Representation of Locally Convex Partial *-algebraic Modules

TL;DR

This paper introduces a new representation concept for locally convex partial *-algebraic modules, extending the framework of inner product modules over C*-algebras to a more general setting.

Contribution

It proposes a novel notion of representation for these modules as concrete spaces of maps, advancing the theoretical understanding of their structure.

Findings

Defined a new representation framework for locally convex partial *-algebraic modules

Extended the theory of inner product modules over C*-algebras to a broader class

Laid groundwork for future applications in operator algebra theory

Abstract

In this paper, we introduce a new notion of representation for a locally convex partial *-algebraic module as a concrete space of maps. This is a continuation of our systematic study of locally convex partial *-algebraic modules, which are generalizations of inner product modules over C*-algebras.

Equations

$$\Vert x{\Vert }_{\alpha }=|⟨x,x{⟩}_{\mathcal{B}}{|}_{\alpha }^{1/2},x\in \mathcal{D},\alpha \in \Delta .$$

$$\frac{1}{2}(|⟨x,y{⟩}_{\mathcal{B}}{|}_{\alpha }+|⟨y,x{⟩}_{\mathcal{B}}{|}_{\alpha })\le \Vert x{\Vert }_{\alpha }\Vert y{\Vert }_{\alpha },\forall x,y\in \mathcal{D},\alpha \in \Delta .$$

$$|⟨x,y{⟩}_{\mathcal{B}}{|}_{\alpha }\le \Vert x{\Vert }_{\alpha }\Vert y{\Vert }_{\alpha },\forall x,y\in \mathcal{D},\alpha \in \Delta .$$

$${\mathcal{M}}_{\mathcal{D}}=\mathrm{span}\{⟨x,y{⟩}_{\mathcal{B}}:x,y\in \mathcal{D}\}\bigcap R(\mathcal{B}).$$

$$⟨tx,y{⟩}_{\mathcal{Y},\mathcal{B}}=⟨x,t^{\ast }y{⟩}_{\mathcal{X},\mathcal{B}},\forall x\in \mathcal{D},y\in \mathcal{Y}$$

$$\Gamma =$$

$$t\circ s=$$

$${\pi }_{x,y}^{\mathcal{B}}(z)=x.⟨y,z{⟩}_{\mathcal{B}}.$$

$${\left({\pi }_{x,y}^{\mathcal{B}}\right)}^{+}:={\left({\pi }_{x,y}^{\mathcal{B}}\right)}^{\ast }\upharpoonright \mathcal{D}={\pi }_{y,x}^{\mathcal{B}}$$

$$⟨\vartheta (x),\vartheta (y){⟩}_{{\mathcal{L}}^{+}}=\pi (⟨x,y{⟩}_{\mathcal{B}})$$

$$\sum _{j,k=1}^n{⟨{\xi }_j,\sigma (x_j,x_k){\xi }_k⟩}_{\mathcal{H}}$$

$$=\sum _{j,k=1}^n{⟨[\vartheta (x_k{)}^{+}\circ \vartheta (x_j)]{\xi }_j,{\xi }_k⟩}_{\mathcal{H}}=\sum _{j,k=1}^n{⟨\vartheta (x_k{)}^{\ast }\vartheta (x_j){\xi }_j,{\xi }_k⟩}_{\mathcal{H}}$$

$$\displaystyle=\sum_{j,k=1}^{n}\left\langle\vartheta(x_{j})\xi_{j},\vartheta(x_{k})\xi_{k}\right\rangle_{{\mathcal{H}}}=\Big{\langle}\sum_{j=1}^{n}\vartheta(x_{j})\xi_{j},\sum_{k=1}^{n}\vartheta(x_{k})\xi_{k}\Big{\rangle}_{{\mathcal{H}}}$$

$$\displaystyle=\bigg{\|}\sum_{j=1}^{n}\vartheta(x_{j})\xi_{j}\bigg{\|}^{2}\geq 0$$

$$⟨\vartheta (x.b),\vartheta (y){⟩}_{{\mathcal{L}}^{+}}$$

$$=\pi (⟨y,x{⟩}_{\mathcal{B}}b{)}^{+}=(\pi (⟨y,x{⟩}_{\mathcal{B}})\circ \pi (b){)}^{+}$$

$$=\pi (b{)}^{+}\circ \pi (⟨y,x{⟩}_{\mathcal{B}}{)}^{+}=\pi (b{)}^{+}\circ ⟨\vartheta (y),\vartheta (x){⟩}_{{\mathcal{L}}^{+}}^{+}$$

$$=\pi (b{)}^{+}\circ ⟨\vartheta (x),\vartheta (y){⟩}_{{\mathcal{L}}^{+}}=⟨\vartheta (x)\circ \pi (b),\vartheta (y){⟩}_{{\mathcal{L}}^{+}}$$

$$\vartheta (x.b)=\vartheta (x)\circ \pi (b),\forall x\in \mathcal{X}\text{mbox}andb\in {\mathcal{B}}_0.$$

$$⟨\vartheta (x.b),\vartheta (y){⟩}_{{\mathcal{L}}^{+}}$$

$$=\pi (b^{\ast })\circ \pi (⟨x,y{⟩}_{\mathcal{B}})=\pi (b^{\ast })\circ ⟨\vartheta (x),\vartheta (y){⟩}_{{\mathcal{L}}^{+}}$$

$$=⟨\vartheta (x)\circ \pi (b^{\ast }{)}^{+},\vartheta (y){⟩}_{{\mathcal{L}}^{+}}$$

$$\vartheta (x.b)=\vartheta (x)\circ \pi (b^{\ast }{)}^{+},\forall x\in \mathcal{X}\text{mbox}andb\in {\mathcal{B}}_0$$

$$⟨\vartheta (x+y),\vartheta (z){⟩}_{{\mathcal{L}}^{+}}$$

$$=\pi (⟨x,z{⟩}_{\mathcal{B}})+\pi (⟨y,z{⟩}_{\mathcal{B}})=⟨\vartheta (x),\vartheta (z){⟩}_{{\mathcal{L}}^{+}}+⟨\vartheta (y),\vartheta (z){⟩}_{{\mathcal{L}}^{+}}$$

$$=⟨\vartheta (x)+\vartheta (y),\vartheta (z){⟩}_{{\mathcal{L}}^{+}}$$

Full text

Representation of Locally Convex Partial ∗-algebraic Modules

F. A. Tsav

Department of Mathematics and Computer Science, Benue State University, Makurdi, Nigeria

Abstract: In this paper, we introduce a new notion of representation for a locally convex partial ∗-algebraic module as a concrete space of maps. This is a continuation of our systematic study of locally convex partial ∗-algebraic modules, which are generalizations of inner product modules over C∗-algebras.

Keywords: Partial ∗-algebra, locally convex $({\mathcal{B}},\tau_{{\mathcal{B}}})$-module, adjointable map, partial-inner product.

AMS 2010 Subject Classification: 46A03, 46C99, 47C99

1 Introduction

The notion of a locally convex partial ∗-algebraic module was introduced by Ekhaguere [2] in his study of the representation of completely positive maps between partial ∗-algebras. Locally convex partial ∗-algebraic modules are generalizations of inner product modules over B∗-algebras [8]. These inner product modules [9], now generally known as pre-Hilbert C∗-modules, provide a natural generalization of the Hilbert space in which the complex field of scalars is replaced by a C∗-algebra. Although the theory of Hilbert C∗-modules, in the case of commutative unital C∗-algebras, can be traced back to the work of Kaplansky [4], where he proved that derivations of type I AW∗-algebras are inner, it was Paschke [8] who gave the general framework. Apart from being interesting on its own, the theory of Hilbert C∗-modules has had several areas of applications. For example, the work of Kasparov on KK-theory [5, 6], the work of Rieffel on induced representations and Morita equivalence [9, 10], and the work of Woronowicz on C∗-algebraic quantum group theory [13], etc. For a more detailed bibliography of the theory of Hilbert C∗-modules, see [3]. In this paper, we continue the systematic development of some of the properties of locally convex partial ∗-algebraic modules, which was begun in [12]. At this stage, we introduce a new notion of representation for a locally convex partial ∗-algebraic module as a concrete space of maps. This result, which is interesting in its own, generalizes the notion of a representation of a Hilbert C∗-module as a concrete space of operators[7] and would be useful elsewhere to develop an analogue of Stinespring theorem for locally convex partial ∗-algebraic modules, by extending a number of results from the theory of Hilbert C∗-modules.

The paper is organized as follows. In section 2, we outline some of the fundamental notions used in the sequel. We establish our notations in four subsections under this section: The basic notion of a partial ∗-algebra is given in subsection 2.1. Subsection 2.2 outlines the notions of a locally convex partial ∗-algebraic modules, while subsections 2.3 and 2.4 outline some properties of these objects and their adjointable maps, respectively. See [1, 2, 11, 12], for more details of these notions. Finally, in section 3 we introduce a new notion of a ${\mathcal{B}}$-valued inner product, which extends the notion of locally convex partial ∗-algebraic modules to include partial ∗-algebras themselves. We prove our main result in this new setting.

2 Fundamental Notions

For more details of these notions, see [1, 2, 11, 12].

2.1 Partial ∗-algebra

A partial ∗-algebra is simply a complex involutive linear space ${\mathcal{A}}$ with a multiplication that is defined only for certain pairs of compatible elements determined by a relation on ${\mathcal{A}}$. More precisely, there is the following definition.

Definition 2.1

A partial ∗-algebra is a quadruple $({\mathcal{A}},\Gamma,\diamond,*)$ comprising:

1. (a)

a linear space ${\mathcal{A}}$ over $I\!\!\!\!C$; 2. (b)

a relation $\Gamma\subseteq{\mathcal{A}}\times{\mathcal{A}}$; 3. (c)

a partial multiplication, $\diamond$, such that

1. (c$1$)

$(x,y)\in\Gamma$* if and only if $x\diamond y\in{\mathcal{A}}$;* 2. (c$2$)

$(x,y),(x,z)\in\Gamma$* implies $(x,\lambda y+\mu z)\in\Gamma$ and then*

$x\diamond(\lambda y+\mu z)=\lambda(x\diamond y)+\mu(x\diamond z),\forall\,\lambda,\mu\in I\!\!\!\!C$; and 4. (d)

an involution ($x\mapsto x^{*}$) such that

1. (d$1$)

$(x+\lambda y)^{*}=x^{*}+\overline{\lambda}y^{*},\forall x,y\in{\mathcal{A}},\,\lambda\in I\!\!\!\!C$* and $x^{**}=x,\forall\,x\in{\mathcal{A}}$;* 2. (d$2$)

$(x,y)\in\Gamma$* if and only if $(y^{*},x^{*})\in\Gamma$ and then $(x\diamond y)^{*}=y^{*}\diamond x^{*}$.*

Definition 2.2

*An element $e$ of a partial ∗-algebra ${\mathcal{B}}$ is called a unit, and ${\mathcal{B}}$ is said to be unital, if $(e,x),(x,e)\in\Gamma$, and then $e^{*}=e$, and $e\diamond x=x\diamond e=x$, for every $x\in{\mathcal{B}}$.

${\mathcal{B}}$ is said to be abelian if, for all $x,y\in{\mathcal{B}}$, $(x,y),(y,x)\in\Gamma$, and then $x\diamond y=y\diamond x$.*

Remark 2.1

Partial ∗-algebras are studied by means of their spaces of multipliers.

Definition 2.3

*Let $({\mathcal{A}},\Gamma,\diamond,*)$ be a partial ∗-algebra, ${\mathcal{M}}\subset{\mathcal{A}}$ and $x\in{\mathcal{A}}$. Put $L(x)=\{y\in{\mathcal{A}}:(y,x)\in\Gamma\}$(resp., $R(x)=\{y\in{\mathcal{A}}:(x,y)\in\Gamma\}$,

$L({\mathcal{M}})=\bigcap_{x\in{\mathcal{M}}}L(x)\equiv\{y\in{\mathcal{A}}:y\in L(x),\forall x\in{\mathcal{M}}\}$,

$R({\mathcal{M}})=\bigcap_{x\in{\mathcal{M}}}R(x)\equiv\{y\in{\mathcal{A}}:y\in R(x),\forall x\in{\mathcal{M}}\})$.

Then $L(x)$ (resp., $R(x)$, $L({\mathcal{M}})$, $R({\mathcal{M}})$) is called the space of left multipliers of $x$ (resp., right multipliers of $x$, left multipliers of ${\mathcal{M}}$, right multipliers of ${\mathcal{M}}$). In particular, elements of $L({\mathcal{A}})$ (resp., $R({\mathcal{A}})$) are called universal left (resp., universal right) multipliers.

$M({\mathcal{A}})\equiv L({\mathcal{A}})\cap R({\mathcal{A}})$ is the so-called universal multipliers of ${\mathcal{A}}$.*

Definition 2.4

A partial ∗-algebra ${\mathcal{B}}$ is said to be semi-associative if $y\in R(x)$ implies $y\diamond z\in R(x)$ for every $z\in R({\mathcal{B}})$ and $(x\diamond y)\diamond z=x\diamond(y\diamond z)$.

Remark 2.2

If a partial ∗-algebra ${\mathcal{B}}$ is semi-associative, then $L({\mathcal{B}})$ and $R({\mathcal{B}})$ are algebras, while $M({\mathcal{B}})$ is a ∗-algebra.

Definition 2.5

The positive cone of a partial ∗-algebra ${\mathcal{A}}$ is the set ${\mathcal{A}}_{+}$ given by ${\mathcal{A}}_{+}:=\{\sum_{j=1}^{n}x_{j}^{*}\diamond x_{j}:x_{j}\in R({\mathcal{A}}),n\in{\mathbb{N}}\}$. We say that $x\in{\mathcal{A}}$ is positive if $x\in{\mathcal{A}}_{+}$ and write $x\geq 0$.

Definition 2.6

Given a Hausdorff locally convex topology $\tau$ on ${\mathcal{A}}$, we call the pair $({\mathcal{A}},\tau)$ a locally convex partial ∗-algebra if and only if:

1. (i)

$({\mathcal{A}}_{0},\tau)$* is a Hausdorff locally convex space, where ${\mathcal{A}}_{0}$ is the underlying linear space of ${\mathcal{A}}$,* 2. (ii)

the map $x\in{\mathcal{A}}\mapsto x^{*}\in{\mathcal{A}}$ is $\tau$-continuous, 3. (iii)

the map $x\in{\mathcal{A}}\mapsto a\diamond x\in{\mathcal{A}}$ is $\tau$-continuous, for all $a\in L({\mathcal{A}})$ and 4. (iv)

the map $x\in{\mathcal{A}}\mapsto x\diamond b\in{\mathcal{A}}$ is $\tau$-continuous, for all $b\in R({\mathcal{A}})$.

Definition 2.7

Let ${\mathcal{B}}$ be a complex linear space and ${\mathcal{B}}_{0}$ a ∗-algebra contained in ${\mathcal{B}}$. ${\mathcal{B}}$ is said to be a quasi ∗-algebra with distinguised ∗-algebra ${\mathcal{B}}_{0}$ if

1. (i)

${\mathcal{B}}$* is a bimodule over ${\mathcal{B}}_{0}$ for which the module action extends the multiplication of ${\mathcal{B}}_{0}$ such that $x.(y.b)=(x.y).b$ and $x.(b.y)=(x.b).y$, for all $b\in{\mathcal{B}}$ and $x,y\in{\mathcal{B}}_{0}$;* 2. (ii)

the involution ∗ on ${\mathcal{B}}$ extends the involution of ${\mathcal{B}}_{0}$ such that $(x.b)^{*}=b^{*}.x^{*}$ and $(b.x)^{*}=x^{*}.b^{*}$, for all $b\in{\mathcal{B}}$ and $x\in{\mathcal{B}}_{0}$.

If ${\mathcal{B}}$ is a locally convex space with a locally convex topology $\tau$ such that

1. (i)

${\mathcal{B}}_{0}$* is $\tau$-dense in ${\mathcal{B}}$;* 2. (ii)

the involution ∗ is $\tau$-continuous; 3. (iii)

the left and right module actions are separately $\tau$-continuous,

then $({\mathcal{B}},{\mathcal{B}}_{0})$ is said to be a locally convex quasi ∗-algebra.

Remark 2.3

Every quasi ∗-algebra is a semi-associative partial ∗-algbera

2.2 Locally Convex Partial ∗-Algebraic Modules

As in [2], let $({\mathcal{B}},\tau_{{\mathcal{B}}})$ be a locally convex partial ∗-algebra, with involution ∗ and partial multiplication written as juxtaposition. Let $\tau_{{\mathcal{B}}}$ be generated by a family $\{|\cdot|_{\alpha}:\alpha\in\Delta\}$ of seminorms. In what follows, we assume, without loss of generality, that the family $\{|\cdot|_{\alpha}:\alpha\in\Delta\}$ of seminorms is directed. Let ${\mathcal{D}}$ be a linear space which is also a right $R({\mathcal{B}})$-module in the sense that $x.a+y.b\in{\mathcal{D}}$, whenever $x,y\in{\mathcal{D}}$ and $a,b\in R({\mathcal{B}})$, where the action of $R({\mathcal{B}})$ on ${\mathcal{D}}$ is written as $z.c$ for $z\in{\mathcal{D}}$, $c\in R({\mathcal{B}})$. Locally convex partial ∗-algebraic modules were introduced in [2] as follows.

Definition 2.8

*A ${\mathcal{B}}$-valued inner product on ${\mathcal{D}}$ is a conjugate-bilinear map

$\langle\cdot,\cdot\rangle_{{\mathcal{B}}}:{\mathcal{D}}\times{\mathcal{D}}\longrightarrow{\mathcal{B}}$ satisfying the following:*

1. ($i$)

$\langle x,x\rangle_{{\mathcal{B}}}\in{\mathcal{B}}_{+},\forall x\in{\mathcal{D}}$* and $\langle x,x\rangle_{{\mathcal{B}}}=0$ only if $x=0$,* 2. ($ii$)

$\langle x,y\rangle_{{\mathcal{B}}}=\langle y,x\rangle_{{\mathcal{B}}}^{*},\forall x,y\in{\mathcal{D}}$, 3. ($iii$)

$\langle x,y.b\rangle_{{\mathcal{B}}}=\langle x,y\rangle_{{\mathcal{B}}}b,\forall x,y\in{\mathcal{D}},b\in R({\mathcal{B}})$**

Lemma 2.1

Let $\langle\cdot,\cdot\rangle_{{\mathcal{B}}}$ be a ${\mathcal{B}}$-valued inner product on ${\mathcal{D}}$. Define $\|\cdot\|_{\alpha}:{\mathcal{D}}\longrightarrow[0,\infty)$ by

[TABLE]

Then, the following inequality holds:

[TABLE]

Moreover, if $|\cdot|_{\alpha}$ is ∗-ivariant, i.e., if $|a^{*}|_{\alpha}=|a|_{\alpha},\forall a\in{\mathcal{B}},\alpha\in\Delta$, then the inequality (2) reduces to

[TABLE]

Corollary 2.2

If $\|\cdot\|_{\alpha}:{\mathcal{D}}\longrightarrow[0,\infty)$ is defined as in Equation (1), then $\|\cdot\|_{\alpha}$ is a seminorm on ${\mathcal{D}}$ for each $\alpha\in\Delta$.

Remark 2.4

We observe that the family $\{\|\cdot\|_{\alpha}:\alpha\in\Delta\}$ of seminorms is directed.

Definition 2.9

A locally convex $({\mathcal{B}},\tau_{{\mathcal{B}}})$-module is a triple $({\mathcal{D}},\langle\cdot,\cdot\rangle_{{\mathcal{B}}},\tau_{{\mathcal{D}},{\mathcal{B}}})$ comprising:

1. (a)

a linear space ${\mathcal{D}}$ which is also a right $R({\mathcal{B}})$-module; 2. (b)

a ${\mathcal{B}}$-valued inner product $\langle\cdot,\cdot\rangle_{{\mathcal{B}}}:{\mathcal{D}}\times{\mathcal{D}}\longrightarrow{\mathcal{B}}$; and 3. (c)

a locally convex topology $\tau_{{\mathcal{D}},{\mathcal{B}}}$ on ${\mathcal{D}}$ generated by the family $\{\|\cdot\|_{\alpha}:\alpha\in\Delta\}$ of seminorms given by (1) and, with respect to this topology, the map $l_{R}(b):{\mathcal{D}}\longrightarrow{\mathcal{D}}$ given by $l_{R}(b)x=x.b,\forall x\in{\mathcal{D}}$, is continuous for each $b\in R({\mathcal{B}})$; i.e., for each $\alpha\in\Delta,\exists$ a $\beta(\alpha)\in\Delta$ and $K_{\alpha,b}>0$ such that $\|l_{R}(b)x\|_{\alpha}\leq K_{\alpha,b}\|x\|_{\beta(\alpha)}$

2.3 Some Basic Properties of Locally Convex $({\mathcal{B}},\tau_{{\mathcal{B}}})$-modules

Definition 2.10

Let ${\mathcal{A}}$ be a partial ∗-algebra and ${\mathcal{B}}$ a linear subspace of ${\mathcal{A}}$. Then ${\mathcal{B}}$ is said to be a left (resp., right) ideal in ${\mathcal{A}}$, if $a\in L({\mathcal{A}})$ and $b\in{\mathcal{B}}$ (resp., $a\in R({\mathcal{A}})$ and $b\in{\mathcal{B}}$) implies $ab\in{\mathcal{B}}$ (resp., $ba\in{\mathcal{B}}$). If ${\mathcal{B}}$ is both a left and a right ideal in ${\mathcal{A}}$, then ${\mathcal{B}}$ is called a two-sided ideal, or simply, an ideal in ${\mathcal{A}}$.

Remark 2.5

From the definition above, if ${\mathcal{A}}$ is a semi-associative partial ∗-algebra, then:

1. (i)

$L({\mathcal{A}})$* is a left ideal in ${\mathcal{A}}$* 2. (ii)

$R({\mathcal{A}})$* is a right ideal in ${\mathcal{A}}$* 3. (iii)

$M({\mathcal{A}})$* is an ideal in ${\mathcal{A}}$*

Proposition 2.1

Let $({\mathcal{D}},\langle\cdot,\cdot\rangle_{{\mathcal{B}}},\tau_{{\mathcal{D}},{\mathcal{B}}})$ be a locally convex $({\mathcal{B}},\tau_{{\mathcal{B}}})$-module. Define the linear subspace, ${\mathcal{M}}_{{\mathcal{D}}}$ of ${\mathcal{B}}$ by

[TABLE]

Then ${\mathcal{M}}_{{\mathcal{D}}}$ is an ideal in ${\mathcal{B}}$.

2.4 Adjointable Maps on Locally Convex $({\mathcal{B}},\tau_{{\mathcal{B}}})$-modules

Definition 2.11

*A map $t:{\mathcal{X}}\to{\mathcal{Y}}$ is called a $({\mathcal{B}},\tau_{{\mathcal{B}}})$-module map (or simply, a module map) if and only if $t(x.b)=(tx).b,\,\forall x\in{\mathcal{D}},b\in R({\mathcal{B}})$.

One also says that $t$ is a $({\mathcal{B}},\tau_{{\mathcal{B}}})$-linear map. We denote by ${\mathcal{L}}_{{\mathcal{B}}}({\mathcal{X}},{\mathcal{Y}})$, the set of all linear $({\mathcal{B}},\tau_{{\mathcal{B}}})$-module maps from ${\mathcal{X}}$ to ${\mathcal{Y}}$.*

Definition 2.12

*We call a map $t:{\mathcal{X}}\to{\mathcal{Y}}$ adjointable if there exists a map

$t^{*}:{\mathcal{Y}}\to{\mathcal{X}}$ such that*

[TABLE]

The map $t^{*}$ will be called the adjoint of $t$.

Proposition 2.2

If the map $t:{\mathcal{X}}\to{\mathcal{Y}}$ is adjointable, then $t\in{\mathcal{L}}_{{\mathcal{B}}}({\mathcal{X}},{\mathcal{Y}})$.

Notation 2.1

*Let $({\mathcal{D}},\langle\cdot,\cdot\rangle_{{\mathcal{B}}},\tau_{{\mathcal{D}},{\mathcal{B}}})$ be a dense locally convex $({\mathcal{B}},\tau_{{\mathcal{B}}})$-submodule of $({\mathcal{X}},\langle\cdot,\cdot\rangle_{{\mathcal{B}}},\tau_{{\mathcal{X}},{\mathcal{B}}})$. ${\mathcal{L}}_{{\mathcal{B}}}({\mathcal{D}},{\mathcal{X}})$ becomes a linear space when furnished with the usual (pointwise) operations of vector addition, $t+s$ and scalar multiplication, $\lambda t$, $t,s\in{\mathcal{L}}_{{\mathcal{B}}}({\mathcal{D}},{\mathcal{X}}),\,\lambda\in I\!\!\!\!C$. Now set ${\mathcal{L}}_{{\mathcal{B}}}^{*}({\mathcal{D}},{\mathcal{X}}):=\{t\in{\mathcal{L}}_{{\mathcal{B}}}({\mathcal{D}},{\mathcal{X}}):t\,\text{is continuous and adjointable}\}$. Since ${\mathcal{D}}$ is dense in ${\mathcal{X}}$, $t^{*}$ is uniquely determined, and hence, well-defined. It follows that ${\mathcal{L}}_{{\mathcal{B}}}^{*}({\mathcal{D}},{\mathcal{X}})$ is a ∗-invariant linear subspace of ${\mathcal{L}}_{{\mathcal{B}}}({\mathcal{D}},{\mathcal{X}})$. It is not a ∗-algebra, except

${\mathcal{L}}_{{\mathcal{B}}}^{*}({\mathcal{D}},{\mathcal{X}})\equiv{\mathcal{L}}_{{\mathcal{B}}}^{*}({\mathcal{D}},{\mathcal{D}})={\mathcal{L}}_{{\mathcal{B}}}^{*}({\mathcal{D}}):=\{t\in{\mathcal{L}}_{{\mathcal{B}}}^{*}({\mathcal{D}},{\mathcal{X}}):t{\mathcal{D}}\subseteq{\mathcal{D}}\,\,\mbox{and}\,\,t^{*}{\mathcal{D}}\subseteq{\mathcal{D}}\}$. However, if one sets ${\mathcal{L}}_{{\mathcal{B}}}^{+}({\mathcal{D}},{\mathcal{X}}):=\{t\in{\mathcal{L}}_{{\mathcal{B}}}^{*}({\mathcal{D}},{\mathcal{X}}):\mbox{dom}(t^{*})\supseteq{\mathcal{D}}\}$, then:*

Proposition 2.3

The linear space ${\mathcal{L}}_{{\mathcal{B}}}^{+}({\mathcal{D}},{\mathcal{X}})$ is a partial ∗-algebra with:

1. (i)

involution: $t\mapsto t^{+}:=t^{*}\upharpoonright{\mathcal{D}}$, for all $t\in{\mathcal{L}}_{{\mathcal{B}}}^{+}({\mathcal{D}},{\mathcal{X}})$ and 2. (ii)

partial multiplication, specified by

[TABLE]

Definition 2.13

A +-invariant linear subspace ${\mathcal{M}}$ of ${\mathcal{L}}_{{\mathcal{B}}}^{+}({\mathcal{D}},{\mathcal{X}})$ is called a partial ∗-subalgebra of ${\mathcal{L}}_{{\mathcal{B}}}^{+}({\mathcal{D}},{\mathcal{X}})$ if $t,s\in{\mathcal{M}}$, with $t\in L(s)$ implies $t\circ s\in{\mathcal{M}}$.

Remark 2.6

Let ${\mathcal{X}}$ be a complete locally convex $({\mathcal{B}},\tau_{{\mathcal{B}}})$-module. Set ${\mathcal{D}}=\{z\in{\mathcal{X}}:\langle x,z\rangle_{{\mathcal{B}}}\in R({\mathcal{B}}),\forall x,\in{\mathcal{X}}\}$. In what follows, we shall assume that ${\mathcal{D}}$ is dense in ${\mathcal{X}}$.

Proposition 2.4

For $x,y\in{\mathcal{X}}$, define the map $\pi_{x,y}^{{\mathcal{B}}}:{\mathcal{D}}\to{\mathcal{X}}$ as

[TABLE]

Then the map $\pi_{x,y}^{{\mathcal{B}}}$ is continuous and adjointable with adjoint

[TABLE]

Remark 2.7

*From the preceding, we note that, since ${\mathcal{D}}\subseteq\mbox{dom}((\pi_{x,y}^{{\mathcal{B}}})^{*})$ we have, for $n\in{\mathbb{N}}$, dom$((\sum_{j=1}^{n}\pi_{x_{j},y_{j}}^{{\mathcal{B}}})^{*})\supseteq\mbox{dom}((\pi_{x_{1},y_{1}}^{{\mathcal{B}}})^{*})\bigcap\mbox{dom}((\pi_{x_{2},y_{2}}^{{\mathcal{B}}})^{*})\bigcap\cdots$

$\bigcap\mbox{dom}((\pi_{x_{n},y_{n}}^{{\mathcal{B}}})^{*})\supseteq{\mathcal{D}}$. It follows that $\sum_{j=1}^{n}\pi_{x_{j},y_{j}}^{{\mathcal{B}}}\in{\mathcal{L}}_{{\mathcal{B}}}^{+}({\mathcal{D}},{\mathcal{X}})$. Also, for $\alpha\in I\!\!\!\!C$ and $\pi_{x,y}^{{\mathcal{B}}}\in{\mathcal{L}}_{{\mathcal{B}}}^{+}({\mathcal{D}},{\mathcal{X}})$, it is clear that $\alpha\pi_{x,y}^{{\mathcal{B}}}\in{\mathcal{L}}_{{\mathcal{B}}}^{+}({\mathcal{D}},{\mathcal{X}})$. So we introduce the linear subspace

${\mathcal{K}}_{{\mathcal{B}}}^{+}({\mathcal{D}},{\mathcal{X}})={\rm span}\{\pi_{x,y}^{{\mathcal{B}}}\in{\mathcal{L}}_{{\mathcal{B}}}^{+}({\mathcal{D}},{\mathcal{X}}):x,y\in{\mathcal{X}}\}$ of ${\mathcal{L}}_{{\mathcal{B}}}^{+}({\mathcal{D}},{\mathcal{X}})$.*

Proposition 2.5

${\mathcal{K}}_{{\mathcal{B}}}^{+}({\mathcal{D}},{\mathcal{X}})$* is a partial ∗-subalgebra and an ideal of ${\mathcal{L}}_{{\mathcal{B}}}^{+}({\mathcal{D}},{\mathcal{X}})$.*

3 Main Result

We introduce another ${\mathcal{B}}$-valued inner product as follows. Let $({\mathcal{B}},\tau_{{\mathcal{B}}})$ be a locally convex partial ∗-algebra, with involution ∗ and partial multiplication written as juxtaposition. Let $\tau_{{\mathcal{B}}}$ be generated by a family $\{|\cdot|_{\alpha}:\alpha\in\Delta\}$ of seminorms. Let ${\mathcal{B}}_{0}\subseteq R({\mathcal{B}})$ be a linear subspace of ${\mathcal{B}}$ and $({\mathcal{X}},{\mathcal{D}})$ a pair comprising:

1. (i)

a linear space ${\mathcal{X}}$ which is also a right ${\mathcal{B}}_{0}$-module; i.e., $x.a+y.b\in{\mathcal{X}}$, whenever $x,y\in{\mathcal{X}}$ and $a,b\in{\mathcal{B}}_{0}$, where the action of ${\mathcal{B}}_{0}$ on ${\mathcal{X}}$ is written as $z.c$ for $z\in{\mathcal{X}}$, $c\in{\mathcal{B}}_{0}$ and 2. (ii)

a right ${\mathcal{B}}_{0}$-submodule ${\mathcal{D}}$ of ${\mathcal{X}}$; i.e., $x.b\in{\mathcal{D}}$, for $x\in{\mathcal{D}}$, $b\in{\mathcal{B}}_{0}$.

We shall call the pair $({\mathcal{X}},{\mathcal{D}})$ a right partial ${\mathcal{B}}_{0}$-module. A left partial ${\mathcal{B}}_{0}$-module can be defined in a similar way; in this case, ${\mathcal{B}}_{0}$ would be a subset of $L({\mathcal{B}})$. If ${\mathcal{B}}_{0}=R({\mathcal{B}})$, then we shall call the pair $({\mathcal{X}},{\mathcal{D}})$ a right partial $R({\mathcal{B}})$-module.

Definition 3.1

A ${\mathcal{B}}$-valued partial-inner product on a right partial ${\mathcal{B}}_{0}$-module $({\mathcal{X}},{\mathcal{D}})$ is a conjugate-bilinear map $\langle\cdot,\cdot\rangle_{{\mathcal{B}}}:{\mathcal{X}}\times{\mathcal{X}}\to{\mathcal{B}}$ such that $\langle x,y\rangle_{{\mathcal{B}}}\in{\mathcal{B}}$ if and only if $x\in{\mathcal{D}}$ or $y\in{\mathcal{D}}$ and satisfying:

1. (i)

$\langle x,x\rangle_{{\mathcal{B}}}\in{\mathcal{B}}_{+},\quad\forall x\in{\mathcal{D}}$* and $\langle x,x\rangle_{{\mathcal{B}}}=0$ only if $x=0$;* 2. (ii)

$\langle x,y\rangle_{{\mathcal{B}}}^{*}=\langle y,x\rangle_{{\mathcal{B}}},\quad\forall x\in{\mathcal{D}}$* or $\forall y\in{\mathcal{D}}$;* 3. (iii)

$\langle x,y.b\rangle_{{\mathcal{B}}}=\langle x,y\rangle_{{\mathcal{B}}}b,\quad\forall x\in{\mathcal{D}}$* or $\forall y\in{\mathcal{D}}$, $b\in{\mathcal{B}}_{0}$.*

Example 3.1

Let $({\mathcal{B}},\tau_{{\mathcal{B}}})$ be a locally convex, semi-associative partial ∗-algebra. Then ${\mathcal{B}}$ is a right $R({\mathcal{B}})$-module (resp., a right $M({\mathcal{B}})$-module). By semi-associativity of ${\mathcal{B}}$, $R({\mathcal{B}})$ is an algebra (resp., $M({\mathcal{B}})$ is a ∗-algebra). It follows that $R({\mathcal{B}})$ is itself a right $R({\mathcal{B}})$-submodule of ${\mathcal{B}}$ (resp., $M({\mathcal{B}})$ is itself a right $M({\mathcal{B}})$-submodule of ${\mathcal{B}}$). Thus the pair $(({\mathcal{B}},\tau_{{\mathcal{B}}}),R({\mathcal{B}}))$ is a right partial $R({\mathcal{B}})$-module (resp., $(({\mathcal{B}},\tau_{{\mathcal{B}}}),M({\mathcal{B}}))$ is a right partial $M({\mathcal{B}})$-module). Define $\langle\cdot,\cdot\rangle_{{\mathcal{B}}}$ on $(({\mathcal{B}},\tau_{{\mathcal{B}}}),R({\mathcal{B}}))$ (resp., $(({\mathcal{B}},\tau_{{\mathcal{B}}}),M({\mathcal{B}}))$) by $\langle x,y\rangle_{{\mathcal{B}}}=x^{*}y$. Then $x^{*}y\in{\mathcal{B}}$ if and only if $x\in R({\mathcal{B}})$ or $y\in R({\mathcal{B}})$ (resp., if and only if $x\in M({\mathcal{B}})$ or $y\in M({\mathcal{B}})$). $\langle\cdot,\cdot\rangle_{{\mathcal{B}}}$ is a ${\mathcal{B}}$-valued partial-inner product on $(({\mathcal{B}},\tau_{{\mathcal{B}}}),R({\mathcal{B}}))$ (resp., $(({\mathcal{B}},\tau_{{\mathcal{B}}}),M({\mathcal{B}}))$). Indeed:

1. (i)

$\langle x,x\rangle_{{\mathcal{B}}}=x^{*}x\in{\mathcal{B}}_{+},\forall x\in R({\mathcal{B}})$ (resp., $x\in M({\mathcal{B}})$) and if $x=0$, then $\langle x,x\rangle_{{\mathcal{B}}}=x^{*}x=0$; 2. (ii)

$\langle y,x\rangle_{{\mathcal{B}}}^{*}=(y^{*}x)^{*}=x^{*}y=\langle x,y\rangle_{{\mathcal{B}}},\forall x\in R({\mathcal{B}})$ or $y\in R({\mathcal{B}})$ (resp., $\forall x\in M({\mathcal{B}})$ or $y\in M({\mathcal{B}})$), 3. (iii)

$\langle x,y.b\rangle_{{\mathcal{B}}}=x^{*}(y.b)=(x^{*}y)b=\langle x,y\rangle_{{\mathcal{B}}}b,\forall x\in R({\mathcal{B}})$ or $y\in R({\mathcal{B}})$ and $b\in R({\mathcal{B}})$ (resp., $\forall x\in M({\mathcal{B}})$ or $y\in M({\mathcal{B}})$ and $b\in M({\mathcal{B}})$).

Remark 3.1

*Let ${\mathcal{H}}$ be a Hilbert space and $D$ a dense subspace of ${\mathcal{H}}$. Let ${\mathcal{L}}^{+}(D,\,{\mathcal{H}})$ be the partial ∗-algebra of closable operators $t$ with $\mbox{dom}(t)=D$ and partial multiplication $\circ$, given by $t\circ s:=t^{+*}s$, defined if and only if $sD\subseteq\mbox{dom}(t^{+*})$ and $t^{+}D\subseteq\mbox{dom}(s^{*})$. Let ${\mathcal{L}}^{+}(D)=\{t\in{\mathcal{L}}^{+}(D,{\mathcal{H}}):tD\subset D\,\,\mbox{and}\,\,t^{*}D\subset D\}$. Then ${\mathcal{L}}^{+}(D)$ is a linear subspace of ${\mathcal{L}}^{+}(D,{\mathcal{H}})$ and a ∗-algebra with respect to the usual operations. It follows that ${\mathcal{L}}^{+}(D,{\mathcal{H}})$ is a right ${\mathcal{L}}^{+}(D)$-module and ${\mathcal{L}}^{+}(D)$ is itself a right ${\mathcal{L}}^{+}(D)$-module. So the pair $({\mathcal{L}}^{+}(D,{\mathcal{H}}),{\mathcal{L}}^{+}(D))$ is a right partial ${\mathcal{L}}^{+}(D)$-module. Endow ${\mathcal{L}}^{+}(D,{\mathcal{H}})$ with the weak topology $\tau_{w}$, which is the locally convex topology defined by the family of seminorms: $|t|_{\eta,\xi}:=|\langle\eta,t\xi\rangle_{{\mathcal{H}}}|,\,\eta,\xi\in D$. Define an ${\mathcal{L}}^{+}(D,{\mathcal{H}})$-valued partial-inner product $\langle\cdot,\cdot\rangle_{{\mathcal{L}}^{+}}\equiv\langle\cdot,\cdot\rangle_{{\mathcal{L}}^{+}(D,{\mathcal{H}})}$ on $({\mathcal{L}}^{+}(D,{\mathcal{H}}),{\mathcal{L}}^{+}(D))$ by $\langle x,y\rangle_{{\mathcal{L}}^{+}}=x^{+}\circ y$. Then $({\mathcal{L}}^{+}(D,{\mathcal{H}}),{\mathcal{L}}^{+}(D))$ is a locally convex $({\mathcal{L}}^{+}(D,{\mathcal{H}}),\tau_{w})$-module with the locally convex topology given by the family of seminorms: $\|x\|_{\eta,\xi}:=|\langle x,x\rangle_{{\mathcal{L}}^{+}}|_{\eta,\xi}^{\frac{1}{2}}$.

By the preceding we now give the following definition.*

Definition 3.2

Let $({\mathcal{B}},\tau_{{\mathcal{B}}})$ be a locally convex partial ∗-algebra, ${\mathcal{B}}_{0}\subseteq R({\mathcal{B}})$ a linear subspace of ${\mathcal{B}}$ and $\pi:{\mathcal{B}}\to{\mathcal{L}}^{+}(D,{\mathcal{H}})$ a ∗-representation such that $\pi({\mathcal{B}}_{0})\subseteq{\mathcal{L}}^{+}(D)$. Let the pair $({\mathcal{X}},{\mathcal{D}})$ be such that ${\mathcal{X}}$ is an $O^{*}$-vector space on $D$ and ${\mathcal{D}}$ a linear subspace of ${\mathcal{L}}^{+}(D)$. Then $({\mathcal{X}},{\mathcal{D}})$ will be called a concrete locally convex partial ∗-algebraic module if the following conditions are satisfied:

1. (i)

$x\circ b\in{\mathcal{X}}$, for all $x\in{\mathcal{X}}$ and $b\in\pi({\mathcal{B}}_{0})$; 2. (ii)

$x\circ b\in{\mathcal{D}}$, for all $x\in{\mathcal{D}}$ and $b\in\pi({\mathcal{B}}_{0})$; 3. (iii)

the partial-inner product $\langle x,y\rangle_{{\mathcal{B}}}=x^{+}\circ y\in\pi({\mathcal{B}})$, for all $x\in{\mathcal{D}}$ or $y\in{\mathcal{D}}$.

Definition 3.3

*Let $\sigma:{\mathcal{D}}\times{\mathcal{D}}\to{\mathcal{L}}^{+}(D,{\mathcal{H}})$ be a conjugate-bilinear map. We say that the map $\sigma$ is positive definite if and only if, for every positive integer $n$,

$\sum_{j,k=1}^{n}\left\langle\xi_{j},\sigma(x_{j},x_{k})\xi_{k}\right\rangle_{{\mathcal{H}}}\geq 0$ for all $x_{1},\cdots,x_{n}\in{\mathcal{D}}$, and $\xi_{1},\cdots,\xi_{n}\in D$, where $\langle\cdot,\cdot\rangle_{{\mathcal{H}}}$ is the inner product of the Hilbert space ${\mathcal{H}}$.*

Proposition 3.1

Let $({\mathcal{B}},\tau_{{\mathcal{B}}})$ be a locally convex partial ∗-algebra and $({\mathcal{X}},{\mathcal{D}})$ a locally convex $({\mathcal{B}},\tau_{{\mathcal{B}}})$-module. Suppose that there exist a representation $\pi$ of ${\mathcal{B}}$ onto some partial $O^{*}$-algebra of closable operators on $D$ in the Hilbert space ${\mathcal{H}}$ and a map $\vartheta$ from $({\mathcal{X}},{\mathcal{D}})$ onto a concrete locally convex $\pi({\mathcal{B}})$-module $({\mathcal{Y}},{\mathcal{E}})$ of closable operators on $D$ in ${\mathcal{H}}$. Then

[TABLE]

If (7) holds, then:

1. (i)

$\vartheta(x.b)=\vartheta(x)\circ\pi(b)$, for all $x\in{\mathcal{X}}$ and $b\in{\mathcal{B}}_{0}\subseteq R({\mathcal{B}})$; 2. (ii)

the representation $\pi:{\mathcal{B}}\to{\mathcal{L}}^{+}(D,{\mathcal{H}})$ is (necessarily) a ∗-map, i.e., $\pi(b^{*})=\pi(b)^{+}$, for all $b\in{\mathcal{B}}$; 3. (iii)

the map $\vartheta$ is linear, and if $\pi$ is faithful, then $\vartheta$ is injective.

**Proof. ** Let $\pi:{\mathcal{B}}\to{\mathcal{L}}^{+}(D,{\mathcal{H}})$ be any representation of ${\mathcal{B}}$ into some partial $O^{*}$-algebra. Let $\vartheta:{\mathcal{D}}\to{\mathcal{L}}^{+}(D,{\mathcal{H}})$ be any map and let $\sigma:{\mathcal{X}}\times{\mathcal{X}}\to{\mathcal{L}}^{+}(D,{\mathcal{H}})$ be a conjugate-bilinear map defined by $\sigma(x,y)=\vartheta(x)^{+}\circ\vartheta(y)$. Then we have, for all $x_{1},\cdots,x_{n}\in{\mathcal{D}}$, and $\xi_{1},\cdots,\xi_{n}\in D$, $n\in\mathbb{N}$, that

[TABLE]

It follows that the map $\sigma$ is positive definite and so $\vartheta(x)^{+}\circ\vartheta(x)$ is positive. Now set $\vartheta(x)^{+}\circ\vartheta(y)=\pi(\langle x,y\rangle_{{\mathcal{B}}})$, where $\vartheta(x)^{+}\circ\vartheta(y)=\langle\vartheta(x),\vartheta(y)\rangle_{{\mathcal{L}}^{+}}$. Then

$\langle\vartheta(x),\vartheta(y)\rangle_{{\mathcal{L}}^{+}}=\pi(\langle x,y\rangle_{{\mathcal{B}}})$, for all $x\in{\mathcal{D}}$ or $y\in{\mathcal{D}}$.

1. (i)

For all $x\in{\mathcal{X}}$, $y\in{\mathcal{D}}$ and $b\in{\mathcal{B}}_{0}$, we have

[TABLE]

i.e.,

[TABLE] 2. (ii)

On the other hand, we have

[TABLE]

i.e.,

[TABLE]

Now (8) and (9) imply that $\pi(b^{*})=\pi(b)^{+}$, for all $b\in{\mathcal{B}}_{0}$. But since

$\langle\vartheta(x),\vartheta(y)\rangle_{{\mathcal{L}}^{+}}^{+}=\langle\vartheta(y),\vartheta(x)\rangle_{{\mathcal{L}}^{+}}=\pi(\langle y,x\rangle_{{\mathcal{B}}})=\pi(\langle x,y\rangle_{{\mathcal{B}}}^{*})$ and

$\langle\vartheta(x),\vartheta(y)\rangle_{{\mathcal{L}}^{+}}^{+}=\pi(\langle x,y\rangle_{{\mathcal{B}}})^{+}$, it follows that $\pi(b^{*})=\pi(b)^{+}$, for all $b\in{\mathcal{B}}$. 3. (iii)

For all $x,y\in{\mathcal{X}}$ and $z\in{\mathcal{D}}$,

[TABLE]

i.e., $\vartheta(x+y)=\vartheta(x)+\vartheta(y)$, for all $x,y\in{\mathcal{X}}$.

It remains to show that $\vartheta$ is injective if $\pi$ is faithful. Now let $\vartheta$ be a faithful representation and suppose $\vartheta(x)=\vartheta(y)$, for all $x,y\in{\mathcal{D}}$. Since $\vartheta$ is linear, this implies that $\vartheta(x-y)=0$. It follows that $0=\langle\vartheta(x-y),\vartheta(x-y)\rangle_{{\mathcal{L}}^{+}}=\pi(\langle x-y,x-y\rangle_{{\mathcal{B}}})$, and since $\pi$ is faithful we have $\langle x-y,x-y\rangle_{{\mathcal{B}}}=0$, whence $x=y$. Hence $\vartheta$ is injective.

Definition 3.4

Let $(({\mathcal{X}},{\mathcal{D}}),\langle\cdot,\cdot\rangle_{{\mathcal{A}}},\tau_{{\mathcal{D}},{\mathcal{A}}})$ and $(({\mathcal{Y}},{\mathcal{E}}),\langle\cdot,\cdot\rangle_{{\mathcal{B}}},\tau_{{\mathcal{E}},{\mathcal{B}}})$ be two (complete) locally convex partial ∗-algebraic modules such that $\varphi:{\mathcal{A}}\to{\mathcal{B}}$ is any linear map. A map $\vartheta:{\mathcal{X}}\to{\mathcal{Y}}$ will be called a $\varphi$-map if $\vartheta$ and $\varphi$ satisfy $\langle\vartheta(x),\vartheta(y)\rangle_{{\mathcal{B}}}=\varphi(\langle x,y\rangle_{{\mathcal{A}}})$, for all $x\in{\mathcal{D}}$ or $y\in{\mathcal{D}}$. If $\varphi$ is a homomorphism and $\vartheta$ is a $\varphi$-map, then $\vartheta$ will be called a $\varphi$-homomorphism. Finally, If $\varphi=\pi$ is a representation and $\vartheta$ is a $\pi$-map, then $\vartheta$ will be called a $\pi$-representation.