#  Structure of octonionic Hilbert spaces with applications in the Parseval
equality and Cayley-Dickson algebras

Qinghai Huo<EMAIL_ADDRESS>Department of Mathematics, University of
Science and Technology of China, Hefei 230026, China  and  Guangbin Ren
<EMAIL_ADDRESS>Department of Mathematics, University of Science and
Technology of China, Hefei 230026, China

###### Abstract.

Contrary to the simple structure of the tensor product of the quaternionic
Hilbert space, the octonionic situation becomes more involved. It turns out
that an octonionic Hilbert space can be decomposed as an orthogonal direct sum
of two subspaces, each of them isomorphic to a tensor product of an
irreducible octonionic Hilbert space with a real Hilbert space. As an
application, we find that for a given orthogonal basis the octonionic Parseval
equality holds if and only if the basis is weak associative. Fortunately,
there always exists a weak associative orthogonal basis in an octonionic
Hilbert space. This completely removes the obstacles caused by the failure of
the octonionic Parseval equality. As another application, we provide a new
approach to study the Cayley-Dickson algebras, which turn out to be specific
examples of octonionic Hilbert spaces. An explicit weak associative
orthonormal basis is constructed in each Cayley-Dickson algebra.

###### Key words and phrases:

Octonionic Hilbert space; sedenions; Cayley-Dickson algebras; tensor product;
Parseval’s equality

###### 2010 Mathematics Subject Classification:

Primary: 17A60;46S10

This work was supported by the NNSF of China (11771412).

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Tensor product decompositions
  4. 4 Weak associative orthonormal basis and Parseval’s theorem
  5. 5 Cayley-Dickson algebras as $\mathbb{O}$-Hilbert spaces

##  1\. Introduction

The theory of octonionic Hilbert spaces is initiated by Goldstine and Horwitz
[5] in 1964. It has many developments in the spectral theory [15], operator
theory [14], and mathematical physics [1, 8, 7, 17].

However, for the normed division algebras, many results are restricted to the
quaternionic case and remain open in the octonionic case [21].

A quaternionic Hilbert space is of a simple structure:

$\bullet$ A Hilbert quaternionic bimodule is the tensor product of the
quaternion algebra $\mathbb{H}$ with a real Hilbert space [16].

$\bullet$ A Hilbert left quaternionic module is also a tensor product up to an
isomorphism. This is because the compatible right multiplication always exits
which depends on a choice of a Hilbert basis as shown in [4, Section 3.1].

A natural question for the octonionic case is whether a Hilbert left
octonionic module is the tensor product of the octonionic algebra $\mathbb{O}$
with some real Hilbert space. If the answer was affirmative, then it would
extremely simplify the theory of Hilbert left octonionic modules. We refer to
[14] for such a development of the theory of octonionic Hilbert spaces under
the assumption of such a tensor product decomposition.

In this article, we give a negative answer to the question above. It turns out
that a Hilbert left octonionic module can be decomposed into an orthogonal sum
of two subspaces, each with a tensor product decomposition.

To state our results in detail, we first recall the definition of Hilbert left
$\mathbb{O}$-modules due to Goldstine and Horwitz [5].

######  Definition 1.1 ([5]).

A Hilbert left $\mathbb{O}$-module $H$ is a left $\mathbb{O}$-module with an
$\mathbb{O}$-inner product

$\left\langle\cdot,\cdot\right\rangle:H\times H\rightarrow\mathbb{O}$

such that $(H,\text{Re}\,\left\langle\cdot,\cdot\right\rangle)$ is a real
Hilbert space. Here the $\mathbb{O}$-inner product satisfies the following
axioms for all $u,v\in H$ and $p\in\mathbb{O}$:

  1. (a)

$\left\langle u+v,w\right\rangle=\left\langle u,w\right\rangle+\left\langle
v,w\right\rangle$;

  2. (b)

$\left\langle u,v\right\rangle=\overline{\left\langle v,u\right\rangle}$;

  3. (c)

$\left\langle u,u\right\rangle\in\mathbb{R}^{+}$; and $\left\langle
u,u\right\rangle=0$ if and only if $u=0$;

  4. (d)

$\left\langle tu,v\right\rangle=t\left\langle u,v\right\rangle$ for
$t\in\mathbb{R}$;

  5. (e)

$\text{Re}\,\left\langle pu,v\right\rangle=\text{Re}\,(p\left\langle
u,v\right\rangle)$;

  6. (f)

$\left\langle pu,u\right\rangle=p\left\langle u,u\right\rangle$.

Recently, we find that axiom $(f)$ is non-independent [11]. Since the
functionals induced from the $\mathbb{O}$-inner product are not
$\mathbb{O}$-linear, this motivates us to introduce the para-linearity instead
[11]. It turns out that para-linearity is the main object of octonionic
functional analysis, other than octonionic linearity.

We first consider the underline module structure of a Hilbert left
$\mathbb{O}$-module $H$. We refer to [20, 22] for the structure of alternative
modules, which take $\mathbb{O}$-modules as specific examples.

It is known that there are only two irreducible left $\mathbb{O}$-modules,
i.e., $\mathbb{O}$ and $\overline{\mathbb{O}}$. Here the conjugate regular
module $\overline{\mathbb{O}}$ coincides with $\mathbb{O}$ as a set and its
left module structure is defined by

$p\hat{\cdot}x:=\overline{p}x,$

for any $p\in\mathbb{O}$ and $x\in\mathbb{O}$. Moreover, any left
$\mathbb{O}$-module $H$ admits a decomposition [10]

(1.1) $\displaystyle
H=\mathbb{O}\mathscr{A}(H)\oplus{\mathbb{O}}\mathscr{A}^{-}(H),$

where

$\mathscr{A}(H):=\\{m\in H\mid(pq)m-p(qm)=0\text{ for all
}p,q\in\mathbb{O}\\}$

is the set of associative elements, called the nucleus of $M$, and

$\mathscr{A}^{-}(H):=\\{m\in H\mid(pq)m-q(pm)=0\text{ for all
}p,q\in\mathbb{O}\\}$

is the set of conjugate associative elements. Here, we denote

$\mathbb{O}U:=\left\\{\sum_{i=1}^{N}p_{i}u_{i}\mid p_{i}\in\mathbb{O},\
u_{i}\in U,\ N\in\mathbb{N}\right\\}$

for any $U\subseteq H$.

With the $\mathbb{O}$-module structure of $\mathbb{O}$ and
$\overline{\mathbb{O}}$ at hands, we can endow a natural $\mathbb{O}$-module
structure on the set

(1.2)
$\displaystyle\Big{(}\mathbb{O}\otimes_{\mathbb{R}}\mathscr{A}(H)\Big{)}\oplus\Big{(}\overline{\mathbb{O}}\otimes_{\mathbb{R}}\mathscr{A}^{-}(H)\Big{)}$

The decomposition (1.1) can be modified as

$H\cong\Big{(}\mathbb{O}\otimes_{\mathbb{R}}\mathscr{A}(H)\Big{)}\oplus\Big{(}\overline{\mathbb{O}}\otimes_{\mathbb{R}}\mathscr{A}^{-}(H)\Big{)},$

which is a natural isomorphism of $\mathbb{O}$-modules.

Notice that this isomorphism is only at the level of left
$\mathbb{O}$-modules. We need to seek further an isomorphism at the level of
Hilbert left $\mathbb{O}$-modules. To this end, we will endow the left
$\mathbb{O}$-module in (1.2) with a canonical $\mathbb{O}$-inner product. It
is shown that the submodules in the two summands in (1.2) are orthogonal to
each other, and the $\mathbb{O}$-inner products restricted to both
$\mathscr{A}(H)$ and $\mathscr{A}^{-}(H)$ are all real-valued, respectively.
This leads to the tensor product decomposition of Hilbert left
$\mathbb{O}$-modules.

######  Theorem 1.2 (Tensor product decomposition).

Let $H$ be a Hilbert left $\mathbb{O}$-module. Then there exists an
$\mathbb{O}$-isomorphism of $\mathbb{O}$-Hilbert spaces

(1.3) $\displaystyle
H\cong\Big{(}\mathbb{O}\otimes_{\mathbb{R}}\mathscr{A}(H)\Big{)}\oplus\Big{(}\overline{\mathbb{O}}\otimes_{\mathbb{R}}\mathscr{A}^{-}(H)\Big{)}.$

Theorem 1.2 can be applied to study the Parseval theorem in
$\mathbb{O}$-Hilbert spaces. As observed by Goldstine and Horwitz in the
appendix of [6], the Parseval equality may fail for arbitrary orthonormal
basis in an $\mathbb{O}$-Hilbert space. We find for a given orthonormal basis,
the related Parseval equality holds if and only if the orthonormal basis is
weak associative (see Definition 4.7). Fortunately, a weak associative
orthonormal basis always exists in an $\mathbb{O}$-Hilbert space. This makes
the Parseval equality applicable in the octonionic setting.

Theorem 1.2 can also be applied to study the Cayley-Dickson algebra
$\mathbb{A}_{n}$. We refer to [2, 9] for the theory of Cayley-Dickson
algebras. In this article, we introduce a new approach to study the Cayley-
Dickson algebra $\mathbb{A}_{n}$. That is, we regard $\mathbb{A}_{n}$ as a
Hilbert left $\mathbb{O}$-module when $n\geqslant 4$. It leads to an
isomorphism of Hilbert left $\mathbb{O}$-modules

$\mathbb{A}_{n}\cong\mathbb{O}^{2^{n-4}}\oplus\overline{\mathbb{O}}^{2^{n-4}}.$

In particular, for the algebra of sedenions $\mathbb{S}$, we have

$\mathbb{S}=\mathbb{O}\oplus\overline{\mathbb{O}}.$

We can construct an explicit weak associative orthonormal basis in
$\mathbb{A}_{n}$. Note that a Cayley-Dickson algebra is generally not a
quaternionic vector space. The fact that $\mathbb{A}_{n}$ is a Hilbert left
$\mathbb{O}$-module provides a new method to study Cayley-Dickson algebras.

##  2\. Preliminaries

In this section we review some definitions and basic properties about the
algebra $\mathbb{O}$ of octonions, $\mathbb{O}$-modules and
$\mathbb{O}$-Hilbert spaces.

###  2.1. Octonions

The algebra $\mathbb{O}$ of octonions is the $8$-dimensional non-associative,
non-commutative, normed division algebra over $\mathbb{R}$. For convenience,
we denote $e_{0}=1$. Let $e_{0},e_{1},\dots,e_{7}$ be a basis of $\mathbb{O}$
as a real vector space subject to the multiplication rule

(2.1) $\displaystyle e_{i}e_{j}=\epsilon_{ijk}e_{k}-\delta_{ij},\quad
i,j=1,\dots,7,$

where $\delta_{ij}$ is the Kronecker delta and $\epsilon_{ijk}$ is a
completely skew-symmetric with value 1 precisely when

$ijk=123,145,176,246,257,347,365.$

An octonion can be written as

$x=x_{0}+\sum_{i=1}^{7}x_{i}e_{i},$

where $x_{i}\in\mathbb{R}$ for $i=0,\dots,7$. Its conjugate is defined by

$\overline{x}:=x_{0}-\sum_{i=1}^{7}x_{i}e_{i},$

its norm equals $|x|=\sqrt{x\overline{x}}\in\mathbb{R}$, and its real part is
$\text{Re}\,{x}=x_{0}=\frac{1}{2}(x+\overline{x})$. We denote by
$\mathbb{S}_{6}$ the set of imaginary units in $\mathbb{O}$, i.e.,

$\mathbb{S}_{6}:=\\{J\in\mathbb{O}\mid J^{2}=-1\\}.$

Then there is a book structure on octonions (see [13]):

$\mathbb{O}=\bigcup\mathbb{H}_{I,J},$

where the sum runs over all $I,J\in\mathbb{S}_{6}$ which are mutually
orthogonal and $\mathbb{H}_{I,J}$ denotes the quaternionic algebra spanned by
$\\{1,I,J,IJ\\}$.

###  2.2. $\mathbb{O}$-modules

An octonionic module is a specific alternative module since the octonionic
algebra is an alternative algebra. The general representation theory over
alternative algebras has been fully studied; see [18, 12, 20, 19, 10]. We
recall some basic notations and results on $\mathbb{O}$-modules in this
subsection.

######  Definition 2.1.

An $\mathbb{R}$-vector space $M$ is called a left $\mathbb{O}$-module, if
there is an $\mathbb{R}$-linear map

$\displaystyle L:\mathbb{O}$ $\displaystyle\rightarrow$
$\displaystyle\text{End}_{\mathbb{R}}M$ $\displaystyle p$
$\displaystyle\mapsto$ $\displaystyle L_{p}$

satisfying $L_{1}=id_{M}$ and

(2.2) $\displaystyle[p,q,x]=-[q,p,x],$

for all $p,q\in\mathbb{O}$ and $x\in M$. Here

$[p,q,x]:=(pq)x-p(qx)=L_{pq}(x)-L_{p}L_{q}(x)$

is called the left associator of $M$. The definition of right
$\mathbb{O}$-module is similar.

It is a useful fact that (2.2) is equivalent to the following statement

$r(rm)=r^{2}m$

for all $r\in\mathbb{O}$ and all $m\in M$.

The structure of a left $\mathbb{O}$-module is characterized by its
associative elements and conjugate associative elements as follows. The
associative elements of a left $\mathbb{O}$-module $M$ form a set

$\mathscr{A}(M):=\\{m\in M\mid[p,q,m]=0\text{ for all }p,q\in\mathbb{O}\\},$

called the nucleus of $\mathbb{O}$-module $M$. The conjugate associative
elements of a left $\mathbb{O}$-module $M$ form a set denoted by

$\mathscr{A}^{-}(M):=\\{m\in M\mid(pq)m=q(pm)\text{ for all
}p,q\in\mathbb{O}\\}.$

######  Theorem 2.2 ([10]).

Let $M$ be a left $\mathbb{O}$-module. Then

$M=\mathbb{O}\mathscr{A}(M)\oplus{\mathbb{O}}\mathscr{A}^{-}(M).$

The category of left $\mathbb{O}$-modules is shown to be isomorphic to the
category of left $C\ell_{7}$-modules [10]. For a left $\mathbb{O}$-module $M$,
we can endow it with a natural $C\ell_{7}$-module structure, denoted by
$M_{C\ell_{7}}$. Let $M$, $M^{\prime}$ be two left $\mathbb{O}$-module. We
denote

$\text{Hom}_{\mathbb{O}}(M,M^{\prime}):=\\{f\in\text{Hom}_{\mathbb{R}}(M,M^{\prime})\mid
f(px)=pf(x)\text{ for all }p\in\mathbb{O}\\}.$

Then it follows from Schur’s lemma that

(2.3)
$\displaystyle\text{Hom}_{\mathbb{O}}(\mathbb{O},\overline{\mathbb{O}})\cong\text{Hom}_{C\ell_{7}}(\mathbb{O}_{C\ell_{7}},\overline{\mathbb{O}}_{C\ell_{7}})=\\{0\\}.$

We restate this fact in a more precise way as follows.

######  Lemma 2.3.

If $f\in End_{\mathbb{R}}(\mathbb{O})$ satisfies $f(px)=\overline{p}f(x)$ for
all $p,x\in\mathbb{O}$, then $f=0$.

A simple variant of Lemma 2.3 is as follows.

######  Lemma 2.4.

Let $f\in\text{End}_{\mathbb{R}}(\mathbb{O})$. If it holds $f(xq)=qf(x)$ for
all $q,x\in\mathbb{O}$, then $f=0$.

###### Proof.

We define $g(x):=f(\overline{x})$. Then we obtain

$g(px)=f(\overline{x}\
\overline{p})=\overline{p}f(\overline{x})=\overline{p}g({x}).$

It thus follows from Lemma 2.3 that $g=0$, i.e., $f=0$. ∎

###  2.3. $\mathbb{O}$-Hilbert spaces

Octonionic Hilbert spaces are introduced by Goldstine and Horwitz [5].
Recently, we [11] develop this theory by using a new notion of
$\mathbb{O}$-para-linearity.

######  Definition 2.5 ([11]).

Let $M$ be a left $\mathbb{O}$-module. A real linear map
$f:M\xlongrightarrow[]{}\mathbb{O}$ is called an $\mathbb{O}$-para-linear
function if it satisfies

$\text{Re}\,[p,x,f]=0$

for any $p\in\mathbb{O}$ and $x\in M$. Here

$[p,x,f]:=f(px)-pf(x),$

which is called the second associator.

The definition of $\mathbb{O}$-Hilbert spaces can be restated in terms of
para-linearity.

######  Definition 2.6 ([11]).

A left $\mathbb{O}$-module $H$ is called a  pre-Hilbert $\mathbb{O}$-module if
there exists an $\mathbb{R}$-bilinear map
$\left\langle\cdot,\cdot\right\rangle:H\times H\rightarrow\mathbb{O}$, which
is referred to as an $\mathbb{O}$-inner product, satisfying:

  1. (${\mathrm{i}}$)

($\mathbb{O}$-para-linearity) $\left\langle\cdot,u\right\rangle$ is (left)
$\mathbb{O}$-para-linear for all $u\in H$.

  2. (${\mathrm{ii}}$)

(Octonion hermiticity) $\left\langle u,v\right\rangle=\overline{\left\langle
v,u\right\rangle}$ for all $u,v\in H$.

  3. (${\mathrm{iii}}$)

(Positivity) $\left\langle u,u\right\rangle\in\mathbb{R}^{+}$ and
$\left\langle u,u\right\rangle=0$ if and only if $u=0$.

We can define the norm
$\left|\left|\cdot\right|\right|:H\rightarrow\mathbb{R}^{+}$ by setting

(2.4) $\displaystyle\left|\left|u\right|\right|=\sqrt{\left<u,u\right>}.$

A pre-Hilbert left $\mathbb{O}$-module $H$ is said to be a Hilbert left
$\mathbb{O}$-module if it is complete with respect to its natural distance
induced by the norm.

######  Example 2.7.

We give two typical examples of Hilbert left $\mathbb{O}$-modules.

  1. (${\mathrm{i}}$)

$\mathbb{O}$ is a Hilbert left $\mathbb{O}$-module with the $\mathbb{O}$-inner
product $\left<\cdot,\cdot\right>_{\mathbb{O}}$ defined by

$\left<x,y\right>_{\mathbb{O}}=x\overline{y}$

for all $x,y\in\mathbb{O}$.

  2. (${\mathrm{ii}}$)

The conjugate regular module $\overline{\mathbb{O}}$ is a Hilbert left
$\mathbb{O}$-module with the $\mathbb{O}$-inner product
$\left<\cdot,\cdot\right>_{\overline{\mathbb{O}}}$ defined by

$\left<x,y\right>_{\overline{\mathbb{O}}}=y\overline{x}$

for all $x,y\in\mathbb{O}$. We remark that its left module structure is
defined by

$p\hat{\cdot}x:=\overline{p}x,$

for any $p\in\mathbb{O}$ and $x\in\overline{\mathbb{O}}$.

In an $\mathbb{O}$-Hilbert space $H$, we can regard $v\in H$ as an
$\mathbb{O}$-para-linear map induced by the $\mathbb{O}$-inner product. This
motivates us to introduce the second associator of $H$ as

$[p,u,v]:=[p,u,\langle\cdot,v\rangle]=\left\langle
pu,v\right\rangle-p\left\langle u,v\right\rangle$

for $u,v\in H$ and $p\in\mathbb{O}$,

From now on, we shall denote

$\text{Re}\,\left<u,v\right>:=\left<u,v\right>_{\mathbb{R}}$

for all $u,v\in H$. We assemble some identities concerning second associators
for later use.

######  Lemma 2.8 ([11]).

For all $u,v\in H$ and all $p,q\in\mathbb{O}$, the following hold

(2.5) $\displaystyle\left\langle[p,q,u],v\right\rangle_{\mathbb{R}}$
$\displaystyle=$ $\displaystyle-\left\langle
u,[p,q,v]\right\rangle_{\mathbb{R}};$ (2.6) $\displaystyle{[p,v,u]}$
$\displaystyle=$ $\displaystyle-[p,u,v];$ (2.7) $\displaystyle{[pq,v,u]}$
$\displaystyle=$
$\displaystyle\left<[p,q,v],u\right>-[p,q,\left<v,u\right>]+p[q,v,u]+[p,qv,u].$

The following characterization of associative elements is very useful in the
sequel.

######  Lemma 2.9 ([11]).

An element $x\in\mathscr{A}(H)$ if and only if

(2.8) $[p,x,y]=0$

for all $p\in\mathbb{O}$ and $y\in H.$

In contrast to the complex and quaternionic cases, the $\mathbb{O}$-inner
products are $\mathbb{O}$-para-linear, rather than $\mathbb{O}$-linear. The
following lemma reveals the close relations among $\mathbb{O}$-inner product,
$\mathbb{O}$-scalar product, and the second associator.

######  Lemma 2.10 ([11]).

Let $H$ be a pre-Hilbert left $\mathbb{O}$-module. Then for all $u,v\in H$ and
all $p,q\in\mathbb{O}$, the following hold:

(2.9) $\displaystyle\left<u,pv\right>$ $\displaystyle=$
$\displaystyle\left<u,v\right>\overline{p}+[p,u,v];$ (2.10)
$\displaystyle\left<pu,qv\right>$ $\displaystyle=$
$\displaystyle(p\left<u,v\right>)\overline{q}+[{pq},u,v]+\left<[p,q,v],u\right>.$

##  3\. Tensor product decompositions

In this section, we provide a Hilbert left $\mathbb{O}$-module with a tensor
product decomposition structure. We first define the tensor product of a
Hilbert left $\mathbb{O}$-module with a real Hilbert space.

######  Definition 3.1.

Let $(H_{1},\left<\cdot,\cdot\right>_{1})$ be a Hilbert left
$\mathbb{O}$-module and $(H_{2},\left<\cdot,\cdot\right>_{2})$ be a real
Hilbert space. Then we endow the tensor product
$H_{1}\otimes_{\mathbb{R}}H_{2}$ with a canonical left $\mathbb{O}$-module
structure

$\displaystyle\mathbb{O}\times\big{(}H_{1}\otimes_{\mathbb{R}}H_{2}\big{)}$
$\displaystyle\to$ $\displaystyle H_{1}\otimes_{\mathbb{R}}H_{2}$
$\displaystyle(p,u\otimes_{\mathbb{R}}x)$ $\displaystyle\mapsto$
$\displaystyle(pu)\otimes_{\mathbb{R}}x$

and an $\mathbb{O}$-inner product

(3.1)
$\displaystyle\left<u\otimes_{\mathbb{R}}x,v\otimes_{\mathbb{R}}y\right>:=\left<u,v\right>_{1}\left<x,y\right>_{2}$

for all $u,v\in H_{1}$ and all $x,y\in H_{2}$. Then one can check that the
tensor product $(H_{1}\otimes_{\mathbb{R}}H_{2},\left<\cdot,\cdot\right>)$ is
a Hilbert left $\mathbb{O}$-module.

######  Definition 3.2.

Let $(H_{1},\left<\cdot,\cdot\right>_{1})$ and
$(H_{2},\left<\cdot,\cdot\right>_{2})$ be two Hilbert left
$\mathbb{O}$-modules. We endow the direct sum $H_{1}\oplus H_{2}$ with the
canonical left $\mathbb{O}$-module structure and the $\mathbb{O}$-inner
product

$\left<(x,u),(y,v)\right>:=\left<x,y\right>_{1}+\left<u,v\right>_{2}$

for all $x,y\in H_{1}$ and $u,v\in H_{2}$. One can check that the direct sum
$(H_{1}\oplus H_{2},\left<\cdot,\cdot\right>)$ is a Hilbert left
$\mathbb{O}$-module. By definition we know that $H_{1}$ is orthogonal to
$H_{2}$ if we regard $H_{1}$ and $H_{2}$ as canonical subsets of $H_{1}\oplus
H_{2}$.

######  Definition 3.3.

Let $(H_{1},\left<\cdot,\cdot\right>_{1})$ and
$(H_{2},\left<\cdot,\cdot\right>_{2})$ be two Hilbert left
$\mathbb{O}$-modules. A map

$f:H_{1}\to H_{2}$

is called an isomorphism of Hilbert left $\mathbb{O}$-modules if

  1. (${\mathrm{i}}$)

for all $p\in\mathbb{O}$ and $x\in H_{1}$,

$f(px)=pf(x);$

  2. (${\mathrm{ii}}$)

for all $x,y\in H_{1}$,

$\left<f(x),f(y)\right>_{2}=\left<x,y\right>_{1}.$

It is useful to know the behavior of the $\mathbb{O}$-inner product on
associative and conjugate associative elements in an $\mathbb{O}$-Hilbert
space.

######  Lemma 3.4.

Let $H$ be a Hilbert left $\mathbb{O}$-module. Then we have

  1. (${\mathrm{i}}$)

for all $u,v\in\mathscr{A}(H)$,

(3.2) $\displaystyle\left<u,v\right>$ $\displaystyle\in\mathbb{R};$

  2. (${\mathrm{ii}}$)

for all $u\in\mathscr{A}(H)$ and $v\in\mathscr{A}^{-}(H)$,

(3.3) $\displaystyle\left<u,v\right>=0.$

###### Proof.

We first prove assertion $({\mathrm{i}})$. Recall identity (2.7):

${[pq,v,u]}=\left<[p,q,v],u\right>-[p,q,\left<v,u\right>]+p[q,v,u]+[p,qv,u]$

for all $p,\,q\in\mathbb{O}$ and $u,v\in H$. If $u,\,v\in\mathscr{A}(H)$, then
it follows from Lemma 2.9 that

$[p,{q},\left<v,u\right>]=0$

for all $p,\,q\in\mathbb{O}$. This implies $\left<v,u\right>\in\mathbb{R}$ as
desired.

Now we prove assertion $({\mathrm{ii}})$. Let $u\in\mathscr{A}(H)$ and
$v\in\mathscr{A}^{-}(H)$. By definition, we have

$\displaystyle[p,q,v]$ $\displaystyle=$ $\displaystyle(pq)v-p(qv)$
$\displaystyle=$ $\displaystyle(pq)v-(qp)v$ $\displaystyle=$
$\displaystyle[p,q]v.$

According to identity (2.7) and Lemma 2.9, we obtain

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\left<[p,q,v],u\right>-[p,q,\left<v,u\right>]$ $\displaystyle=$
$\displaystyle\left<[p,q]v,u\right>-[p,q,\left<v,u\right>]$ $\displaystyle=$
$\displaystyle[p,q]\left<v,u\right>+[[p,q],v,u]-[p,q,\left<v,u\right>]$
$\displaystyle=$ $\displaystyle(pq-
qp)\left<v,u\right>-(pq)\left<v,u\right>+p(q\left<v,u\right>)$
$\displaystyle=$ $\displaystyle p(q\left<v,u\right>)-(qp)\left<v,u\right>.$

This means the map

$\displaystyle f:\mathbb{O}$ $\displaystyle\to$ $\displaystyle\mathbb{O}$
$\displaystyle x$ $\displaystyle\mapsto$ $\displaystyle x\left<v,u\right>$

satisfies

$pf(q)=f(qp)$

for all $p,q\in\mathbb{O}$. By Lemma 2.4, we obtain that $f=0$ and hence
$\left<v,u\right>=0$. ∎

######  Lemma 3.5.

Let $H$ be a Hilbert left $\mathbb{O}$-module. If $\mathscr{A}^{-}(H)=0$, then
there exists an isomorphism of Hilbert left $\mathbb{O}$-modules

(3.4) $\displaystyle H\cong\mathbb{O}\otimes_{\mathbb{R}}\mathscr{A}(H).$

Here $\mathbb{O}$ is a Hilbert left $\mathbb{O}$-module as defined in Example
2.7, and $\mathscr{A}(H)$ is a real Hilbert space with the inner product
induced from $H$ by restriction. More precisely, for any $p,q\in\mathbb{O}$
and $x,y\in\mathscr{A}(H)$, we have

(3.5)
$\displaystyle\left<px,qy\right>=\left<p,q\right>_{\mathbb{O}}\left<x,y\right>.$

###### Proof.

The $\mathbb{O}$-Hilbert space structure of
$\mathbb{O}\otimes_{\mathbb{R}}\mathscr{A}(H)$ is given as in Definition 3.1.
That is,

(3.6)
$\displaystyle\Big{\langle}{\sum_{i=0}^{7}e_{i}\otimes_{\mathbb{R}}x_{i}},\ \
{\sum_{i=0}^{7}e_{i}\otimes_{\mathbb{R}}y_{i}}\Big{\rangle}_{\mathbb{O}\otimes_{\mathbb{R}}\mathscr{A}(H)}=\sum_{i,j=0}^{7}(e_{i}\overline{e_{j}})\left<x_{i},y_{j}\right>$

for any $x_{i},y_{i}\in\mathscr{A}(H)$.

Since $\mathscr{A}^{-}(H)=0$, it follows from Theorem 2.2 that

$H=\mathbb{O}\mathscr{A}(H).$

We consider the canonical map

$\displaystyle\varphi:\quad\mathbb{O}\otimes_{\mathbb{R}}\mathscr{A}(H)$
$\displaystyle\to$ $\displaystyle H$
$\displaystyle\sum_{i=0}^{7}e_{i}\otimes_{\mathbb{R}}x_{i}$
$\displaystyle\mapsto$ $\displaystyle\sum_{i=0}^{7}e_{i}x_{i}.$

One can easily check that this is an $\mathbb{O}$-isomorphism of left
$\mathbb{O}$-modules. It follows from identity (2.10) and Lemma 3.4 that

$\displaystyle\Big{\langle}{\sum_{i=0}^{7}e_{i}x_{i}},\ \
{\sum_{i=0}^{7}e_{i}y_{i}}\Big{\rangle}$ $\displaystyle=$
$\displaystyle\sum_{i,j=0}^{7}\left<e_{i}x_{i},e_{j}y_{j}\right>$
$\displaystyle=$
$\displaystyle\sum_{i,j=0}^{7}\Big{(}e_{i}(\left<x_{i},y_{j}\right>\overline{e_{j}})+[{e_{i}e_{j}},x_{i},y_{j}]+\left<[e_{i},e_{j},y_{j}],x_{i}\right>\Big{)}$
$\displaystyle=$
$\displaystyle\sum_{i,j=0}^{7}(e_{i}\overline{e_{j}})\left<x_{i},y_{j}\right>.$

This means that

$\Big{\langle}{\sum_{i=0}^{7}e_{i}x_{i}},\ \
{\sum_{i=0}^{7}e_{i}y_{i}}\Big{\rangle}=\Big{\langle}{\sum_{i=0}^{7}e_{i}\otimes_{\mathbb{R}}x_{i}},\
\
{\sum_{i=0}^{7}e_{i}\otimes_{\mathbb{R}}y_{i}}\Big{\rangle}_{\mathbb{O}\otimes_{\mathbb{R}}\mathscr{A}(H)}.$

Hence $\varphi$ is an isomorphism of $\mathbb{O}$-Hilbert spaces. ∎

Every Hilbert left $\mathbb{O}$-module can be associated with another Hilbert
left $\mathbb{O}$-module structure.

######  Definition 3.6.

Let $(H,\left<\cdot,\cdot\right>)$ be a Hilbert left $\mathbb{O}$-module. We
endow $H$ with another left $\mathbb{O}$-multiplication

$\displaystyle\mathbb{O}\times H$ $\displaystyle\to$ $\displaystyle H$
$\displaystyle(p,x)$ $\displaystyle\mapsto$ $\displaystyle
p\hat{\cdot}x:=\overline{p}x.$

This is also a left $\mathbb{O}$-module and we denote it by $H^{-}$. We define

$\left<x,y\right>_{H^{-}}:=\left<y,x\right>$

for any $x,y\in H$.

We now check that $(H^{-},\left<\cdot,\cdot\right>_{H^{-}})$ is indeed a
Hilbert left $\mathbb{O}$-module.

######  Lemma 3.7.

Let $(H,\left<\cdot,\cdot\right>)$ be a Hilbert left $\mathbb{O}$-module. Then
$(H^{-},\left<\cdot,\cdot\right>_{H^{-}})$ is a Hilbert left
$\mathbb{O}$-module. Moreover, it holds

(3.7) $\displaystyle\mathscr{A}(H)$ $\displaystyle=$
$\displaystyle\mathscr{A}^{-}(H^{-}),$ (3.8) $\displaystyle\mathscr{A}(H^{-})$
$\displaystyle=$ $\displaystyle\mathscr{A}^{-}(H).$

The two Hilbert left $\mathbb{O}$-modules have a close relation between their
second associators

(3.9) $\displaystyle[p,x,y]_{H^{-}}=[p,x,y]+[\left<x,y\right>,p]$

for all $p\in\mathbb{O}$ and $x,y\in H$.

###### Proof.

By direct calculations, we have

$\displaystyle\text{Re}\,\left(\left<p\hat{\cdot}x,y\right>_{H^{-}}-p\left<x,y\right>_{H^{-}}\right)$
$\displaystyle=$
$\displaystyle\text{Re}\,\left(\left<y,\overline{p}x\right>-p\left<y,x\right>\right)$
$\displaystyle=$
$\displaystyle\text{Re}\,\left(\left<y,x\right>p-[\overline{p},y,x]-p\left<y,x\right>\right)$
$\displaystyle=$
$\displaystyle\text{Re}\,\left([\left<y,x\right>,p]-[\overline{p},y,x]\right)$
$\displaystyle=$ $\displaystyle 0.$

This shows the para-linearity. The proof of hermiticity and positivity is
trivial.

We come to prove (3.7). By definition,

$\displaystyle\mathscr{A}^{-}(H^{-})$ $\displaystyle=$
$\displaystyle\\{x\mid(pq)\hat{\cdot}x=q\hat{\cdot}(p\hat{\cdot}x),\forall
p,q\in\mathbb{O}\\}$ $\displaystyle=$ $\displaystyle\\{x\mid(\overline{q}\
\overline{p})x=\overline{q}(\overline{p}x),\forall p,q\in\mathbb{O}\\}$
$\displaystyle=$ $\displaystyle\\{x\mid[p,q,x]=0,\forall p,q\in\mathbb{O}\\}$
$\displaystyle=$ $\displaystyle\mathscr{A}(H).$

This shows

$\mathscr{A}^{-}(H^{-})=\mathscr{A}(H).$

It is easy to see that

$(H^{-})^{-}=H.$

Hence we get

$\mathscr{A}(H^{-})=\mathscr{A}^{-}((H^{-})^{-})=\mathscr{A}^{-}(H).$

Finally, we compute the second associators of $H^{-}$. For any
$p\in\mathbb{O}$ and $x,y\in H$, we have

$\displaystyle[p,x,y]_{H^{-}}$ $\displaystyle=$
$\displaystyle\left<p\hat{\cdot}x,y\right>_{H^{-}}-p\left<x,y\right>_{H^{-}}$
$\displaystyle=$ $\displaystyle\left<y,\overline{p}x\right>-p\left<y,x\right>$
$\displaystyle=$ $\displaystyle\left<y,x\right>p-[p,y,x]-p\left<y,x\right>$
$\displaystyle=$ $\displaystyle[p,x,y]+[\left<x,y\right>,p].$

This proves the lemma. ∎

By the duality in Lemma 3.7, we can extend the result in Lemma 3.4 to
$\mathscr{A}^{-}(H)$.

######  Lemma 3.8.

Let $H$ be a Hilbert left $\mathbb{O}$-module. Then

(3.10) $\displaystyle\left<x,y\right>\in\mathbb{R}$

for all $x,y\in\mathscr{A}^{-}(H)$.

###### Proof.

It is a direct consequence of Lemma 3.7 with $H$ replaced by $H^{-}$. ∎

######  Lemma 3.9.

Let $H$ be a Hilbert left $\mathbb{O}$-module. If $\mathscr{A}(H)=0$, then
there exists an $\mathbb{O}$-isomorphism of $\mathbb{O}$-Hilbert spaces

(3.11) $\displaystyle
H\cong\overline{\mathbb{O}}\otimes_{\mathbb{R}}\mathscr{A}^{-}(H).$

Here $\overline{\mathbb{O}}$ is a Hilbert left $\mathbb{O}$-module as defined
in Example 2.7, and $\mathscr{A}^{-}(H)$ is a real Hilbert space with the
inner product induced from that of $H$. More precisely, for any
$p,q\in\mathbb{O}$ and $x,y\in\mathscr{A}^{-}(H)$, we have

(3.12)
$\displaystyle\left<px,qy\right>=\left<\overline{p},\overline{q}\right>_{\overline{\mathbb{O}}}\left<x,y\right>.$

###### Proof.

We consider the $\mathbb{O}$-isomorphism

$\displaystyle\phi:\quad\overline{\mathbb{O}}\otimes_{\mathbb{R}}\mathscr{A}^{-}(H)$
$\displaystyle\to$ $\displaystyle H$
$\displaystyle\sum_{i=0}^{7}e_{i}\otimes_{\mathbb{R}}x_{i}$
$\displaystyle\mapsto$ $\displaystyle\sum_{i=0}^{7}\overline{e_{i}}x_{i}.$

Note that the left $\mathbb{O}$-module structure of
$\overline{\mathbb{O}}\otimes_{\mathbb{R}}\mathscr{A}^{-}(H)$ is determined by
the $\mathbb{O}$-multiplication of $\overline{\mathbb{O}}$ as follows:

$p\Big{(}\sum_{i=0}^{7}e_{i}\otimes_{\mathbb{R}}x_{i}\Big{)}:=\sum_{i=0}^{7}(\overline{p}e_{i})\otimes_{\mathbb{R}}x_{i}.$

Hence it follows from $x_{i}\in\mathscr{A}^{-}(H)$ that

$\displaystyle\phi\Big{(}p\Big{(}\sum_{i=0}^{7}e_{i}\otimes_{\mathbb{R}}x_{i}\Big{)}\Big{)}$
$\displaystyle=$
$\displaystyle\sum_{i=0}^{7}\overline{(\overline{p}e_{i})}x_{i}$
$\displaystyle=$ $\displaystyle\sum_{i=0}^{7}(\overline{e_{i}}p)x_{i}$
$\displaystyle=$ $\displaystyle p\sum_{i=0}^{7}\overline{e_{i}}x_{i}$
$\displaystyle=$ $\displaystyle
p\phi\Big{(}\sum_{i=0}^{7}e_{i}\otimes_{\mathbb{R}}x_{i}\Big{)}.$

This proves that $\phi$ is indeed an $\mathbb{O}$-isomorphism.

We next consider the induced module $H^{-}$. By (3.7) and (3.8), we have

$\mathscr{A}^{-}(H^{-})=\mathscr{A}(H)=0$

and

$\mathscr{A}(H^{-})=\mathscr{A}^{-}(H).$

Using Lemma 3.5, we obtain

$H^{-}\cong{\mathbb{O}}\otimes_{\mathbb{R}}\mathscr{A}^{-}(H).$

That is, for any $p,q\in\mathbb{O}$ and $x,y\in\mathscr{A}^{-}(H)$,

$\displaystyle\left<p\hat{\cdot}x,q\hat{\cdot}y\right>_{H^{-}}=\left<p,q\right>_{\mathbb{O}}\left<x,y\right>_{H^{-}},$

or equivalently,

$\displaystyle\left<px,qy\right>=(\overline{q}p)\left<x,y\right>=\left<\overline{p},\overline{q}\right>_{\overline{\mathbb{O}}}\left<x,y\right>.$

Therefore,

$\displaystyle\Big{\langle}{\phi\Big{(}\sum_{i=0}^{7}e_{i}\otimes_{\mathbb{R}}x_{i}\Big{)}},\
\
{\phi\Big{(}\sum_{i=0}^{7}e_{i}\otimes_{\mathbb{R}}y_{i}\Big{)}}\Big{\rangle}$
$\displaystyle=$
$\displaystyle\Big{\langle}{\sum_{i=0}^{7}\overline{e_{i}}x_{i}},\ \
{\sum_{i=0}^{7}\overline{e_{i}}y_{i}}\Big{\rangle}$ $\displaystyle=$
$\displaystyle\sum_{i,j=0}^{7}\left<\overline{e_{i}}x_{i},\overline{e_{j}}y_{j}\right>$
$\displaystyle=$
$\displaystyle\sum_{i,j=0}^{7}(e_{j}\overline{e_{i}})\left<x_{i},y_{j}\right>$
$\displaystyle=$
$\displaystyle\sum_{i,j=0}^{7}\left<e_{i},e_{j}\right>_{\overline{\mathbb{O}}}\left<x_{i},y_{j}\right>$
$\displaystyle=$
$\displaystyle\Big{\langle}{\sum_{i=0}^{7}e_{i}\otimes_{\mathbb{R}}x_{i}},\ \
{\sum_{i=0}^{7}e_{i}\otimes_{\mathbb{R}}y_{i}}\Big{\rangle}_{\overline{\mathbb{O}}\otimes_{\mathbb{R}}\mathscr{A}^{-}(H)}.$

This proves that $\phi$ is an isomorphism of $\mathbb{O}$-Hilbert spaces. ∎

######  Theorem 3.10 (Tensor product decomposition).

Let $H$ be a Hilbert left $\mathbb{O}$-module. Then there exists an
$\mathbb{O}$-isomorphism of $\mathbb{O}$-Hilbert spaces

(3.13) $\displaystyle
H\cong\Big{(}\mathbb{O}\otimes_{\mathbb{R}}\mathscr{A}(H)\Big{)}\oplus\Big{(}\overline{\mathbb{O}}\otimes_{\mathbb{R}}\mathscr{A}^{-}(H)\Big{)}.$

###### Proof.

By Theorem 2.2, we have

$H=\mathbb{O}\mathscr{A}(H)\oplus\mathbb{O}\mathscr{A}^{-}(H).$

Denote $H_{1}=\mathbb{O}\mathscr{A}(H)$ and
$H_{2}=\mathbb{O}\mathscr{A}^{-}(H)$. Then by Lemmas 3.5 and 3.9,

$H_{1}\cong\mathbb{O}\otimes_{\mathbb{R}}\mathscr{A}(H),\quad
H_{2}\cong\overline{\mathbb{O}}\otimes_{\mathbb{R}}\mathscr{A}^{-}(H).$

It remains to show that $H_{1}$ is orthogonal to $H_{2}$. Recall identity
(2.10):

$\left<pu,qv\right>=(p\left<u,v\right>)\overline{q}+[{pq},u,v]+\left<[p,q,v],u\right>.$

If $u\in\mathscr{A}^{-}(H)$ and $v\in\mathscr{A}(H)$, then it follows from
(3.3) and Lemma 2.9 that

$\left<pu,qv\right>=0.$

This shows that $H_{1}$ is orthogonal to $H_{2}$ as desired. ∎

######  Remark 3.11.

Note that the right-hand side of equality (3.13) with $\mathscr{A}(H)$ and
$\mathscr{A}^{-}(H)$ replaced by any two real Hilbert spaces is always an
$\mathbb{O}$-Hilbert space. Hence this theorem gives a complete classification
of Hilbert left $\mathbb{O}$-modules.

##  4\. Weak associative orthonormal basis and Parseval’s theorem

As observed by Goldstine and Horwitz in the appendix of [6], the Parseval
theorem may fail for any orthonormal basis in an $\mathbb{O}$-Hilbert space.
We introduce a new notion of weak associative orthonormal basis and show that
the Parseval equality holds for an orthonormal basis if and only if it is weak
associative.

######  Definition 4.1.

A subset $S=\\{x_{\alpha}\\}_{\alpha\in\Lambda}$ of an $\mathbb{O}$-Hilbert
space is said to be an orthonormal system if

$\left<x_{\alpha},x_{\beta}\right>=\delta_{\alpha\beta}.$

An orthonormal system $S=\\{x_{\alpha}\\}_{\alpha\in\Lambda}$ is said to be an
orthonormal basis if there does not exist other orthonormal system
$S^{\prime}$ such that $S\subsetneqq S^{\prime}$.

As usual, we have the following characterizations of an orthonormal basis. The
proof runs in the same manner as in the classical case and is omitted here.

######  Lemma 4.2.

Let $S$ be an orthonormal system. Then $S$ is an orthonormal basis if and only
if there does not exist non-zero element which is orthogonal to every element
of $S$.

In contrast to the classical cases, it is meaningless to define the dimension
of an $\mathbb{O}$-Hilbert space as the cardinality of an orthonormal basis as
shown in the following special case of $\mathbb{O}^{2}$.

######  Example 4.3.

The $\mathbb{O}$-Hilbert space $\mathbb{O}^{2}$ admits two sets of basis. One
has cardinality $2$ and the other has cardinality $4$.

(1). The $\mathbb{O}$-Hilbert space $\mathbb{O}^{2}$ admits an orthonormal
basis given by

$(0,1),\quad(1,0).$

(2). The $\mathbb{O}$-Hilbert space $\mathbb{O}^{2}$ admits an alternative
orthonormal basis given by

(4.1) $\displaystyle x_{1}=\dfrac{1}{\sqrt{2}}(e_{1},e_{2}),\quad
x_{2}=\dfrac{1}{\sqrt{2}}(e_{4},e_{7}),\quad
x_{3}=\dfrac{1}{\sqrt{2}}(e_{6},e_{5}),\quad
x_{4}=\dfrac{1}{\sqrt{2}}(1,e_{3}).$

To show that (4.1) is an orthonormal basis, we first observe that
$\\{x_{n}\\}_{n=1}^{4}$ is an orthonormal system of $\mathbb{O}^{2}$. If there
exists an element $x=(a,b)\in\mathbb{O}^{2}$ orthogonal to
$x_{1},\dots,x_{4}$, then from the assumption that $\left<x,x_{4}\right>=0$ we
have

$a=be_{3}.$

Since $\left<x,x_{1}\right>=0$, we get

(4.2) $\displaystyle(be_{3})e_{1}+be_{2}=2be_{2}+[b,e_{3},e_{1}]=0.$

Using identity (2.5) in the $\mathbb{O}$-Hilbert space $\mathbb{O}$, we obtain

(4.3) $\displaystyle[I,J,x]\in{\mathbb{H}_{I,J}}^{\perp_{\mathbb{R}}}$

for any mutually orthogonal pair $(I,J)$ of $\mathbb{S}_{6}$ and all
$x\in\mathbb{O}$. Here the notation $\mathbb{H}_{I,J}$ stands for the subset
spanned by $\\{1,I,J,IJ\\}$, which was introduced in Subsection 2.1. Combining
(4.3) with (4.2), we get

$be_{2}\in{\mathbb{H}_{e_{1},e_{3}}}^{\perp_{\mathbb{R}}}.$

This yields that $b_{0}=b_{1}=b_{2}=b_{3}=0$ if we write

$b=b_{0}+\sum_{i=1}^{7}e_{i}b_{i}.$

The assumptions that $\left<x,x_{2}\right>=\left<x,x_{3}\right>=0$ imply that

$\displaystyle 2be_{7}+[b,e_{3},e_{4}]=0,$ $\displaystyle
2be_{5}+[b,e_{3},e_{6}]=0.$

By similar argument, we conclude that $b_{4}=b_{5}=b_{6}=b_{7}=0$ and hence
$b=0$. This means that $x=0$. We thus conclude from Lemma 4.2 that
$\\{x_{n}\\}_{n=1}^{4}$ is an orthonormal basis.

The Parseval equality may not hold for the orthonormal basis (4.1). Indeed, we
take $y=(1,0)\in\mathbb{O}^{2}$. By direct calculation, we have

$\left<y,x_{1}\right>=\dfrac{1}{\sqrt{2}}\overline{e_{1}},\quad\left<y,x_{2}\right>=\dfrac{1}{\sqrt{2}}\overline{e_{4}},\quad\left<y,x_{3}\right>=\dfrac{1}{\sqrt{2}}\overline{e_{6}},\quad\left<y,x_{4}\right>=\dfrac{1}{\sqrt{2}}.$

It follows that

$y-\sum_{n=1}^{4}\left<y,x_{n}\right>x_{n}=(-1,e_{3})$

and

$\sum_{n=1}^{4}\left|\left|\left<y,x_{n}\right>\right|\right|^{2}\neq\left|\left|y\right|\right|^{2}.$

Denote $p_{i}=\left<y,x_{i}\right>$ for $i=1,\dots,4$, Using identity (2.5)
and the left alternativity of associators, we obtain

$\displaystyle\sum_{m,n=1}^{4}\left<[\left<y,x_{n}\right>,\left<y,x_{m}\right>,x_{m}],x_{n}\right>_{\mathbb{R}}$
$\displaystyle=2\sum_{m<n}\left<[\left<y,x_{n}\right>,\left<y,x_{m}\right>,x_{m}],x_{n}\right>_{\mathbb{R}}$
$\displaystyle=2\big{(}\left<[p_{1},p_{2},x_{2}],x_{1}\right>_{\mathbb{R}}+\left<[p_{1},p_{3},x_{3}],x_{1}\right>_{\mathbb{R}}+\left<[p_{2},p_{3},x_{3}],x_{2}\right>_{\mathbb{R}}\big{)}$
$\displaystyle=\dfrac{1}{2}\text{Re}\,([{e_{1}},{e_{4}},e_{7}]\overline{e_{2}}+[{e_{1}},e_{6},{e_{5}}]\overline{e_{2}}+[{e_{4}},{e_{6}},e_{5}]\overline{e_{7}})$
$\displaystyle=3.$

Therefore, we have

$\sum_{n=1}^{4}\left|\left<y,x_{n}\right>\right|^{2}+\Big{\|}{y-\sum_{n=1}^{4}\left<y,x_{n}\right>x_{n}}\Big{\|}^{2}-\sum_{m,n=1}^{2}\left<[\left<y,x_{n}\right>,\left<y,x_{m}\right>,x_{m}],x_{n}\right>_{\mathbb{R}}=\left|\left|y\right|\right|^{2}.$

In fact, the above equality even holds generally.

######  Lemma 4.4.

Let $\\{x_{n}\\}_{n=1}^{N}$ be an orthonormal system of an
$\mathbb{O}$-Hilbert space $H$. Then for all $x\in H$ we have

(4.4)
$\displaystyle\left|\left|x\right|\right|^{2}=\sum_{n=1}^{N}\left|\left<x,x_{n}\right>\right|^{2}+\Big{\|}{x-\sum_{n=1}^{N}\left<x,x_{n}\right>x_{n}}\Big{\|}^{2}-\sum_{m,n=1}^{N}\left<[\left<x,x_{n}\right>,\left<x,x_{m}\right>,x_{m}],x_{n}\right>_{\mathbb{R}}.$

In particular, there holds the octonionic version of Bessel’s inequality:

(4.5)
$\displaystyle\sum_{n=1}^{N}\left|\left<x,x_{n}\right>\right|^{2}-\sum_{m,n=1}^{N}\left<[\left<x,x_{n}\right>,\left<x,x_{m}\right>,x_{m}],x_{n}\right>_{\mathbb{R}}\leqslant\left|\left|x\right|\right|^{2}.$

###### Proof.

Set

$p_{n}=\left<x,x_{n}\right>,\quad u=\sum_{n=1}^{N}p_{n}x_{n},\quad v=x-u.$

By calculation, we have

$\displaystyle\left|\left|x\right|\right|^{2}$
$\displaystyle=\left<u+v,u+v\right>$
$\displaystyle=\left|\left|u\right|\right|^{2}+\left|\left|v\right|\right|^{2}+2\text{Re}\,\left<u,x-u\right>$
(4.6)
$\displaystyle=\left|\left|v\right|\right|^{2}-\left|\left|u\right|\right|^{2}+2\text{Re}\,\left<u,x\right>.$

Since

$\displaystyle\left<u,x\right>$
$\displaystyle=\Big{\langle}{\sum_{i=1}^{N}p_{n}x_{n}},\ \ {x}\Big{\rangle}$
$\displaystyle=\sum_{n=1}^{N}p_{n}\left<x_{n},x\right>+[{p_{n}},x_{n},x]$
$\displaystyle=\sum_{n=1}^{N}\left|\left<x,x_{n}\right>\right|^{2}+\sum_{n=1}^{N}[{p_{n}},x_{n},x],$

we conclude that

$\text{Re}\,\left<u,x\right>=\sum_{n=1}^{N}\left|\left<x,x_{n}\right>\right|^{2}.$

By (2.10), we obtain

$\displaystyle\left|\left|u\right|\right|^{2}$ $\displaystyle=$
$\displaystyle\Big{\langle}{\sum_{n=1}^{N}p_{n}x_{n}},\ \
{\sum_{m=1}^{N}p_{m}x_{m}}\Big{\rangle}$ $\displaystyle=$
$\displaystyle\sum_{m,n=1}^{N}\big{(}p_{n}\left<x_{n},x_{m}\right>\big{)}\overline{p_{m}}+[{p_{n}p_{m}},x_{n},x_{m}]+\left<[p_{n},p_{m},x_{m}],x_{n}\right>$
$\displaystyle=$
$\displaystyle\sum_{n=1}^{N}\left|\left<x,x_{n}\right>\right|^{2}+\sum_{m,n=1}^{N}[{p_{n}p_{m}},x_{n},x_{m}]+\left<[\left<x,x_{n}\right>,\left<x,x_{m}\right>,x_{m}],x_{n}\right>,$

so that

$\displaystyle\left|\left|u\right|\right|^{2}=\text{Re}\,\left|\left|u\right|\right|^{2}$
$\displaystyle=\sum_{n=1}^{N}\left|\left<x,x_{n}\right>\right|^{2}+\sum_{m,n=1}^{N}\left<[\left<x,x_{n}\right>,\left<x,x_{m}\right>,x_{m}],x_{n}\right>_{\mathbb{R}}.$

Hence (4) becomes

$\displaystyle\left|\left|x\right|\right|^{2}$ $\displaystyle=$
$\displaystyle\left|\left|v\right|\right|^{2}-\Big{(}\sum_{n=1}^{N}\left|\left<x,x_{n}\right>\right|^{2}+\sum_{m,n=1}^{N}\left<[\left<x,x_{n}\right>,\left<x,x_{m}\right>,x_{m}],x_{n}\right>_{\mathbb{R}}\Big{)}+2\sum_{n=1}^{N}\left|\left<x,x_{n}\right>\right|^{2}$
$\displaystyle=$
$\displaystyle\sum_{n=1}^{N}\left|\left<x,x_{n}\right>\right|^{2}+\Big{\|}{x-\sum_{n=1}^{N}\left<x,x_{n}\right>x_{n}}\Big{\|}^{2}-\sum_{m,n=1}^{N}\left<[\left<x,x_{n}\right>,\left<x,x_{m}\right>,x_{m}],x_{n}\right>_{\mathbb{R}}.$

This proves the lemma. ∎

######  Remark 4.5.

In Example 4.3, it is worth pointing out that $\\{x_{n}\\}_{n=1}^{4}$ is not
$\mathbb{O}$-linearly independent. Indeed, if we denote

$x_{5}=\dfrac{1}{\sqrt{2}}(1,-e_{3}),$

then we have

$x_{1}=e_{1}x_{5},\quad x_{2}=e_{4}x_{5},\quad x_{3}=e_{6}x_{5}.$

Moreover, it is easy to check that $\\{x_{n}\\}_{n=4}^{5}$ is also an
orthonormal basis. By direct calculation, for $y=(1,0)$ we have

$y-\sum_{i=4}^{5}\left<y,x_{i}\right>x_{i}=0$

and

$\left|\left|y\right|\right|^{2}=\sum_{i=4}^{5}\left|\left<y,x_{i}\right>\right|^{2}.$

In fact, this is not accidental in terms of the corollary below. The point is
that the orthonormal basis $\\{x_{n}\\}_{n=4}^{5}$ satisfies

$[p,x_{4},x_{5}]=0$

for all $p\in\mathbb{O}$.

######  Corollary 4.6.

Under the assumption of Lemma 4.4 and suppose $\\{x_{n}\\}_{n=1}^{N}$
satisfies

(4.7) $\displaystyle[p,x_{n},x_{m}]=0$

for all $p\in\mathbb{O}$ and for each $m,n=1,\dots,N$. Then we have

(4.8)
$\displaystyle\left|\left|x\right|\right|^{2}=\sum_{n=1}^{N}\left|\left<x,x_{n}\right>\right|^{2}+\Big{\|}{x-\sum_{n=1}^{N}\left<x,x_{n}\right>x_{n}}\Big{\|}^{2}$

and in this case we obtain the classical Bessel’s inequality

(4.9)
$\displaystyle\sum_{n=1}^{N}\left|\left<x,x_{n}\right>\right|^{2}\leqslant\left|\left|x\right|\right|^{2}.$

###### Proof.

If $[p,x_{n},x_{m}]=0$, then by taking the real part on both sides of (2.7),
we have

$\sum_{m,n=1}^{N}\left<[\left<x,x_{n}\right>,\left<x,x_{m}\right>,x_{m}],x_{n}\right>_{\mathbb{R}}=-\sum_{m,n=1}^{N}\text{Re}\,\left(\left<x,x_{n}\right>[\left<x,x_{m}\right>,x_{n},x_{m}]\right)=0.$

Therefore, (4.8) follows from (4.4) directly. This completes the proof. ∎

This inspires us to introduce the following notion.

######  Definition 4.7.

An orthonormal basis $S=\\{x_{\alpha}\\}_{\alpha\in\Lambda}$ is said to be a
weak associative orthonormal basis if $S$ satisfies condition (4.7) in
Corollary 4.6.

######  Lemma 4.8.

If $S=\\{x_{\alpha}\\}_{\alpha\in\Lambda}$ is a weak associative orthonormal
basis in a Hilbert left $\mathbb{O}$-module $H$, then

$\widetilde{S}:=\\{e_{i}x_{\alpha}\mid i=0,\dots,7,\ \alpha\in\Lambda\\}$

is a real orthonormal basis of the real Hilbert space
$(H,\left<\cdot,\cdot\right>_{\mathbb{R}})$.

###### Proof.

According to identities (2.10) and (2.7), we have

$\displaystyle\left<e_{i}x_{\alpha},e_{j}x_{\beta}\right>_{\mathbb{R}}$
$\displaystyle=\text{Re}\,\left(e_{i}\left<x_{\alpha},x_{\beta}\right>\right)\overline{e_{j}}+[e_{i}{e_{j}},x_{\alpha},x_{\beta}]+\left<[e_{i},e_{j},x_{\beta}],x_{\alpha}\right>_{\mathbb{R}}$
$\displaystyle=\text{Re}\,(e_{i}\overline{e_{j}}\delta_{\alpha\beta})-\text{Re}\,(e_{i}[e_{j},x_{\beta},x_{\alpha}])$
$\displaystyle=\delta_{ij}\delta_{\alpha\beta}$

for each $i,j=0,\dots,7$ and each $\alpha,\beta\in\Lambda$. Hence
$\widetilde{S}$ is a real orthonormal system of
$(H,\left<\cdot,\cdot\right>_{\mathbb{R}})$. We next show that $\widetilde{S}$
is a real orthonormal basis. If not, then there exists a non-zero element
$x\in H$ such that $x$ is orthogonal to $\widetilde{S}$ with respect to the
real inner product $\left<\cdot,\cdot\right>_{\mathbb{R}}$. Thus we have

$\left<x_{\alpha},x\right>=\left\langle
x_{\alpha},x\right\rangle_{\mathbb{R}}-\sum_{i=0}^{7}\left\langle
e_{i}x_{\alpha},x\right\rangle_{\mathbb{R}}e_{i}=0.$

In view of Lemma 4.2, this is impossible. ∎

Now we are in position to establish the Parseval theorem for weak associative
orthonormal bases.

######  Theorem 4.9 (Parseval theorem).

Let $H$ be an $\mathbb{O}$-Hilbert space and
$S=\\{x_{\alpha}\\}_{\alpha\in\Lambda}$ be a weak associative orthonormal
basis. Then any $x\in H$ can be uniquely expressed as

$x=\sum_{\alpha\in\Lambda}\left<x,x_{\alpha}\right>x_{\alpha}$

and there holds

$\left|\left|x\right|\right|^{2}=\sum_{\alpha\in\Lambda}|\left<x,x_{\alpha}\right>|^{2}.$

###### Proof.

In view of Cauchy-Schwarz inequality, we get

$\left|\left<x,x_{\alpha}\right>\right|\leqslant\left|\left|x\right|\right|$

for each $\alpha\in\Lambda$. Note that

$\bigcup_{k=1}^{\infty}\big{[}\dfrac{1}{k+1}\left|\left|x\right|\right|,\dfrac{1}{k}\left|\left|x\right|\right|\big{]}=(\;0,\
\left|\left|x\right|\right|\;].$

Thanks to Corollary 4.6, there are only countable $\alpha$, say
$(\alpha_{j})_{j=1}^{\infty}$, such that $\left<x,x_{\alpha_{j}}\right>\neq
0$. Then by Bessel’s inequality (4.9) in Corollary 4.6, we have

(4.10)
$\displaystyle\sum_{j=1}^{\infty}\left|\left<x,x_{\alpha_{j}}\right>\right|^{2}<+\infty.$

Set

$y_{n}=\sum_{j=1}^{n}\left<x,x_{\alpha_{j}}\right>x_{\alpha_{j}}.$

For any $m<n$, we have

$y_{n}-y_{m}=\sum_{j=m+1}^{n}\left<x,x_{\alpha_{j}}\right>x_{\alpha_{j}}.$

Since $S$ is a weak associative orthonormal basis, we get

$\displaystyle\left<y_{n}-y_{m},x_{\alpha_{k}}\right>$ $\displaystyle=$
$\displaystyle\sum_{j=m+1}^{n}\left<\left<x,x_{\alpha_{j}}\right>x_{\alpha_{j}},x_{\alpha_{k}}\right>$
$\displaystyle=$
$\displaystyle\sum_{j=m+1}^{n}\left<x,x_{\alpha_{j}}\right>\left<x_{\alpha_{j}},x_{\alpha_{k}}\right>+[{\left<x,x_{\alpha_{j}}\right>},x_{\alpha_{j}},x_{\alpha_{k}}]$
$\displaystyle=$ $\displaystyle\left<x,x_{\alpha_{k}}\right>.$

It then follows from Corollary 4.6 again that

$\displaystyle\left|\left|y_{n}-y_{m}\right|\right|^{2}$ $\displaystyle=$
$\displaystyle\sum_{k=m+1}^{n}\left|\left<y_{n}-y_{m},x_{\alpha_{k}}\right>\right|^{2}+\Big{\|}{(y_{n}-y_{m})-\sum_{k=m+1}^{n}\left<y_{n}-y_{m},x_{\alpha_{k}}\right>x_{\alpha_{k}}}\Big{\|}^{2}$
$\displaystyle=$
$\displaystyle\sum_{k=m+1}^{n}\left|\left<x,x_{\alpha_{k}}\right>\right|^{2}+\Big{\|}{\sum_{j=m+1}^{n}\left<x,x_{\alpha_{j}}\right>x_{\alpha_{j}}-\sum_{k=m+1}^{n}\left<x,x_{\alpha_{k}}\right>x_{\alpha_{k}}}\Big{\|}^{2}$
$\displaystyle=$
$\displaystyle\sum_{k=m+1}^{n}\left|\left<x,x_{\alpha_{k}}\right>\right|^{2}.$

Thus we conclude from (4.10) that $\\{y_{n}\\}_{n=1}^{\infty}$ is a Cauchy
sequence. From the completeness of $H$, there exists an element $x^{\prime}\in
H$ such that

$\lim_{n\rightarrow+\infty}y_{n}=x^{\prime}.$

We next prove that $x^{\prime}=x$. In view of Lemma 4.8, it suffices to prove

(4.11) $\displaystyle\left<x-x^{\prime},e_{i}x_{\alpha}\right>_{\mathbb{R}}$
$\displaystyle=0$

for each $\alpha\in\Lambda$ and $i=1,\dots,7$. For any $x_{\alpha_{k}}$, we
get

$\displaystyle\left<x-x^{\prime},e_{i}x_{\alpha_{k}}\right>_{\mathbb{R}}$
$\displaystyle=$
$\displaystyle\text{Re}\,\big{(}\left<x-x^{\prime},x_{\alpha_{k}}\right>\overline{e_{i}}+[{e_{i}},x-x^{\prime},x_{\alpha_{k}}]\big{)}$
$\displaystyle=$
$\displaystyle\text{Re}\,\Big{(}\left<x,x_{\alpha_{k}}\right>\overline{e_{i}}-\lim_{n\rightarrow+\infty}\sum_{j=1}^{n}\left<\left<x,x_{\alpha_{j}}\right>x_{\alpha_{j}},{x_{\alpha_{k}}}\right>\overline{e_{i}}\Big{)}$
$\displaystyle=$
$\displaystyle\text{Re}\,\Big{(}\left<x,x_{\alpha_{k}}\right>\overline{e_{i}}-\lim_{n\rightarrow+\infty}\sum_{j=1}^{n}\big{(}\left<x,x_{\alpha_{j}}\right>\left<x_{\alpha_{j}},x_{\alpha_{k}}\right>\big{)}\overline{e_{i}}\Big{)}$
$\displaystyle=$ $\displaystyle 0.$

For any $x_{\alpha}$ with $\alpha\neq\alpha_{j},\ j=1,2,\dots,$ we also have

$\displaystyle\left<x-x^{\prime},e_{i}x_{\alpha}\right>_{\mathbb{R}}$
$\displaystyle=$
$\displaystyle\text{Re}\,\big{(}\left<x-x^{\prime},x_{\alpha}\right>\overline{e_{i}}+[{e_{i}},x-x^{\prime},x_{\alpha}]\big{)}$
$\displaystyle=$
$\displaystyle\text{Re}\,\Big{(}\left<x,x_{\alpha}\right>\overline{e_{i}}-\lim_{n\rightarrow+\infty}\sum_{j=1}^{n}\left<\left<x,x_{\alpha_{j}}\right>x_{\alpha_{j}},{x_{\alpha}}\right>\overline{e_{i}}\Big{)}$
$\displaystyle=$
$\displaystyle\text{Re}\,\Big{(}-\lim_{n\rightarrow+\infty}\sum_{j=1}^{n}\big{(}\left<x,x_{\alpha_{j}}\right>\left<x_{\alpha_{j}},x_{\alpha}\right>\big{)}\overline{e_{i}}\Big{)}$
$\displaystyle=$ $\displaystyle 0.$

This proves $x=x^{\prime}$. Hence we have

$x=\sum_{j=1}^{\infty}\left<x,x_{\alpha_{j}}\right>x_{\alpha_{j}}=\sum_{\alpha\in\Lambda}\left<x,x_{\alpha}\right>x_{\alpha}.$

By Corollary 4.6 we obtain

$\left|\left|x\right|\right|^{2}=\sum_{j=1}^{n}\Big{|}\left<x,x_{\alpha_{j}}\right>\Big{|}^{2}+\Big{\|}{x-\sum_{j=1}^{n}\left<x,x_{\alpha_{j}}\right>x_{\alpha_{j}}}\Big{\|}^{2}$

for any $n\in\mathbb{N}$. Letting $n\rightarrow\infty$, we get

$\left|\left|x\right|\right|^{2}=\sum_{j=1}^{\infty}\left|\left<x,x_{\alpha_{j}}\right>\right|^{2}=\sum_{\alpha\in\Lambda}|\left<x,x_{\alpha}\right>|^{2}.$

The uniqueness of the decomposition follows from the condition (4.7). This
completes the proof. ∎

######  Remark 4.10.

For any $\mathbb{O}$-Hilbert space, there always exists a weak associative
orthonormal basis. This is because there always exists an orthonormal basis in
a real Hilbert space. Note that by Theorem 3.10, an $\mathbb{O}$-Hilbert space
$H$ can be decomposed into an orthogonal sum. Let
$\\{x_{\alpha}\\}_{\alpha\in\Lambda}$ be an orthonormal basis of the real
Hilbert space $(\mathscr{A}(H),\left<\cdot,\cdot\right>_{\mathbb{R}})$ and
$\\{x_{\beta}\\}_{\beta\in\Lambda^{\prime}}$ be an orthonormal basis of the
real Hilbert space
$(\mathscr{A}^{-}(H),\left<\cdot,\cdot\right>_{\mathbb{R}})$. Then

$S:=\\{x_{\gamma}\\}_{\gamma\in\Lambda\cup\Lambda^{\prime}}$

is a weak associative orthonormal basis.

At last, we show that every orthonormal basis for which the Parseval theorem
holds is also weak associative.

######  Definition 4.11.

An orthonormal basis $S=\\{x_{\alpha}\\}_{\alpha\in\Lambda}$ is said to be a
Hilbert basis, if for any $x\in H$ it holds

$\left|\left|x\right|\right|^{2}=\sum_{\alpha\in\Lambda}|\left<x,x_{\alpha}\right>|^{2}.$

######  Theorem 4.12.

Let $S$ be an orthonormal basis. Then $S$ is a Hilbert basis if and only if
$S$ is weak associative.

###### Proof.

In view of Theorem 4.9, it suffices to show that every Hilbert basis is weak
associative. Let $S=\\{x_{\alpha}\\}_{\alpha\in\Lambda}$ be a Hilbert basis.
Then for $x=px_{\beta}$, we have

$\displaystyle\left|\left|x\right|\right|^{2}$
$\displaystyle=\sum_{\alpha\in\Lambda}\left|\left<px_{\beta},x_{\alpha}\right>\right|^{2}$
$\displaystyle=\sum_{\alpha\in\Lambda}\left|(p\delta_{\alpha\beta}+[{p},x_{\beta},x_{\alpha}])\right|^{2}$
$\displaystyle=\left|p\right|^{2}+\sum_{\alpha\neq\beta}\left|[{p},x_{\beta},x_{\alpha}]\right|^{2}.$

It follows from $\left|\left|x\right|\right|=\left|p\right|$ that

$\sum_{\alpha\neq\beta}\left|[{p},x_{\beta},x_{\alpha}]\right|^{2}=0.$

This implies that

$[{p},x_{\beta},x_{\alpha}]=0$

for all $x_{\beta},x_{\alpha}\in S$. This proves that $S$ is weak associative.
∎

##  5\. Cayley-Dickson algebras as $\mathbb{O}$-Hilbert spaces

Any Cayley–Dickson algebra can be endowed with a weak $\mathbb{O}$-bimodule
structure and an $\mathbb{O}$-inner product.

At first, we introduce the notion of weak $\mathbb{O}$-bimodules. Let $M$ be a
left $\mathbb{O}$-module. There associates an involution

$C:M\to M,$

called conjugate map. Since any $x\in M$ can be written as

$x=\sum_{i=0}^{7}e_{i}x_{i}+\sum_{i=0}^{7}e_{i}x^{-}_{i},$

where $x_{i}\in\mathscr{A}(M)$ and $x^{-}_{i}\in\mathscr{A}^{-}(M)$, we define
the conjugate of $x$ as

$C(x)=\overline{x}:=\sum_{i=0}^{7}\overline{e_{i}}x_{i}-\sum_{i=0}^{7}e_{i}x^{-}_{i}.$

Then by definition, we have

$\displaystyle\overline{x}=\begin{cases}x,\qquad&x\in\mathscr{A}(M),\\\ \\\
-x,\qquad&x\in\mathscr{A}^{-}(M).\end{cases}$

This means

(5.1) $\displaystyle C(\mathscr{A}(M))=\mathscr{A}(M),\quad
C(\mathscr{A}^{-}(M))=\mathscr{A}^{-}(M).$

######  Definition 5.1.

We call $M$ a weak $\mathbb{O}$-bimodule if $M$ is both a left
$\mathbb{O}$-module and a right $\mathbb{O}$-module satisfying the
compatibility conditions

(5.2) $\displaystyle\overline{px}=\overline{x}\ \overline{p}$

for all $x\in M$ and $p\in\mathbb{O}$.

######  Remark 5.2.

The notion of a bimodule for a class of algebras has been introduced by
Eilenberg [3]. Following [12, 18], we call $M$ an $\mathbb{O}$-bimodule if $M$
is both a left $\mathbb{O}$-module and a right $\mathbb{O}$-module satisfying
the compatibility conditions

$[p,q,x]=[x,p,q]=[q,x,p]$

for all $p,q\in\mathbb{O}$ and $x\in M$. The regular bimodule $\mathbb{O}$ is
known to be the only irreducible $\mathbb{O}$-bimodule [12, 18]. We remark
that a weak $\mathbb{O}$-bimodule may not be an $\mathbb{O}$-bimodule in
general.

######  Remark 5.3.

Let $M$ be a weak $\mathbb{O}$-bimodule.

  1. (${\mathrm{i}}$)

For all $p\in\mathbb{O}$ and $x\in M$, we have

(5.3) $\displaystyle\overline{xp}=\overline{p}\ \overline{x}.$

Indeed, it follows from (5.2) that

$px=\overline{\overline{px}}=\overline{\overline{x}\ \overline{p}}$

which becomes (5.3) after replacing $p,x$ by $\overline{p},\overline{x}$,
respectively.

  2. (${\mathrm{ii}}$)

The right multiplication of $M$ is determined by its left multiplication due
to (5.3). Conversely, every left $\mathbb{O}$-module can be viewed as a weak
$\mathbb{O}$-bimodule by endowed with such right multiplication. That is, a
weak $\mathbb{O}$-bimodule is nothing but a left $\mathbb{O}$-module endowed
with a certain right multiplication.

  3. (${\mathrm{iii}}$)

We denote the left nucleus by

$\mathscr{A}_{L}(M):=\\{x\in M\mid[p,q,x]=0,\forall p,q\in\mathbb{O}\\}$

and the right nucleus by

$\mathscr{A}_{R}(M):=\\{x\in M\mid[x,p,q]=0,\forall p,q\in\mathbb{O}\\}.$

Similar notations $\mathscr{A}^{-}_{L}(M),\ \mathscr{A}^{-}_{R}(M)$ can be
also defined. One can check that

$\mathscr{A}_{R}(M)=C(\mathscr{A}_{L}(M)),\quad\mathscr{A}^{-}_{R}(M)=C(\mathscr{A}^{-}_{L}(M)).$

Combining this with (5.1), we have

(5.4)
$\displaystyle\mathscr{A}_{R}(M)=\mathscr{A}_{L}(M),\quad\mathscr{A}^{-}_{R}(M)=\mathscr{A}^{-}_{L}(M).$

Henceforth we shall denote

$\displaystyle\mathscr{A}(M):=\mathscr{A}_{R}(M)=\mathscr{A}_{L}(M),\quad\mathscr{A}^{-}(M):=\mathscr{A}^{-}_{R}(M)=\mathscr{A}^{-}_{L}(M).$

We next consider the algebra of sedenions. The algebra of sedenions
$\mathbb{S}$ is constructed from the algebra of octonions through the
Cayley–Dickson construction [2]. The elements in the algebra of sedenions
$\mathbb{S}$ take the form

$s=\sum_{i=0}^{15}x_{i}e_{i},$

where $x_{i}$ are reals, $e_{0}=1$ and $e_{1},e_{2},\dots,e_{15}$ are
imaginary units, i.e.,

$e_{i}^{2}=-1$

for all $i=1,\dots,15$. The conjugate of $s$ is defined as

$\overline{s}=x_{0}-\sum_{i=1}^{15}x_{i}e_{i}.$

We regard $\mathbb{O}$ as the subalgebra of $\mathbb{S}$ generated by
$e_{0},e_{1},\dots,e_{7}$. By the Cayley-Dickson construction, any
$s\in\mathbb{S}$ can be expressed uniquely as

$s=a+be_{8}$

with $a,b\in\mathbb{O}$ so that the conjugate of $s$ becomes

$\overline{s}=\overline{a}-be_{8}$

and the multiplication in $\mathbb{S}$ is defined as

(5.5)
$\displaystyle(a+be_{8})(c+de_{8})=(ac-\overline{d}b)+(da+b\overline{c})e_{8}.$

We want to show that $\mathbb{S}$ is a weak $\mathbb{O}$-bimodule. At first,
we show that $\mathbb{S}$ is a left $\mathbb{O}$-module with multiplication

$\displaystyle\mathbb{O}\times\mathbb{S}$ $\displaystyle\to$
$\displaystyle\mathbb{S}$ $\displaystyle(p,x)$ $\displaystyle\mapsto$
$\displaystyle px.$

Indeed, it follows from (5.5) that

(5.6) $\displaystyle a(de_{8})=(da)e_{8}$

for all $a,d\in\mathbb{O}$. Hence for all $p\in\mathbb{O}$ and
$x=a+be_{8}\in\mathbb{S}$, we have

$p(px)=p(pa+(bp)e_{8})=p^{2}a+(bp^{2})e_{8}=p^{2}x.$

This shows that $\mathbb{S}$ is a left $\mathbb{O}$-module.

One can easily check that the conjugate of $s$ in sedenions coincides with the
conjugate of $s$ when $\mathbb{S}$ is regarded as a left $\mathbb{O}$-module.
Hence the right multiplication induced by (5.2) is

$\displaystyle\mathbb{S}\times\mathbb{O}$ $\displaystyle\to$
$\displaystyle\mathbb{S}$ $\displaystyle(x,p)$ $\displaystyle\mapsto$
$\displaystyle\overline{\overline{p}\ \overline{x}}=xp.$

It can be seen from (5.6) that

$e_{8}\in\mathscr{A}^{-}(\mathbb{S}).$

Therefore we get

$\mathbb{S}\cong\mathbb{O}\oplus\overline{\mathbb{O}}$

as left $\mathbb{O}$-modules.

We now define

$\left<a+be_{8},c+de_{8}\right>_{\mathbb{S}}:=a\overline{c}+\overline{d}b.$

In other words, if we denote by

$\pi_{\mathbb{O}}:\mathbb{S}\to\mathbb{O}$

the projection from $\mathbb{S}$ to $\mathbb{O}$, then for any
$s,w\in\mathbb{S}$,

$\left<s,w\right>_{\mathbb{S}}:=\pi_{\mathbb{O}}(s\overline{w}).$

One can check that this is an $\mathbb{O}$-inner product.

We move on our discussion to arbitrary Cayley–Dickson algebra
$\mathbb{A}_{n}$. For $n=1,2,3$,

$\mathbb{A}_{1}=\mathbb{C},\quad\mathbb{A}_{2}=\mathbb{H},\quad\mathbb{A}_{3}=\mathbb{O}.$

We shall apply the induction to endow $\mathbb{A}_{n}$ with a Hilbert
$\mathbb{O}$-module structure.

The Cayley-Dickson algebra $\mathbb{A}_{n}$ is defined by induction. An
element $s$ in $\mathbb{A}_{n}$ can be written as

$s=a+be_{2^{n-1}},$

where $a,b\in\mathbb{A}_{n-1}$. Its conjugate is defined as

$\overline{a+be_{2^{n-1}}}=\overline{a}-be_{2^{n-1}}.$

the multiplication of $\mathbb{A}_{n}$ is defined as

(5.7)
$\displaystyle(a+be_{2^{n-1}})(c+de_{2^{n-1}})=(ac-\overline{d}b)+(da+b\overline{c})e_{2^{n-1}},$

where $a,b,c,d\in\mathbb{A}_{n-1}$.

As a real vector space, $\mathbb{A}_{n}$ admits a canonical basis

$e_{0},e_{1},\dots,e_{2^{n}-1},$

where

(5.8) $\displaystyle e_{2^{n-1}+j}:=e_{j}e_{2^{n-1}}$

for any $j=1,\dots,2^{n-1}-1$. The octonions $\mathbb{O}$ can be viewed as a
subset of $\mathbb{A}_{n}$ when $n\geqslant 4$ generated by
$e_{0},e_{1},\cdots,e_{7}$.

The left $\mathbb{O}$-module structure on $\mathbb{A}_{n}$ is induced by the
multiplication of $\mathbb{A}_{n}$:

$\displaystyle\mathbb{O}\times\mathbb{A}_{n}$ $\displaystyle\to$
$\displaystyle\mathbb{A}_{n}$ $\displaystyle(p,x)$ $\displaystyle\mapsto$
$\displaystyle px.$

It turns out that $\mathbb{A}_{n}$ is a weak $\mathbb{O}$-bimodule for
$n\geqslant 4$.

######  Lemma 5.4.

If $n\geqslant 4$, then $\mathbb{A}_{n}$ is a weak $\mathbb{O}$-bimodule and
the two conjugates coincide when $\mathbb{A}_{n}$ is regarded as either a weak
$\mathbb{O}$-bimodule or a Cayley–Dickson algebra. Moreover, we have an
isomorphism of left $\mathbb{O}$-modules

$\mathbb{A}_{n}\cong\mathbb{O}^{2^{n-4}}\oplus\overline{\mathbb{O}}^{2^{n-4}}.$

###### Proof.

When $n=4$, we have $\mathbb{A}_{4}=\mathbb{S}$ so the result has been proved.
Suppose it holds for $n$, we come to prove the case of $n+1$.

It follows from (5.7) that

$p(qe_{2^{n}})=(qp)e_{2^{n}}$

for all $p,q\in\mathbb{O}$ and $n\geqslant 4$.

Now we check that the subset $\mathbb{A}_{n}e_{2n}$ of $\mathbb{A}_{n+1}$ is
also a left $\mathbb{O}$-module as well as a right $\mathbb{O}$-module. For
any $p\in\mathbb{O}$ and $x\in\mathbb{A}_{n}$,

(5.9) $\displaystyle p(p(xe_{2^{n}}))=p((xp)e_{2^{n}})=((xp)p)e_{2^{n}}.$

By induction, $\mathbb{A}_{n}$ is a right $\mathbb{O}$-module so that
$(xp)p=xp^{2}$. Therefore (5.9) becomes

$p(p(xe_{2^{n}}))=(xp^{2})e_{2^{n}}=p^{2}(xe_{2^{n}}).$

This shows that $\mathbb{A}_{n}e_{2^{n}}$ is a left $\mathbb{O}$-module. It
can be proved that $\mathbb{A}_{n}e_{2^{n}}$ is a right $\mathbb{O}$-module
similarly.

Moreover, it is easy to check that

(5.10)
$\displaystyle\mathscr{A}_{L}{(\mathbb{A}_{n}e_{2^{n}})}=\mathscr{A}^{-}_{R}({\mathbb{A}_{n}})e_{2^{n}},\quad\mathscr{A}^{-}_{L}{(\mathbb{A}_{n}e_{2^{n}})}=\mathscr{A}_{R}{(\mathbb{A}_{n})}e_{2^{n}}.$

Combining (5.10) and (5.4), we conclude that

(5.11)
$\displaystyle\mathscr{A}{(\mathbb{A}_{n}e_{2^{n}})}=\mathscr{A}^{-}({\mathbb{A}_{n}})e_{2^{n}},\quad\mathscr{A}^{-}{(\mathbb{A}_{n}e_{2^{n}})}=\mathscr{A}{(\mathbb{A}_{n})}e_{2^{n}}.$

By induction, as left modules

$\mathbb{A}_{n}\cong\mathbb{O}^{2^{n-4}}\oplus\overline{\mathbb{O}}^{2^{n-4}}.$

This implies that

$\mathbb{A}_{n}e_{2^{n}}\cong\mathbb{O}^{2^{n-4}}\oplus\overline{\mathbb{O}}^{2^{n-4}}.$

Hence we obtain

$\displaystyle\mathbb{A}_{n+1}$ $\displaystyle=$
$\displaystyle\mathbb{A}_{n}\oplus\mathbb{A}_{n}e_{2^{n}}$
$\displaystyle\cong$
$\displaystyle\mathbb{O}^{2^{n+1-4}}\oplus\overline{\mathbb{O}}^{2^{n+1-4}}.$

This completes the proof. ∎

We finally define the $\mathbb{O}$-inner product on $\mathbb{A}_{n}$. We
denote by

$\pi_{\mathbb{O}}:\mathbb{A}_{n}\to\mathbb{O}$

the projection. For any $s,w\in\mathbb{A}_{n}$, we define

(5.12) $\displaystyle\left<s,w\right>_{n}:=\pi_{\mathbb{O}}(s\overline{w})$

######  Theorem 5.5.

$(\mathbb{A}_{n},\left<\cdot,\cdot\right>_{n})$ is a Hilbert left
$\mathbb{O}$-module for any $n\geqslant 4$.

###### Proof.

We have shown that $\left<s,w\right>_{4}$ is an $\mathbb{O}$-inner product.
Suppose $\left<s,w\right>_{n-1}$ is also an $\mathbb{O}$-inner product on
$\mathbb{A}_{n-1}$. We denote by

$\pi_{n-1}:\mathbb{A}_{n}\to\mathbb{A}_{n-1}$

the orthogonal projection. Then for any
$s=a+be_{2^{n-1}},w=c+de_{2^{n-1}}\in\mathbb{A}_{n}$, we have

$\displaystyle\left<s,w\right>_{n}$ $\displaystyle=$
$\displaystyle\pi_{\mathbb{O}}(s\overline{w})$ $\displaystyle=$
$\displaystyle\pi_{3}\pi_{4}\dots\pi_{n-1}(s\overline{w})$ $\displaystyle=$
$\displaystyle\pi_{3}\pi_{4}\dots\pi_{n-2}(a\overline{c}+\overline{d}b)$
$\displaystyle=$
$\displaystyle\left<a,c\right>_{n-1}+\left<\overline{d},\overline{b}\right>_{n-1}.$

Note that for any $p\in\mathbb{O}$,

$ps=pa+(bp)e_{2^{n-1}}.$

Hence we have

$\displaystyle\left<ps,w\right>_{n}-p\left<s,w\right>_{n}$ $\displaystyle=$
$\displaystyle\left<pa,c\right>_{n-1}+\big{\langle}{\overline{d}},\ \
{\overline{(bp)}}_{n-1}\big{\rangle}-p\big{(}\left<a,c\right>_{n-1}+\left<\overline{d},\overline{b}\right>_{n-1}\big{)}$
$\displaystyle=$
$\displaystyle[p,a,c]_{n-1}+[\left<\overline{d},\overline{b}\right>_{n-1},p]-[p,\overline{d},\overline{b}]_{n-1}.$

Here $[p,a,c]_{n-1}$ and $[p,\overline{d},\overline{b}]_{n-1}$ are second
associators in the $\mathbb{O}$-Hilbert space $\mathbb{A}_{n-1}$. Hence by
induction we obtain

$\text{Re}\,\left(\left<ps,w\right>_{n}-p\left<s,w\right>_{n}\right)=0.$

This proves the para-linearity.

To prove the hermiticity, we apply the inductive assumption to get

$\left<w,s\right>_{n}=\left<c,a\right>_{n-1}+\left<\overline{b},\overline{d}\right>_{n-1}=\overline{\left<a,c\right>_{n-1}}+\overline{\left<\overline{d},\overline{b}\right>_{n-1}}=\overline{\left<s,w\right>_{n}}.$

At last, we come to prove the positivity. According to the inductive
assumption, we have

$\left<s,s\right>_{n}=\left<a,a\right>_{n-1}+\left<\overline{b},\overline{b}\right>_{n-1}\geqslant
0$

and

$\left<s,s\right>_{n}=0\iff a=b=0\iff s=0.$

This shows the positivity. ∎

Theorem 5.5 provides us a new approach to study the Cayley–Dickson algebras.
We point out that (5.11) gives a complete description of the associative and
conjugate associative elements of $\mathbb{A}_{n}$ by induction. More
precisely,

$\displaystyle\mathscr{A}{(\mathbb{A}_{n+1})}$ $\displaystyle=$
$\displaystyle\mathscr{A}(\mathbb{A}_{n})\oplus\mathscr{A}^{-}({\mathbb{A}_{n}})e_{2^{n}},$
$\displaystyle\mathscr{A}^{-}{(\mathbb{A}_{n+1})}$ $\displaystyle=$
$\displaystyle\mathscr{A}^{-}(\mathbb{A}_{n})\oplus\mathscr{A}({\mathbb{A}_{n}})e_{2^{n}}.$

For example, we have

$\mathbb{A}_{n}$ | a basis of $\mathscr{A}{(\mathbb{A}_{n})}$ | a basis of $\mathscr{A}{(\mathbb{A}_{n})}$
---|---|---
$\mathbb{A}_{4}$ | $e_{0}$ | $e_{8}$
$\mathbb{A}_{5}$ | $e_{0},\ {e_{8}}e_{16}$ | $e_{8},e_{16}$
$\mathbb{A}_{6}$ | $e_{0},\ {e_{8}}e_{16},\ {e_{8}}e_{32},\ {e_{16}}e_{32}$ | $e_{8},e_{16},e_{32},\ {({e_{8}}e_{16})}e_{32}$

Let us look at a weak associative orthonormal basis in the Cayley-Dickson
algebra as a Hilbert left $\mathbb{O}$-module.

In the specific case $n=5$, the algebra $\mathbb{A}_{n}=\mathbb{A}_{5}$ admits
a weak associative orthonormal basis

$S:=\\{e_{0},\ e_{24},e_{8},e_{16}\\}.$

This means for any $x\in\mathbb{A}_{5}$,

$x=\pi_{\mathbb{O}}(x)+\pi_{\mathbb{O}}(x\overline{e_{8}})e_{8}+\pi_{\mathbb{O}}(x\overline{e_{16}})e_{16}+\pi_{\mathbb{O}}(x\overline{e_{24}})e_{24}.$

In the general $\mathbb{A}_{n}$, due to (5.8) we can take a weak associative
orthonormal basis as

$S:=\\{e_{8i}:\mid i=0,\dots,2^{n-3}-1\\}.$

Any $x\in\mathbb{A}_{n}$ can be expressed in terms of this weak associative
orthonormal basis as

(5.13) $\displaystyle
x=\sum_{i=0}^{2^{n-3}-1}\pi_{\mathbb{O}}(x\overline{e_{8i}})e_{8i}.$

This can be regarded as an octonionic version of the decomposition

(5.14) $\displaystyle
x=\sum_{i=0}^{2^{n}-1}\text{Re}\,(x\overline{e_{i}})e_{i}.$

######  Remark 5.6.

We have shown that the Cayley-Dickson algebra $\mathbb{A}_{n}$ is a Hilbert
left $\mathbb{O}$-module. However the corresponding result does not hold in
the quaternionic case. That is, $\mathbb{A}_{n}$ is not a $\mathbb{H}$-vector
space in general. This is because the definition of $\mathbb{H}$-vector space
(see [16]) requires an associative condition

$p(qx)=(pq)x$

for all $p,q\in\mathbb{H}$ and any element $x$ of that $\mathbb{H}$-vector
space.

## References

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  * [2] L. E. Dickson.  On quaternions and their generalization and the history of the eight square theorem.  Ann. of Math. (2), 20(3):155–171, 1919.
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# Diameter estimation of $(m,\rho)$-quasi Einstein manifolds

Absos Ali Shaikh1∗, Prosenjit Mandal2, Chandan Kumar Mondal3
1,2Department of Mathematics,
The University of Burdwan,Golapbag,
Purba Bardhaman-713101,
West Bengal, India
3School of Sciences,
Netaji Subhas Open University,
Durgapur Regional Center, Durgapur-713214
Paschim Bardhaman,
West Bengal, India<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

This paper aims to study the $(m,\rho)$-quasi Einstein manifold. This article
shows that a complete and connected Riemannian manifold under certain
conditions becomes compact. Also, we have determined an upper bound of the
diameter for such a manifold. It is also exhibited that the potential function
acquiesces to the Hodge-de Rham potential up to a real constant in an
$(m,\rho)$-quasi Einstein manifold. Later, some triviality and integral
conditions are established for a non-compact complete $(m,\rho)$-quasi
Einstein manifold having finite volume. Finally, it is proved that with some
certain constraints, a complete Riemannian manifold admits finite fundamental
group. Furthermore, some conditions for compactness criteria have also been
deduced.

††footnotetext: ∗ Corresponding author.
$\mathbf{2020}$ Mathematics Subject Classification: 53C20; 53C21; 53C25;
53E20.
Key words and phrases: Riemannian manifold; $(m,\rho)$-Quasi Einstein
Manifold; scalar curvature; index form; diameter estimation.

##  1\. Introduction and preliminaries

A non-gradient generalized $m$-quasi-Einstein manifold ([3]) is an
$n(>2)$-dimensional Riemannian manifold $(N,g)$ such that it preserves a
smooth function $\alpha\in C^{\infty}(N)$, with

(1) $\alpha g+\frac{1}{m}W^{\flat}\otimes
W^{\flat}=\frac{1}{2}\mathcal{L}_{W}g+Ric,$

where $W\in\chi{(N)}$, it’s dual 1-form $W^{\flat}$, $\mathcal{L}_{W}g$
indicates the Lie derivative of $g$ along $W$ and $m$ is a scalar with
$0<m\leq\infty$. In $(\ref{mr2})$, if we replace $\alpha$ by $\lambda+\rho R,$
for some real constants $\lambda$, $\rho$ and the scalar curvature $R$ of $N$,
then $(N,g)$ is called a non-gradient $(m,\rho)$-quasi Einstein manifold
([16]), and in this case equation $(\ref{mr2})$ reduces to

(2) $\lambda g+\rho Rg+\frac{1}{m}W^{\flat}\otimes
W^{\flat}=\frac{1}{2}\mathcal{L}_{W}g+Ric.$

If $W$ is the gradient of a real valued smooth function $f$ on $N$, then the
$(m,\rho)$-quasi Einstein manifold is said to be a (gradient) $(m,\rho)$-quasi
Einstein manifold (for details see [7, 10] ), with potential function $f$. In
this case, (2) reduces to

(3) $\lambda g+\rho Rg+\frac{1}{m}df\otimes df=\nabla^{2}f+Ric.$

If $m=\infty$ and $\rho=0$ (resp., $m=\infty$), then (3) compresses to the
Ricci soliton equation (see e.g. [6, 8, 9, 14, 18, 20]) (resp.,
$\rho$-Einstein soliton equation (see e.g. [17, 19])). In (3), if $m$ and
$\lambda$ is taken from $\chi{(N)}$, then $(N,g)$ is known as the generalized
$m$-quasi Einstein manifold [5]. Furthermore, tracing (3), we obtain

(4) $\lambda n+n\rho R+\frac{1}{m}|\nabla f|^{2}=\Delta f+R.$

For a fixed $\phi\in C^{\infty}(N)$, the weighted Laplacian is defined by
(see, [12])

$\Delta_{\phi}f=\Delta f-\langle\nabla\phi,\nabla f\rangle.$

Catino [5] introduced the notion of a generalized quasi-Einstein manifold and
characterized such type of manifolds with harmonic Weyl tensor. For a compact
$m$-quasi Einstein metric having constant scalar curvature, a triviality
result is proved by Case et al. [4]. They also determined that all compact
$2$-dimensional $m$-quasi Einstein manifolds are trivial. For a compact
gradient $\rho$-Einstein soliton with some conditions, Shaikh et al. [19] have
provided a lower bound of the diameter. Huang and Wei [10] have deduced some
rigidity results on a compact $(m,\rho)$-quasi Einstein manifolds, in
particular, by using some conditions on scalar curvature or on the constant
$\rho$, they have showed that such a manifold is trivial. Demirbaǵ and Gúler
[7] have characterized an $(m,\rho)$-quasi Einstein manifold admitting closed
conformal or parallel vector field. Also, they have established some rigidity
results giving some new examples of the $(m,\rho)$-quasi Einstein manifold.

López and Rio [13] have proved a compactness theorem for a complete Riemannian
manifold satisfying $\mathcal{L}_{W}g+Ric\geq\lambda g$, with potential vector
field $W$ having bounded norm. Wylie [22] has proved that the fundamental
group of a complete Riemannian manifold satisfying
$\mathcal{L}_{W}g+Ric\geq\lambda g$, for $\lambda>0$, is finite.

Hence inspiring by the above studies and the study of [2], in this article, we
have showed the following:

In the first theorem we have generalized the work of Limoncu [11], and
obtained an upper bound of the diameter.

######  Theorem 1.1.

Let $(N,g)$ be a complete and connected Riemannian manifold satisfying
$Ric+\nabla^{2}f-\frac{1}{m}df\otimes df\geq\lambda g+\rho Rg$. If $|f|\leq K$
and $\rho R\geq K_{1}$, for some real constants $K,K_{1}$ choosing in such a
way that $(\lambda+K_{1})>0,$ then $N$ is compact and the diameter satisfies

(5) $diam(N)\leq\pi\sqrt{2}\sqrt{\frac{(n-1)+K\sqrt{2}}{\lambda+K_{1}}}.$

Next, we have generalized the work of Li and Wenyi [12], and obtained the
following:

######  Theorem 1.2.

Assume that $(N,g,e^{-f}dvol)$ is a smooth metric measure space satisfying
$Ric+\nabla^{2}f-\frac{1}{m}df\otimes df\geq\lambda g+\rho Rg$, and $u$ is a
positive smooth $f$-harmonic function on $N$, then

(6) $\frac{1}{2}\Delta F\geq(\lambda+\rho
R)F+\frac{1}{n}F^{2}-\frac{2}{n}|\nabla f|F^{\frac{3}{2}}-|\nabla
F|F^{\frac{1}{2}}+\frac{1}{m}\\{df(\nabla\omega)\\}^{2}+\frac{1}{2}\langle\nabla
f,\nabla F\rangle,$

where we denote $F=|\nabla\log u|^{2}$.

######  Theorem 1.3.

If $(N,g)$ is a compact and oriented $(m,\rho)$-quasi Einstein manifold, then
its potential function is given by $f=\sigma+C_{1}$, where $C_{1}$ is a
constant and $\sigma$ is the Hodge-de Rham potential .

Let $(N,g)$ be an oriented Riemannian manifold and $A^{k}(N)$ be the
collection of all $k$-th differential forms in $N$. Now for given integer
$k\geq 0$, the global inner product in $A^{k}(N)$ is defined by (see, [15])

$\langle\zeta,\omega\rangle=\int_{N}\zeta\wedge*\omega,$

for $\zeta,\omega\in A^{k}(N)$. Here $``*"$ denotes the Hodge star operator.
The global norm of $\zeta\in A^{k}(N)$ is given by
$\|\zeta\|^{2}=\langle\zeta,\zeta\rangle$ with $\|\zeta\|^{2}\leq\infty$.

######  Theorem 1.4.

Let $(N,g)$ be a complete $(m,\rho)$-quasi Einstein manifold which is non-
compact and of finite volume. If $\rho>\frac{1}{n}$ and also $W$ admits finite
global norm, then the following holds:

  * (i)

If $R\geq\frac{\lambda n}{1-\rho n}$, then $N$ is trivial,

  * (ii)

If $(\lambda+\rho)R\geq 0$, then
$\frac{1}{m}\int_{N}|W|^{2}dV\leq\int_{N}RdV,$ and

  * (iii)

$\int_{N}RdV\leq\frac{\lambda n}{1-\rho n}Vol(N),$ where $Vol(N)$ represents
the volume of $N.$

By virtue of Theorem 1.4, we deduce the following corollary:

######  Corollary 1.4.1.

Let $(N,g)$ be a complete $(m,\rho)$-quasi Einstein manifold which is non-
compact and of finite volume. If $\rho<\frac{1}{n}$ and $W$ is of finite
global norm, then the following holds:

  * (i)

If $R\leq\frac{\lambda n}{1-\rho n}$, then $N$ is trivial,

  * (ii)

If $(\lambda+\rho)R\leq 0$, then
$\int_{N}RdV\leq\frac{1}{m}\int_{N}|X|^{2}dV,$ and

  * (iii)

$\int_{N}RdV\geq\frac{\lambda n}{1-\rho n}Vol(N).$

Next, we are interested in studying a complete Riemannian manifold $(N,g)$
with a vector field $W$ such that,

(7) $\lambda g+\rho Rg+\frac{1}{m}W^{\flat}\otimes
W^{\flat}\leq\frac{1}{2}\mathcal{L}_{W}g+Ric,$

In $(\ref{a7})$ the equality gives the fundamental equation of an
$(m,\rho)$-quasi Einstein manifold. Here the main aim is to prove the
following results:

######  Theorem 1.5.

If $(N,g)$ is a complete Riemannian manifold satisfying (7) along with
$\int_{0}^{r}\rho R=K_{3}$, for some constant $K_{3}$, then the fundamental
group of $N$ is finite.

######  Theorem 1.6.

Let $(N,g)$ be a complete Riemannian manifold satisfying $(\ref{a7})$ with
$\|W\|$ bounded and $\int_{0}^{r}\rho R=K_{3}$, for some constant $K_{3}$ with
$(\lambda+K_{3})>0$. Then $N$ becomes compact.

##  2\. Proof of the results

###### Proof of Theorem 1.1.

Let $p,q\in N$ and let $\Gamma$ be a minimizing unit speed geodesic segment
from $p$ to $q$ of length $d$. Considering a parallel orthonormal frame
$\\{\xi_{1}=\Gamma^{\prime},\xi_{2},\cdots,\xi_{n}\\}$ along $\Gamma$ and a
smooth function $h\in C^{\infty}([0,d])$ such that $h(0)=h(d)=0$, we get (see,
[11]),

(8) $\sum_{i=2}^{n}\eta(h\xi_{i},h\xi_{i})=\int_{0}^{d}\\{(n-1)h^{\prime
2}-h^{2}Ric(\Gamma^{\prime},\Gamma^{\prime})\\}dt,$

where $\eta$ denotes the index form of $\Gamma$.
With our assumption the relation $(\ref{d1})$, yields

(9) $\displaystyle\sum_{i=2}^{n}\eta(h\xi_{i},h\xi_{i})$ $\displaystyle\leq$
$\displaystyle\int_{0}^{d}\left[(n-1)h^{\prime
2}+h^{2}\\{\nabla^{2}f-\frac{1}{m}df\otimes df-\lambda g-\rho
Rg\\}(\Gamma^{\prime},\Gamma^{\prime})\right]dt$ $\displaystyle=$
$\displaystyle\int_{0}^{d}\left[(n-1)h^{\prime
2}+h^{2}\nabla^{2}f(\Gamma^{\prime},\Gamma^{\prime})-\frac{h^{2}}{m}\left(df(\Gamma^{\prime})\right)^{2}-h^{2}(\lambda+\rho
R)\right]dt.$

Again, since $|f|\leq K$, we obtain (see, [11]),

(10)
$\displaystyle\int_{0}^{d}h^{2}\nabla^{2}f(\Gamma^{\prime},\Gamma^{\prime})dt$
$\displaystyle=$
$\displaystyle\int_{0}^{d}h^{2}g(\nabla_{\Gamma^{\prime}}\nabla
f,\Gamma^{\prime})dt$ $\displaystyle=$
$\displaystyle\int_{0}^{d}h^{2}\Gamma^{\prime}(g(\nabla f,\Gamma^{\prime}))dt$
$\displaystyle\leq$ $\displaystyle
2K\sqrt{d}\left(\int_{0}^{d}\left(\frac{d}{dt}(hh^{\prime})\right)^{2}dt\right)^{\frac{1}{2}}.$

The equations $(\ref{d2})$ and $(\ref{d3})$ together implies

(11) $\displaystyle\sum_{i=2}^{n}\eta(h\xi_{i},h\xi_{i})\leq$
$\displaystyle\int_{0}^{d}(n-1)h^{\prime
2}dt+2K\sqrt{d}\left(\int_{0}^{d}\left(\frac{d}{dt}(hh^{\prime})\right)^{2}dt\right)^{\frac{1}{2}}$
$\displaystyle-\int_{0}^{d}\frac{h^{2}}{m}\left(df(\Gamma^{\prime})\right)^{2}dt-\int_{0}^{d}h^{2}(\lambda+\rho
R)dt.$

If we take $h(t)=sin(\frac{\pi}{d}t)$, then $(\ref{d4})$, yields

$\displaystyle\sum_{i=2}^{n}\eta(h\xi_{i},h\xi_{i})\leq$
$\displaystyle(n-1)\int_{0}^{d}\frac{\pi^{2}}{d^{2}}cos^{2}\left(\frac{\pi}{d}t\right)dt+2K\sqrt{d}\left(\int_{0}^{d}\left(\frac{\pi^{2}}{d^{2}}cos\left(\frac{2\pi}{d}t\right)\right)^{2}dt\right)^{\frac{1}{2}}$
$\displaystyle-\lambda\int_{0}^{d}sin^{2}\left(\frac{\pi}{d}t\right)dt-\int_{0}^{d}sin^{2}\left(\frac{\pi}{d}t\right)\rho
Rdt$ $\displaystyle\leq$
$\displaystyle(n-1)\frac{\pi^{2}}{2d}+\frac{K\pi^{2}\sqrt{2}}{d}-\frac{\lambda}{2}d-\frac{K_{1}}{2}d.$

Since $\Gamma$ is a minimizing geodesic, hence $\eta(h\xi_{i},h\xi_{i})\geq
0$. Consequently,

$0\leq(n-1)\frac{\pi^{2}}{2d}+\frac{K\pi^{2}\sqrt{2}}{d}-\frac{\lambda}{2}d-\frac{K_{1}}{2}d.$

The above relation entails

$d\leq\pi\sqrt{2}\sqrt{\frac{(n-1)+K\sqrt{2}}{\lambda+K_{1}}}.$

This completes the proof. ∎

###### Proof of Theorem 1.2.

Set $\omega=\log u$, then $F=|\nabla\omega|^{2}$ and $\omega$ satisfies the
equation:

$\Delta_{f}\omega+|\nabla\omega|^{2}=0.$

Thus,

$\Delta_{f}\omega=-F.$

Therefore,

(12)
$|\nabla^{2}\omega|^{2}\geq\frac{1}{n}|\Delta\omega|^{2}=\frac{1}{n}|F-\langle\nabla
f,\nabla\omega\rangle|^{2}\geq\frac{1}{n}F^{2}-\frac{2}{n}\langle\nabla
f,\nabla\omega\rangle F,$

and

(13)
$\langle\nabla\omega,\nabla\Delta_{f}\omega\rangle=-\langle\nabla\omega,\nabla
F\rangle.$

Now, the weighted Bochner formula:

$\frac{1}{2}\Delta_{f}|\nabla\omega|^{2}=|\nabla^{2}\omega|^{2}+\langle\nabla\omega,\nabla\Delta_{f}\omega\rangle+Ric_{f}(\nabla\omega,\nabla\omega),$

together with $(\ref{b1})$, $(\ref{b2})$ and our assumption, yields

$\frac{1}{2}\Delta_{f}F\geq\frac{1}{n}F^{2}-\frac{2}{n}\langle\nabla
f,\nabla\omega\rangle F-\langle\nabla\omega,\nabla F\rangle+(\lambda+\rho
R)|\nabla\omega|^{2}+\frac{1}{m}df\otimes df(\nabla\omega,\nabla\omega).$

This implies

(14) $\frac{1}{2}\Delta F-\langle\nabla f,\nabla
F\rangle\geq\frac{1}{n}F^{2}-\frac{2}{n}\langle\nabla f,\nabla\omega\rangle
F-\langle\nabla\omega,\nabla F\rangle+(\lambda+\rho
R)F+\frac{1}{m}\\{df(\nabla\omega)\\}^{2}.$

Applying the Cauchy-Schwarz inequality, we obtain

$\langle\nabla\omega,\nabla f\rangle\leq|\nabla\omega||\nabla f|=|\nabla
f|F^{\frac{1}{2}}.$

Using the last inequality in $(\ref{b3})$, we get the estimation. ∎

###### Proof of Theorem 1.3.

If $W\in\chi(N)$, then as a consequence of Hodge-de Rham decomposition
theorem, (see e.g. [21]), $W$ can be written as

(15) $W-\nabla\sigma=Y,$

where $\sigma\in\chi(N)$ is the Hodge-de Rham potential and $div\ Y=0$. By
considering an $(m,\rho)$-quasi Einstein manifold $(N,g)$ so that the equation
(2) holds, then we obtain

(16) $R+divW=\lambda n+\frac{1}{m}|W|^{2}+\rho Rn.$

Therefore (15) concludes that $divW=\Delta\sigma$ and hence $(\ref{qe1})$
entails

(17) $\lambda n+\frac{1}{m}|W|^{2}+\rho Rn=R+\Delta\sigma.$

Again $(\ref{e0})$ indicates that

(18) $\lambda n+\frac{1}{m}|\nabla f|^{2}+\rho Rn=R+\Delta f.$

From $(\ref{qe2})$ and $(\ref{qe3})$, we have

$\Delta(f-\sigma)=0,$

which follows the result. ∎

###### Proof of Theorem 1.4.

We have, for $l>0$

$\displaystyle\frac{1}{l}\int_{\mathscr{B}(p,2l)}|W|dV$ $\displaystyle\leq$
$\displaystyle\Big{(}\int_{\mathscr{B}(p,2l)}\Big{(}\frac{1}{l}\Big{)}^{2}dV\Big{)}^{1/2}\Big{(}\int_{\mathscr{B}(p,2l)}\langle
W,W\rangle dV\Big{)}^{1/2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{l}\Big{(}Vol(N)\Big{)}^{1/2}\|W\|_{\mathscr{B}(p,2l)},$

where $\mathscr{B}(p,l)$ is the open ball of radius $l$ and center at $p$.
Thus

$\liminf_{\ l\rightarrow\infty}\frac{1}{l}\int_{\mathscr{B}(p,2l)}|W|dV=0.$

Again, a Lipschitz continuous function $\omega_{l}$ exists, (see [23]) such
that for some real constant $K_{2}>0$,

$\displaystyle 0\leq\omega_{l}(x)\leq 1\quad\forall x\in N,$
$\displaystyle\text{ supp }\omega_{l}\subset\mathscr{B}(p,2l),$
$\displaystyle|d\omega_{l}|\leq\frac{K_{2}}{l}\qquad\text{ almost everywhere
on }N\text{ and}$ $\displaystyle\omega_{l}(x)=1\quad\forall
x\in\mathscr{B}(p,l).$

As $\lim\limits_{l\rightarrow\infty}\omega_{l}=1$, using the function
$\omega_{l}$, we get

$\frac{C}{l}\int_{\mathscr{B}(p,2l)}|W|dV\geq\Big{|}\int_{\mathscr{B}(p,2l)}\omega_{l}divWdV\Big{|}.$

From the defining relation of $(m,\rho)$-quasi Einstein manifold, we obtain

(19) $\int_{N}\\{\lambda n+\frac{1}{m}|W|^{2}+(\rho n-1)R\\}dV=0.$

By virtue of equation (19) and our assumption, we get the desired results. ∎

To prove Theorem 1.5, we need the following results:

######  Lemma 2.1 ([22]).

Let $(N,g)$ be a complete Riemannian manifold and $p,q\in N$ with
$r=d(p,q)>1$. If $\Gamma$ is a minimal geodesic joining from $p$ to $q$ and
parametrized by the arc length s, then

(20)
$2(n-1)+\mathscr{E}_{p}+\mathscr{E}_{q}\geq\int_{0}^{r}Ric(\Gamma^{\prime}(s),\Gamma^{\prime}(s))ds,$

where $\mathscr{E}_{q}=max\\{0,sup\\{Ric_{y}(v,v):y\in B(q,1),||v||=1\\}\\},$
for q $\in$ $N$.

######  Lemma 2.2.

If $(N,g)$ is a complete Riemannian manifold satisfying (7), then for any p,q
$\in N$,

(21)
$max\\{1,\frac{1}{K_{3}+\lambda}\big{(}2(n-1)+\mathscr{E}_{p}+\mathscr{E}_{q}+||W_{p}||+||W_{q}||\big{)}\\}\geq
d(p,q).$

###### Proof.

Suppose that $d(p,q)>1$ and $\Gamma$ is a minimal geodesic joining from p to q
and parametrized by arc length s. Then along $\Gamma$, we can easily get

$\displaystyle\mathcal{L}_{W}g(\Gamma^{\prime},\Gamma^{\prime})=2\frac{d}{ds}[g(W,\Gamma^{\prime})].$

Applying Lemma 2.1, we obtain

(22)
$2(n-1)+\mathscr{E}_{p}+\mathscr{E}_{q}\geq\int_{0}^{r}Ric(\Gamma^{\prime}(s),\Gamma^{\prime}(s))ds.$

Using the Cauchy-Schwarz inequality and (7), we have

$\displaystyle\int_{0}^{r}Ric(\Gamma^{\prime}(s),\Gamma^{\prime}(s))ds$
$\displaystyle\geq$ $\displaystyle
g_{p}(W,\Gamma^{\prime}(0))-g_{q}(W,\Gamma^{\prime}(r))+\int_{0}^{r}(W^{\flat}(\Gamma^{\prime}(s)))^{2}ds+\lambda
d(p,q)+\int_{0}^{r}\rho Rds$ $\displaystyle\geq$ $\displaystyle\lambda
d(p,q)-||W_{p}||-||W_{q}||+K_{3}d(p,q).$

In view of (22) and solving for $d(p,q)$, the last inequality entails (21). ∎

###### Proof of Theorem 1.5.

We omit the proof of Theorem 1.5, as it is straightforward from the Lemma 2.2
and following the technique of [22, Theorem 1.1]. ∎

###### Proof of Theorem 1.6.

Let $p\in N$ and $\Gamma:[0,\infty)\rightarrow N$ be any geodesic starting
from $p$ and parametrized by the arc length $s$. Hence (7) and the Cauchy-
Schwarz inequality together imply that

$\displaystyle\int_{0}^{r}Ric(\Gamma^{\prime}(s),\Gamma^{\prime}(s))ds$
$\displaystyle\geq$
$\displaystyle\int_{0}^{r}(W^{\flat}(\Gamma^{\prime}(s)))^{2}+\lambda
r+\int_{0}^{r}\rho
Rg+2g(W_{p},\Gamma^{\prime}(0))-2g(W_{\Gamma(r)},\Gamma^{\prime}(r))ds$
$\displaystyle\geq$ $\displaystyle
2g(W_{p},\Gamma^{\prime}(0))-2\|W_{\Gamma(r)}\|+\lambda r+K_{3}r.$

The boundedness of $\|W\|$ provides

$\int_{0}^{+\infty}Ric(\Gamma^{\prime}(s),\Gamma^{\prime}(s))=+\infty,$

and hence Ambrose’s compactness criteria [1] concludes that $N$ is compact. ∎

## Acknowledgment

The second author gratefully acknowledges to the CSIR(File
No.:09/025(0282)/2019-EMR-I), Govt. of India for the award of JRF. Also the
third author conveys sincere thanks to the Netaji Subhas Open University for
partial financial assistance (Project No.: AC/140/2021-22).

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