P_{k}|s_{l}\rangle=\displaystyle\sum_{l=1}^{m}\delta_{kj}\alpha_{l}P_{k}|s_{l}\rangle=\displaystyle\sum_{l=1}^{m}\alpha_{l}P_{j}|s_{l}\rangle=0.\end{array}$

Since this last equation is valid for every $k=1,\ldots,2^{n-1}$, we have
$\sum_{l=1}^{m}\beta_{l}|s_{l}\rangle=0$. The set
$\\{|s_{l}\rangle\\}_{l=1}^{m}$ being a basis of $W$, for $l=1,\ldots,m$ we
have $\beta_{l}=0$ such that $\alpha_{l}=0$ and
$\\{P_{j}|s_{l}\rangle\\}_{l=1}^{m}$ is a linearly independent set.

We now show that this set span $V$. Let $|\phi\rangle\in V$ and consider the
ket

$|\psi\rangle=\varepsilon_{j}|\phi\rangle\in W.$

Since the multicomplex span of $\\{|s_{l}\rangle\\}_{l=1}^{m}$ is $W$, there
exists $\beta_{l}\in\mathbb{M}_{n}$ such that
$\sum_{l=1}^{m}\beta_{l}|s_{l}\rangle=|\psi\rangle.$ Therefore,

$|\phi\rangle=P_{j}|\psi\rangle=P_{j}\big{(}\sum_{l=1}^{m}\beta_{l}|s_{l}\rangle\big{)}=\sum_{l=1}^{m}P_{j}(\beta_{l})P_{j}|s_{l}\rangle.$

Thus the complex span of $\\{P_{j}|s_{l}\rangle\\}_{l=1}^{m}$ is $V$. ∎

######  Proposition 26.

Let $|\psi\rangle\in W$ and $1\leq j\leq 2^{n-1}$. If
$|\psi\rangle_{\hat{\jmath}}=P_{j}|\psi\rangle=0$ for a given basis of $W$,
then the $j$-th projection of $|\psi\rangle$ is zero for any other basis of
$W$.

###### Proof.

For a given basis $\\{|w_{l}\rangle\\}_{l=1}^{m}$ of $W$, $|\psi\rangle$ can
be written as
$|\psi\rangle=\sum_{k=1}^{2^{n-1}}\varepsilon_{k}|\psi\rangle_{\hat{k}}$,
where $|\psi\rangle_{\hat{k}}=\sum_{l=1}^{m}z_{l,\hat{k}}|w_{l}\rangle$ for
$z_{l,\hat{k}}\in\mathbb{C}$. If
$|\psi\rangle_{\hat{\jmath}}=P_{j}|\psi\rangle=0$ in this basis then
$z_{1,\hat{\jmath}}=\cdots=z_{m,\hat{\jmath}}=0$. Suppose now a second basis
$\\{|\widetilde{w}_{p}\rangle\\}_{p=1}^{m}$ of $W$. In a similar way we find
$|\psi\rangle=\sum_{k=1}^{2^{n-1}}\varepsilon_{k}|\widetilde{\psi}\rangle_{\hat{k}}$,
where
$|\widetilde{\psi}\rangle_{\hat{k}}=\sum_{l=1}^{m}\widetilde{z}_{l,\hat{k}}|\widetilde{w}_{l}\rangle$
for $\widetilde{z}_{l,\hat{k}}\in\mathbb{C}$. We want to show that

$|\widetilde{\psi}\rangle_{\hat{\jmath}}=\sum_{l=1}^{m}\widetilde{z}_{l,\hat{\jmath}}|\widetilde{w}_{l}\rangle=0\qquad\Leftrightarrow\qquad
z_{1,\hat{\jmath}}=\cdots=z_{m,\hat{\jmath}}=0.$ (30)

The two bases are related by a multicomplex nonsingular matrix $(\gamma_{lp})$
[11] such that

$|w_{l}\rangle=\sum_{p=1}^{m}\gamma_{lp}|\widetilde{w}_{p}\rangle,\qquad\gamma_{lp}\in\mathbb{M}_{n}.$

and $|\psi\rangle_{\hat{k}}$ can be expressed as

$|\psi\rangle_{\hat{k}}=\sum_{l=1}^{m}z_{l,\hat{k}}\sum_{p=1}^{m}\gamma_{lp}|\widetilde{w}_{p}\rangle=\sum_{p=1}^{m}\Big{(}\sum_{l=1}^{m}z_{l,\hat{k}}\gamma_{lp}\Big{)}|\widetilde{w}_{p}\rangle.$

Therefore, we have
$\widetilde{z}_{p,\hat{k}}=\sum_{l=1}^{m}z_{l,\hat{k}}\gamma_{lp}$ and, in
particular, we find that (30) is satisfied. ∎

###  4.2 Multicomplex matrices and determinants

A multicomplex $m\times m$ square matrix $A$ is an array of $m^{2}$
multicomplex numbers $A_{ij}$. Each element can be written as

$A_{ij}=\sum_{k=1}^{2^{n-1}}a_{ij\hat{k}}\varepsilon_{k},\qquad
a_{ij\hat{k}}\in\mathbb{C}.$ (31)

The matrix itself is then

$A=(A_{ij})_{m\times
m}=\Big{(}\sum_{k=1}^{2^{n-1}}a_{ij\hat{k}}\varepsilon_{k}\Big{)}_{m\times
m}=\sum_{k=1}^{2^{n-1}}\varepsilon_{k}(a_{ij\hat{k}})_{m\times m}.$ (32)

Applying the projection operator $P_{l}$ on $A$ gives the $m^{2}$ associated
complex matrix:

$A_{\hat{l}}=P_{l}A=(a_{ij\hat{l}})_{m\times m}.$ (33)

Thus any multicomplex square matrix $A$ can be written as

$A=\sum_{k=1}^{2^{n-1}}\varepsilon_{k}A_{\hat{k}},\qquad A_{\hat{k}}\in
M_{m\times m}(\mathbb{C}).$ (34)

######  Theorem 27.

Let $A=\displaystyle\sum_{k}\varepsilon_{k}A_{\hat{k}}$ be an $m\times m$
multicomplex matrix. Then

$\det A=\sum_{k=1}^{2^{n-1}}\varepsilon_{k}\det A_{\hat{k}}.$

###### Proof.

Let $\\{C_{i}\\}_{i=1}^{m}$ be the set of columns of $A$, so that
$A=(C_{1},C_{2},\ldots,C_{m})$. We can write the $i$th column

$C_{i}=\sum_{k=1}^{2^{n-1}}\varepsilon_{k}C_{i}^{(k)},$

where the columns $C_{i}^{(k)}$ are complex. Since the determinant is a
multilinear function, for $C_{1}$ we have

$\det
A=\det\Big{(}\displaystyle\sum_{k=1}^{2^{n-1}}\varepsilon_{k}C_{1}^{(k)},C_{2},\ldots,C_{m}\Big{)}=\displaystyle\sum_{k=1}^{2^{n-1}}\varepsilon_{k}\det\Big{(}C_{1}^{(k)},C_{2},\ldots,C_{m}\Big{)}.$

By applying thi procedure successively for the remaining columns
$C_{2},\ldots,C_{m}$ we find

$\det
A=\det(C_{1},\ldots,C_{m})=\sum_{k=1}^{2^{n-1}}\varepsilon_{k}\det(C_{1}^{(k)},\ldots,C_{m}^{(k)}).$

∎

From the previous theorem we can see that $\det A=0$ if and only if $\det
A_{\hat{l}}=0$ for $l=1,\ldots,m$. Moreover, $\det A$ is in the null cone if
$\det A_{\hat{l}}=0$ for at least one $l$.

######  Definition 28.

A multicomplex square matrix is singular if its determinant is in the null
cone.

######  Theorem 29.

The inverse $A^{-1}$ of a multicomplex square matrix $A$ exists if and only if
$A$ is nonsingular. Then $A^{-1}$ is given by
$A^{-1}=\displaystyle\sum_{k=1}^{2^{n-1}}\varepsilon_{k}(A_{\hat{k}})^{-1}$.

###### Proof.

If $A^{-1}$ exists then $A^{-1}A=I$ such that $1=\det(A^{-1}A)=\det A^{-1}\det
A$. Consequently, $\det A$ is not in the null cone. Conversely, if $A$ is
nonsingular then $\det A_{\hat{l}}\neq 0$ for $l=1,\ldots,2^{n-1}$ and

$\Big{(}\sum_{k=1}^{2^{n-1}}\varepsilon_{k}(A_{\hat{k}})^{-1}\Big{)}\cdot\Big{(}\sum_{k=1}^{2^{n-1}}\varepsilon_{k}A_{\hat{k}}\Big{)}=\sum_{k=1}^{2^{n-1}}\varepsilon_{k}I=I.$

∎

###  4.3 Linear operators

######  Definition 30.

A multicomplex linear operator is a function $A:W\rightarrow W$ such that for
all $\alpha,\beta\in\mathbb{M}_{n}$ and $|\psi\rangle,|\phi\rangle\in W$,

$A(\alpha|\psi\rangle+\beta|\phi\rangle)=\alpha A|\psi\rangle+\beta
A|\phi\rangle.$

For any $|\psi\rangle\in W$, we set
$A_{\hat{\jmath}}|\psi\rangle:=P_{j}A|\psi\rangle$, then we obtain

$\begin{array}[]{rcl}A_{\hat{\jmath}}|\psi\rangle&=&P_{j}A|\psi\rangle=P_{j}A\displaystyle\sum_{k=1}^{2^{n-1}}\varepsilon_{k}|\psi\rangle_{\hat{k}}=P_{j}\Big{(}\sum_{k=1}^{2^{n-1}}\varepsilon_{k}A|\psi\rangle_{\hat{k}}\Big{)}\\\\[8.61108pt]
&=&\displaystyle\sum_{k=1}^{2^{n-1}}P_{j}(\varepsilon_{k})P_{j}A|\psi\rangle_{\hat{k}}=P_{j}A|\psi\rangle_{\hat{\jmath}}=A_{\hat{\jmath}}|\psi\rangle_{\hat{\jmath}}.\end{array}$
(35)

######  Remark 31.

Note that for an arbitrary ket $|\psi\rangle\in W$, we have
$AP_{j}|\psi\rangle\in W$ and $P_{j}A|\psi\rangle\in V$, hence a multicomplex
linear operator $A$ do not commute with $P_{j}$ in general.

######  Theorem 32.

A multicomplex linear operator $A:W\rightarrow W$ commutes with the ket
projector $P_{j}:W\rightarrow V$ for $1\leq j\leq 2^{n-1}$ if and only if
$AW\subseteq V$.

###### Proof.

Suppose $A$ and $P_{j}$ commute, then for all $|\psi\rangle\in W$ we have

$0=(P_{j}A-AP_{j})|\psi\rangle=A_{\hat{\jmath}}|\psi\rangle_{\hat{\jmath}}-A|\psi\rangle_{\hat{\jmath}}$

such that $A=A_{\hat{\jmath}}$ and $A:W\rightarrow V$.

Conversely if we suppose $AW\subseteq V$, for any $|\psi\rangle\in W$ we have

$P_{j}A|\psi\rangle=P_{j}A|\psi\rangle_{\hat{\jmath}}=A|\psi\rangle_{\hat{\jmath}}$

and

$AP_{j}|\psi\rangle=A|\psi\rangle_{\hat{\jmath}}$

such that $[A,P_{j}]=0$. ∎

######  Definition 33.

A multicomplex linear operator $A:W\rightarrow W$ belongs to the null cone if
for at least one $1\leq j\leq 2^{n-1}$ we have $A_{\hat{\jmath}}=0$.

######  Definition 34.

Let $A:W\rightarrow W$ be a multicomplex linear operator and let

$A|\psi\rangle=\lambda|\psi\rangle,\qquad\lambda\in\mathbb{M}_{n}$

where $|\psi\rangle$ is not in the null cone. Then $\lambda$ is called an
eigenvalue of $A$ and $|\psi\rangle$ is the corresponding eigenket.

Expanding the expression $A|\psi\rangle=\lambda|\psi\rangle$ we get

$A\sum_{k=1}^{2^{n-1}}\varepsilon_{k}|\psi\rangle_{\hat{k}}=\lambda\sum_{k=1}^{2^{n-1}}\varepsilon_{k}|\psi\rangle_{\hat{k}}\quad\Rightarrow\quad\sum_{k=1}^{2^{n-1}}\varepsilon_{k}A|\psi\rangle_{\hat{k}}=\sum_{k=1}^{2^{n-1}}\varepsilon_{k}\lambda|\psi\rangle_{\hat{k}}.$
(36)

Applying $P_{j}$ on both sides gives us

$A_{\hat{\jmath}}|\psi\rangle_{\hat{\jmath}}=\lambda_{\hat{\jmath}}|\psi\rangle_{\hat{\jmath}},\qquad\lambda_{\hat{\jmath}}=P_{j}(\lambda).$
(37)

This last equation means that the eigenvalue of a ket projection of $A$ is the
corresponding multicomplex projection of $\lambda$.

######  Theorem 35.

Let $A$ and $B$ be two multicomplex linear operators. Then for all $1\leq
j\leq 2^{n-1}$ we have the following properties:

$P_{j}(A+B)=A_{\hat{\jmath}}+B_{\hat{\jmath}}\qquad\text{and}\qquad
P_{j}(AB)=A_{\hat{\jmath}}B_{\hat{\jmath}}.$

###### Proof.

Let $|\psi\rangle\in W$, then

$\begin{array}[]{r c
l}P_{j}(A+B)|\psi\rangle&=&P_{j}(A|\psi\rangle+B|\psi\rangle)=A_{\hat{\jmath}}|\psi\rangle_{\hat{\jmath}}+B_{\hat{\jmath}}|\psi\rangle_{\hat{\jmath}}=A_{\hat{\jmath}}|\psi\rangle+B_{\hat{\jmath}}|\psi\rangle\\\\[8.61108pt]
&=&(A_{\hat{\jmath}}+B_{\hat{\jmath}})|\psi\rangle.\\\ \end{array}$

Writing $|\phi\rangle:=B|\psi\rangle$, we have

$P_{j}AB|\psi\rangle=P_{j}A|\phi\rangle=A_{\hat{\jmath}}|\phi\rangle_{\hat{\jmath}}=A_{\hat{\jmath}}P_{j}|\phi\rangle=A_{\hat{\jmath}}P_{j}B|\psi\rangle=A_{\hat{\jmath}}B_{\hat{\jmath}}|\psi\rangle.$

∎

Once again, an equivalent proof of the next theorem is found in [11], since
the case with bicomplex numbers is easily generalized to that of multicomplex
numbers from the previous results.

######  Theorem 36.

The action of a linear multicomplex operator on $W$ can be represented by a
multicomplex matrix.

##  5 Multicomplex Hilbert spaces

###  5.1 Scalar product

######  Definition 37.

The multicomplex scalar product is a function associating a multicomplex
number to each pair of elements $|\psi\rangle,|\phi\rangle\in W$ which for all
$|\chi\rangle\in W$ and $\alpha\in\mathbb{M}_{n}$ satisfies:

  1. 1.

$(|\psi\rangle,|\phi\rangle+|\chi\rangle)=(|\psi\rangle,|\phi\rangle)+(|\psi\rangle,|\chi\rangle)$;

  2. 2.

$(|\psi\rangle,\alpha|\phi\rangle)=\alpha(|\psi\rangle,|\phi\rangle)$;

  3. 3.

$(|\psi\rangle,|\phi\rangle)=(|\phi\rangle,|\psi\rangle)^{\Lambda}$;

  4. 4.

$(|\psi\rangle,|\psi\rangle)\in\mathbb{D}_{n}^{+}$ and
$(|\psi\rangle,|\psi\rangle)=0$ if and only if $|\psi\rangle=0$.

Here $\mathbb{D}_{n}^{+}$ is the set of multicomplex numbers with real and
positive components in the idempotent canonical representation. The $j$th
projection of the multicomplex scalar product is denoted

$(\cdot\,,\,\cdot)_{\hat{\jmath}}:=P_{j}(\cdot\,,\,\cdot)$ (38)

and from Definition 37 it follows that any projection is itself a well defined
standard scalar product on the associated vector space $\varepsilon_{j}V$ (as
well as on $V$), i.e.

$\begin{array}[]{l}(|\psi\rangle,|\phi\rangle+|\chi\rangle)_{\hat{\jmath}}=(|\psi\rangle,|\phi\rangle)_{\hat{\jmath}}+(|\psi\rangle,|\chi\rangle)_{\hat{\jmath}},\quad(|\psi\rangle,\alpha|\phi\rangle)_{\hat{\jmath}}=\alpha_{\hat{\jmath}}(|\psi\rangle,|\phi\rangle)_{\hat{\jmath}},\\\\[8.61108pt]
(|\psi\rangle,|\phi\rangle)_{\hat{\jmath}}=\overline{(|\phi\rangle,|\psi\rangle)}_{\hat{\jmath}}\quad\text{and}\quad(|\psi\rangle,|\psi\rangle)_{\hat{\jmath}}\in\mathbb{R}^{+},\quad(|\psi\rangle,|\psi\rangle)_{\hat{\jmath}}=0\
\Leftrightarrow|\psi\rangle_{\hat{\jmath}}=0.\end{array}$ (39)

######  Proposition 38.

A ket $|\psi\rangle\in W$ is in the null cone if and only if its multicomplex
scalar product $(|\psi\rangle,|\psi\rangle)$ is in the null cone.

###### Proof.

It’s a direct consequence of the last equivalence in (39). ∎

######  Theorem 39.

Let $|\psi\rangle,|\phi\rangle\in W$, then

$(|\psi\rangle,|\phi\rangle)=\sum_{k=1}^{2^{n-1}}(|\psi\rangle_{\hat{k}},|\phi\rangle_{\hat{k}})_{\hat{k}}\,\varepsilon_{k}.$

###### Proof.

$\begin{array}[]{r c
l}(|\psi\rangle,|\phi\rangle)&=&\big{(}\displaystyle\sum_{k=1}^{2^{n-1}}|\psi\rangle_{\hat{k}}\varepsilon_{k},\displaystyle\sum_{l=1}^{2^{n-1}}|\phi\rangle_{\hat{l}}\varepsilon_{l}\big{)}=\displaystyle\sum_{k=1}^{2^{n-1}}\displaystyle\sum_{l=1}^{2^{n-1}}(|\psi\rangle_{\hat{k}},|\phi\rangle_{\hat{l}})\,\varepsilon_{k}^{\Lambda}\
\varepsilon_{l}\\\
&=&\displaystyle\sum_{k=1}^{2^{n-1}}(|\psi\rangle_{\hat{k}},|\phi\rangle_{\hat{k}})\varepsilon_{k}=\displaystyle\sum_{k=1}^{2^{n-1}}\Big{(}\displaystyle\sum_{j=1}^{2^{n-1}}(|\psi\rangle_{\hat{k}},|\phi\rangle_{\hat{k}})_{\hat{\jmath}}\,\varepsilon_{j}\Big{)}\varepsilon_{k}\\\
&=&\displaystyle\sum_{k=1}^{2^{n-1}}(|\psi\rangle_{\hat{k}},|\phi\rangle_{\hat{k}})_{\hat{k}}\,\varepsilon_{k}.\end{array}$

∎

We know that $V$ and $\varepsilon_{j}V$ for all $j=1,\ldots,2^{n-1}$ are
finite-dimensional vector spaces on the complex $\mathbb{C}$. Furthermore, for
each $j$, $(\cdot\,,\,\cdot)_{\hat{\jmath}}$ is a standard complex scalar
product on both $V$ and $\varepsilon_{j}V$, which means that each of these
spaces equipped with the projected scalar product is a finite-dimensional
Hilbert space. From Theorem 24, $W$ is a direct sum of the spaces
$\varepsilon_{j}V$ and is thus itself a standard Hilbert space equipped with
the following scalar product :

$\Big{(}\sum_{k=1}^{2^{n-1}}|\psi_{k}\rangle,\sum_{l=1}^{2^{n-1}}|\phi_{l}\rangle\Big{)}_{\mathbb{C}}:=\sum_{k=1}^{2^{n-1}}(|\psi_{k}\rangle,|\phi_{k}\rangle)_{\hat{k}},\qquad|\psi_{k}\rangle,|\phi_{k}\rangle\in\varepsilon_{k}V.$
(40)

This function induce a norm and a metric from which $W$ is a complete metric
space. There is an important distinction to make between the scalar products
defined on $W$ in Definition 37 and in equation (40): one is multicomplex-
valued and the other is complex-valued respectively. Moreover, from expression
(40) the complex-valued scalar product is clearly induced from the
multicomplex one, but the converse is also possible. Indeed, take
$(\cdot\,,\,\cdot)_{\mathbb{C}}$ as an independently defined complex-valued
scalar product on $W$ (considered as a vector space over $\mathbb{C}$), then

$(|\psi\rangle,|\phi\rangle):=\sum_{k=1}^{2^{n-1}}(|\psi\rangle_{\hat{k}},|\phi\rangle_{\hat{k}})_{\mathbb{C}}\,\varepsilon_{k}$
(41)

is a multicomplex scalar product on $W$.

######  Theorem 40.

Any $\mathbb{M}_{n}$-module $W$ is a Hilbert space if and only if $W$ is
equipped with a multicomplex scalar product.

The $\mathbb{M}_{n}$-module is consequently a special case of a module in
which we can coherently mix these two notions together and say that from the
existence of a multicomplex scalar product, $W$ is a multicomplex Hilbert
space.

######  Theorem 41 (Riesz).

Let $f:W\rightarrow\mathbb{M}_{n}$ be a linear functional on $W$. Then there
exists a unique $|\psi\rangle\in W$ such that $\forall|\phi\rangle\in W$,
$f(|\phi\rangle)=(|\psi\rangle,|\phi\rangle)$.

###### Proof.

Any projection $f_{\hat{\jmath}}$ of $f$ is a linear functional on $V$.
Applying the classical Riesz theorem, there exists a unique
$|\psi_{j}\rangle\in V$ such that for all $|\phi_{j}\rangle\in V$,
$f_{\hat{j}}(|\phi_{j}\rangle)=(|\psi_{j}\rangle,|\phi_{j}\rangle)_{\hat{j}}$.
We set $|\psi\rangle:=\sum_{k=1}^{2^{n-1}}|\psi_{k}\rangle\varepsilon_{k}$ and
use theorem 39 to get

$(|\psi\rangle,|\phi\rangle)=\sum_{k=1}^{2^{n-1}}(|\psi_{k}\rangle,|\phi\rangle_{\hat{k}})_{\hat{k}}\varepsilon_{k}=\sum_{k=1}^{2^{n-1}}f_{\hat{k}}(|\phi\rangle_{\hat{k}})\varepsilon_{k}=f(|\phi\rangle).$

∎

From this generalization of Riezs theorem, linear functionals are in one-to-
one correspondence with kets and can be replaced by the scalar product
operation. This allows the use of Dirac’s notation and the alternative writing
of the scalar product as
$\langle\psi|\phi\rangle:=(|\psi\rangle,|\phi\rangle)$.

######  Theorem 42.

Any ket $|\psi\rangle\in W$ not in the null cone can be normalized.

###### Proof.

For any $|\psi\rangle\in W$ not in the null cone, we have
$(|\psi\rangle,|\psi\rangle)\in\mathbb{D}_{n}^{+}$ and this scalar product has
strictly real positive components:

$(|\psi\rangle,|\psi\rangle)=\sum_{k=1}^{2^{n-1}}a_{k}\,\varepsilon_{k},\qquad
a_{k}>0.$

Then the ket

$|\phi\rangle:=\big{(}\sum_{k=1}^{2^{n-1}}\frac{1}{\sqrt{a_{k}}}\varepsilon_{k}\big{)}|\psi\rangle$

satisfies $(|\phi\rangle,|\phi\rangle)=1$. ∎

###  5.2 Spectral decomposition theorem

######  Definition 43.

Let $A$ be a linear operator on $W$. Then the adjoint operator $A^{*}$ of $A$
is defined as an operator on $W$ satisfying the following equality.

$(|\psi\rangle,A|\phi\rangle):=(A^{*}|\psi\rangle,|\phi\rangle),\qquad\forall|\psi\rangle,|\phi\rangle\in
W.$

By the decomposition of $A$ in its components $A_{\hat{k}}$ (linear operators
on $V$) under the canonical idempotent representation, the adjoint always
exists, is unique, and his expression is given by the linear operator for
which each component is the adjoint $A_{\hat{k}}^{*}$ of $A_{\hat{k}}$, i.e.
for

$A=\sum_{k=1}^{2^{n-1}}A_{\hat{k}}\varepsilon_{k}\qquad\text{then}\qquad
A^{*}=\sum_{k=1}^{2^{n-1}}A_{\hat{k}}^{*}\varepsilon_{k}.$

Thus the multicomplex operator satisfies the same basic properties of the
adjoint on an usual complex vector space. Moreover,

$[P_{j}(A)]^{*}=P_{j}(A^{*})\qquad\text{for}\qquad j=1,\ldots,2^{n-1}.$ (42)

Now let $|\psi\rangle,|\phi\rangle\in W$. We define the operator
$|\phi\rangle\langle\psi|$ so that its action on an arbitrary ket
$|\chi\rangle\in W$ is given by

$(|\phi\rangle\langle\psi|)|\chi\rangle:=|\phi\rangle(\langle\psi|\chi\rangle).$
(43)

From the generalized Riesz theorem, the action of $\langle\psi|$ on a ket is a
linear functional and always gives a scalar, which means that the operator
$|\phi\rangle\langle\psi|$ itself is linear.

As in a standard Hilbert space, the identity operator can be written in terms
of any orthonormal basis $\\{|u_{l}\rangle\\}_{l=1}^{m}$ of $W$ :

$\sum_{l=1}^{m}|u_{l}\rangle\langle u_{l}|=\mathrm{id}.$ (44)

Indeed, the actions of the left-hand side on any basis element $|u_{p}\rangle$
is given by

$\big{(}\sum_{l=1}^{m}|u_{l}\rangle\langle
u_{l}|\big{)}|u_{p}\rangle=\sum_{l=1}^{m}|u_{l}\rangle\langle
u_{l}|u_{p}\rangle=\sum_{l=1}^{m}\delta_{lp}|u_{l}\rangle=|u_{p}\rangle.$ (45)

######  Definition 44.

A multicomplex linear operator $A$ is called self-adjoint if $A^{*}=A$.

Supposing that $A$ is a multicomplex self-adjoint operator, then

$A=A^{*}\Leftrightarrow\sum_{k=1}^{2^{n-1}}A_{\hat{k}}\varepsilon_{k}=\sum_{k=1}^{2^{n-1}}A_{\hat{k}}^{*}\varepsilon_{k}\Leftrightarrow
A_{\hat{k}}=A_{\hat{k}}^{*}\quad\text{for}\quad k=1,\ldots,2^{n-1}$ (46)

and for any projection $[P_{j}(A)]^{*}=P_{j}(A^{*})=P_{j}(A)$. It implies that
the projection of a self-adjoint operator on $W$ is itself a self-adjoint
operator on $V$.

######  Theorem 45.

Let $A$ be a self-adjoint operator on $W$. Then the eigenvalues of $A$
associated to an eigenket not in the null cone are all in the set
$\mathbb{D}_{n}$ of multicomplex numbers with real components.

###### Proof.

Let $|\psi\rangle$ be an eigenket of $A$ not in the null cone. From equation
(37),

$A|\psi\rangle=\lambda|\psi\rangle\quad\Rightarrow\quad
A_{\hat{\jmath}}|\psi\rangle_{\hat{\jmath}}=\lambda_{\hat{\jmath}}|\psi\rangle_{\hat{\jmath}},\qquad|\psi\rangle_{\hat{j}}\neq
0,\qquad j=1,\ldots,2^{n-1}.$

If $A$ is self-adjoint then its projections are also self-adjoint and this
implies that $\lambda_{\hat{\jmath}}$ is a real number for
$j=1,\ldots,2^{n-1}$. ∎

######  Theorem 46.

Two eigenkets of a multicomplex self-adjoint operator are orthogonal if the
difference of the two associated eigenvalues is not in the null cone.

###### Proof.

Let $|\psi\rangle$ and $|\phi\rangle$ be two eigenkets of a self-adjoint
operator $A$ on $W$ with associated eigenvalues $\lambda$ and
$\lambda^{\prime}$ respectively. Then

$\displaystyle 0$
$\displaystyle=(|\psi\rangle,A|\phi\rangle)-(|\phi\rangle,A|\psi\rangle)^{\Lambda}$
$\displaystyle=\lambda^{\prime}(|\psi\rangle,|\phi\rangle)-\lambda^{\Lambda}(|\phi\rangle,|\psi\rangle)^{\Lambda}$
$\displaystyle=(\lambda^{\prime}-\lambda^{\Lambda})(|\psi\rangle,|\phi\rangle).$

Since $\lambda\in\mathbb{D}_{n}$, $\lambda^{\Lambda}=\lambda$ and if
$\lambda^{\prime}-\lambda$ is not in the null cone then
$(|\psi\rangle,|\phi\rangle)=0$. ∎

######  Theorem 47 (Spectral decomposition).

Let $W$ be a finite-dimensional free $\mathbb{M}_{n}$-module and let
$A:W\rightarrow W$ be a multicomplex self-adjoint operator. It is always
possible to find a set $\\{|\psi_{l}\rangle\\}_{l=1}^{m}$ of eigenkets of $A$
that makes up an orthonormal basis of $W$. Moreover, $A$ can be expressed as

$A=\sum_{l=1}^{m}\lambda_{l}|\psi_{l}\rangle\langle\psi_{l}|,$

where $\lambda_{l}$ is the eigenvalue of $A$ associated with the eigenket
$|\psi_{l}\rangle$.

###### Proof.

Any projection $P_{j}(A)=A_{\hat{\jmath}}$ is a self-adjoint linear operator
on $V$. Applying the standard spectral decomposition theorem, for all
$j=1,\ldots,2^{n-1}$ there exists an orthonormal set
$\\{|\psi_{l}\rangle_{\hat{\jmath}}\\}_{l=1}^{m}$ of eigenkets of
$A_{\hat{\jmath}}$ which is also a basis of $V$ with respect to the scalar
product $(\cdot\,,\,\cdot)_{\hat{j}}$. For

$|\psi_{l}\rangle:=\sum_{k=1}^{2^{n-1}}|\psi_{l}\rangle_{\hat{k}}\,\varepsilon_{k}$

then the set $\\{|\psi_{l}\rangle\\}_{l=1}^{m}$ satisfies the statement of
this theorem and

$\displaystyle A$
$\displaystyle=\sum_{k=1}^{2^{n-1}}A_{\hat{k}}\,\varepsilon_{k}=\sum_{k=1}^{2^{n-1}}\Big{(}\sum_{l=1}^{m}\lambda_{l,\hat{k}}|\psi_{l}\rangle_{\hat{k}}\langle\psi_{l}|_{\hat{k}}\Big{)}\varepsilon_{k}$
$\displaystyle=\sum_{l=1}^{m}\sum_{k=1}^{2^{n-1}}P_{k}\big{(}\lambda_{l}|\psi_{l}\rangle\langle\psi_{l}|\big{)}\varepsilon_{k}=\sum_{l=1}^{m}\lambda_{l}|\psi_{l}\rangle\langle\psi_{l}|$

where $\lambda_{l,\hat{k}}$ represents the complex eigenvalue associated with
the eigenket $|\psi_{l}\rangle_{\hat{k}}$. ∎

##  6 Conclusion

The multicomplex idempotent canonical basis seems to be the first step to a
full understanding of the multicomplex number space. It is a natural basis to
represent the principal ideals of this structure, but also a way to greatly
simplify the algebra as both addition and multiplication become componentwise.
The composition of conjugates $\Lambda_{n}=\dagger_{1}\cdots\dagger_{n}$
allowing us to introduce a natural convolution and a multicomplex norm which
gives C∗-algebra. The multicomplex version of modules and Hilbert spaces share
the same general properties of their complex counterparts, with some
differences or more specific cases due to the presence of zero divisors.
Following this work, we expect several new results where multicomplex algebra
can be applied in fundamental or applied mathematics as well as in physics.

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1DÉPARTEMENT DE MATHÉMATIQUES ET D’INFORMATIQUE,
UNIVERSITÉ DU QUÉBEC, TROIS-RIVIÈRES, QC, CANADA
Email address<EMAIL_ADDRESS>

2DÉPARTEMENT DE MATHÉMATIQUES ET D’INFORMATIQUE,
UNIVERSITÉ DU QUÉBEC, TROIS-RIVIÈRES, QC, CANADA
Email address<EMAIL_ADDRESS>