数理解析研究所講究録
1077 巻 1999 年 52-61

52

Unbounded$C^{*}$-seminorns and *-representations of$*$-algebras
福岡大学理学部応用数学科
荻 秀和 (Hidekazu Ogi)

1. INTRODUCTION
Unbounded-seminorms on -algebras in the sense that they are-seminorms
defmed on -subalgebras have appeared in many mathematical and physicalsubjects (for example, locally convex *-algebras and the quantum field theory etc.).
But this systematical study has not yet done sufficiently. The main purpose of thispaper is to do a systematical study of unboundedorms and to apply it toa study of unbounded *-representations.
The paper is organized as follows: In Section 2 we construct unbounded
representations of a -algebra from unbounded-seminorms and investigate
them. Let $A$ be-algebra. Let
be a
orm defined on A.
representation of the Hausdorff completion of $(A,p)$ gives rise to a$*$$C^{*}$$C^{*}$$*$$C^{*}- \mathrm{s}\mathrm{e}\mathrm{m}\dot{\mathrm{m}}$$*_{-}$$*$$C^{*}$$\mathrm{a}^{*}$$C^{*}- \mathrm{s}\mathrm{e}\mathrm{m}\dot{\mathrm{m}}$$p$$\mathrm{B}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{y}-*$$*-$representation of $A$ into bounded Hilbert space operators. However, there are anumber of situations in which natural-seminorms are defined on -subalgebras
of $A$ . Then they should lead to unbounded operator representations of $A$ . An
unboundedresp.
)
is a submultiplicative(resp.
seminorm
defmed on a
-subalgebraof $A$ . Then
is a left ideal of $A$ . It is shown that any
representation:of the Hausdorff completionof$*$$C^{*}$$m^{*}-($$*$$C^{*}-)_{\mathrm{S}}\mathrm{e}\dot{\mathrm{m}}\mathrm{n}\mathrm{o}\mathrm{m}$

$*$

$p$

$\equiv\{x\in \mathcal{D}(p);ax\in \mathcal{D}(p),$$\Pi_{p}$$C^{*}-$$\mathcal{D}(p)$$\mathfrak{R}_{p}$$*-$$\forall_{a}\in A\}$

$A_{p}arrow \mathfrak{B}(\mathcal{H})$$A_{p}$

leads to an unbounded *-representation$\pi_{p}$of $A$ such that$(\mathcal{D}(p),p)$$||\overline{\pi_{p}(x)}||\leq p(x)$for all. We denote by Rep $(A,p)$ the set of all such *-representationsof $A$ .
In order to investigate representations in Rep $(A,p)$ in details, we introduce the$x\in \mathcal{D}(p)$$\pi_{p}$notions of nondegenerate, finite,semifmite, semifmite and weakly
orms, and show that if
semifinite unboundedis (weakly) semifinite,then there exists a strongly nondegenerate *-representation
in Rep $(A,p)$ such$\wedge \mathrm{u}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{l}\mathrm{y}$$C^{*}- \mathrm{s}\mathrm{e}\mathrm{m}\dot{\mathrm{m}}$$p$$\pi_{p}$

53

that

$||\overline{\pi_{p}(x)}||=p(x)$for all$x\in \mathcal{D}(p)$. Such a$\pi_{p}$is called well-behaved. In Section

3 we consider the converse direction of Section 2. We construct an unbounded
and a naturaloffrom a -representationon-seminomof constIucted from which is the restriction of the closure
representationis a
is strongly nondegenerate if and only ifof . It is shown thatis a weakly
well-behaved *-representation of . Further, it is shown that if
andis any well-behaved-seminorm on
semifinite unbounded
is a maximal extension of .
representation, then$r_{\pi}$

$A$

$\pi$

$A$

$\pi_{r_{\hslash}}^{N}$$\tilde{\pi}$

$*$

$A$

$C^{*}$$r_{\pi}$

$\pi$

$\pi$

$\pi_{r_{\pi}}^{N}$

$A$

$p$

$*_{-}$$A$$C^{*}$$\pi_{p}$$p$$r_{\pi_{\rho}}$

2. REPRESENTATIONS INDUCED BY UNBOUNDED$C^{*}$-SENImORMS-representations of -algebra $A$ induced
In this section we construct a fanily
orm on $A$ and investigate the properties. We begin
by an unboundedwith the review of (unbounded) *-representations of $A$ . Throughout this section
letbe a dense subspace in a Hilbert ace-algebra. Let
let $A$ bewith the domain $D$ for whichindenote the set of all linear operators
is -algebra under the usual
D. Thenand$\mathrm{a}^{*}$$\mathrm{o}\mathrm{f}*$$C^{*}- \mathrm{s}\mathrm{e}\mathrm{m}\dot{\mathrm{m}}$$\mathcal{D}$$\mathrm{a}^{*}$$\mathcal{H}.\mathrm{a}\mathrm{n}\mathrm{d}$$\mathrm{s}\mathrm{p}_{i}$$X$$L^{\dagger}(\mathcal{D})$$X\mathcal{D}\subset D,$$X^{*}\mathcal{D}\subset$$\mathcal{D}(X^{*})\supset \mathcal{D}$$\dot{\mathcal{H}}$$L^{\dagger}(\mathcal{D})$.

$\mathrm{a}^{*}$-subalgebra of the -algebra-representation of $A$ on a.
inand-homomorphism of $A$ into,$\pi(1)=1$ if $A$ has identity 1, and then we write
and
by
and
is a closed
be -representations of $A$ . If
andrespectively. Let
is said to be anfor each $x\in A$ , thenandsubspace of,and. In particular, if
and denoted byextension ofoperations and the involution-algebra on
is said to be aniswith a domain
Hilbert space$Xarrow X^{\dagger}\equiv X^{*}|\mathcal{D}$$O^{*}$$L^{\dagger}(\mathcal{D})$$\mathcal{D}$$\mathcal{H}$$\mathcal{D}$$\mathcal{H}$$*$$\mathrm{A}*$$\mathrm{A}^{*}$$\pi$$L^{+}(\mathcal{D})$$\mathrm{a}^{*}$$\mathcal{D}$$\mathcal{H}$$\mathcal{D}(\pi)$$\mathcal{H}_{\pi}$$*$$\pi_{1}$$\mathcal{H}_{\pi_{2}}$$\mathcal{H}_{\pi_{1}}$$\pi_{2}$$\pi_{1}(x)\subset\pi_{2}(\chi)$$\pi_{2}$$\pi_{1}\subset\pi_{2}$$\pi_{1}\subset\pi_{2}$$\pi_{1}$$\mathcal{H}_{\pi_{1}}=\mathcal{H}_{\pi_{2}}$as the same Hilbert space. Let be
is said to be an extension ofdefmed byis complete with the graph topology
representation of $A$ . If$||;x\in A\}$ , thenis said to be
the family of seminorms

then

$\pi$

$\pi_{1}$$\pi_{2}$$\mathcal{D}(\pi)$$t_{\pi}$$\{||\bullet||_{\pi(_{X)}}\equiv||\bullet||+||\pi(x)$closed. It is well known thatclosure$\tilde{\pi}$

of

$\pi$

$\pi$

$\pi$

$\bullet$is closed if and only if$\mathcal{D}^{f_{(}}\pi$)$= \bigcap_{AX\in}\mathcal{D}(\overline{\pi(x)})$is defined by$\mathcal{D}(\tilde{\pi})=\bigcap_{X\in A}\mathcal{D}(\overline{\pi(\chi)})$$\mathrm{a}^{*}-$

and$\tilde{\pi}(x)\xi=\overline{\pi(x}\mathrm{g}$for $x\in A,$$\xi\in \mathcal{D}(\tilde{\pi})$

.

. The

54

Then

$\tilde{\pi}$is the smallest closed extension of$\pi$. The weak commutantof$\pi(A)_{w}’$$\pi$

is defined by$\pi(\mathcal{A})_{w}’=\{C\in \mathfrak{B}(\mathcal{H}_{\pi});c\pi(X)\xi=\pi(x^{*})^{*}c\xi,$$\forall_{\chi}\in A,$$\forall_{\xi}\in \mathcal{D}(\pi)\}$,, and it is a weakly
is the set of all bounded linear operators on
where, but it is not necessarily an algebra. It isclosed *-invariant subspace of$\mathfrak{B}(\mathcal{H}_{\pi})$$\mathcal{H}_{\pi}$$\mathfrak{B}(\mathcal{H}_{\pi})$known thatand$\overline{\pi(x)}$$\pi(A)_{w}’D(\pi)\subset \mathcal{D}(\pi)$

if and only if $\pi(A)_{w}’$ is a von Neumann algebra

is affiliated with the von Neumann algebra $(\pi(A)_{w}’)’$ for each $x\in A$ .

Definition 2.1. A mappingof a subspaceof $A$ intoissaid to be an unbounded (semi) norm on $A$ if it is a (semi) norm on, and(semi) norm on $A$ if
is said to be an unboundedis) (semi) norm on
subalgebra of $A$ and is a submultiplicative *-(resp.
.
$\mathcal{D}(p)$$p$$m^{*}-(\mathrm{r}\mathrm{e}\mathrm{s}_{\mathrm{P}}.

C^{*}-)$

If a seminorm
on -algebra $A$ is a
, then it is a
-property $p(x^{*}x)=p(X)^{2},$$C^{*}-\mathrm{s}\mathrm{e}\mathrm{m}\dot{\mathrm{m}}\mathrm{o}\mathrm{m}$$\mathrm{a}^{*}$$p$$\forall_{\chi}\in A$$p(x^{*})=p(x)$ and

Let

$p$

$N_{p}$

be an unbounded

algebrais-ideal of$\mathcal{D}(p)/N_{p}$$(x\in D(p))$$\mathcal{D}(p)/N_{p}$algebra$\mathrm{a}^{*}$$C^{*}$

and$\mathcal{D}(p)$$\mathrm{a}*-$$A_{p}$

Rep

on Hilbert space$\forall_{x,y}\in A$, that is, it satisfies the$m^{*}- \mathrm{s}\mathrm{e}\mathrm{m}\dot{\mathrm{m}}$

om on

$A$

, that is,

.

$*$

.

is a left ideal of $A$ , and the quotient-algebra with the-norm $||x+N_{p}||_{p}\equiv p(x)$$(A_{p})$$*-$$C^{*}$

the$\mathcal{H}_{\Pi_{\rho}}$$\forall_{a}\in A\}-$$\mathfrak{R}_{p}$$C^{*}$-algebra obtained by the completion of

the set of all *-representations$\Pi_{p}$of the$C^{*}-$. Put$\mathrm{F}{\rm Re}_{\mathrm{P}()}A_{p}=\{\Pi_{p}\in \mathrm{R}\mathrm{e}\mathrm{p}(A_{p});\Pi_{p}$is$(A_{p})=\{\Pi_{p}\in \mathrm{R}\mathrm{e}\mathrm{p}(Ap\mathrm{I};\Pi_{p}$It is well known that FNRep$(A)p\neq\emptyset$

bounded *-representationof$\pi_{p}^{0}$$\mathcal{D}(p)$$\mathfrak{R}_{p}\equiv\{x\in \mathcal{D}(p);a\chi\in \mathcal{D}(p),$andis a nomed, and denote by

FNRep

$p$

-seminorm on $A$ . We put$\mathcal{D}(p)$. We denote by$A_{p}$

for$p(xy)\leq p(X)p(\mathcal{Y})$$N_{p}=\{x\in \mathcal{D}(p);p(\chi)=0\}$Then$\mathcal{D}(p)$$C^{*}-$$p$$C^{*}$$\mathbb{R}^{+}=[\mathrm{o},\infty)$$\mathcal{D}(p)$faithful}
is faithful and. For each$\Pi_{p}\in \mathrm{R}\mathrm{e}\mathrm{p}(A_{p})$on the Hilbert space.

$\pi_{p}^{0}(_{X})=\Pi_{p}(x+N_{p}),$$\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e}\}$$\mathcal{H}_{\Pi p}$.

.

we can define aby$x\in \mathcal{D}(p)$The natural question arises: Can we extend the bounded *-representation$\pi_{p}^{0}$of the55$*$-algebra$\mathcal{D}(p)$to a (generally unbounded) *-representation of the *-algebra A?

We show that this question has afflmative answer.

Proposition 2.2. Let, there exists$\Pi_{p}\in \mathrm{R}\mathrm{e}\mathrm{p}(A_{p}\mathrm{I}$such that$||\overline{\pi_{p}(b)}||\leq p(b)$$||\overline{\pi_{p}(X)}||=p(\chi)$for each

be an unbounded$p$$\mathrm{a}^{*}$-representationfor each$b\in \mathcal{D}(p)$-seminorm on $A$ . For anyofon a HilbeIt space$C^{*}$$\mathcal{A}$$\pi_{p}$$\mathcal{H}_{\pi_{p}}$. In particular, if$\Pi_{p}\in \mathrm{F}\mathrm{R}\mathrm{e}\mathrm{p}(A_{p})$, then.

$x\in \mathfrak{R}_{p}$Proof. We put$\mathcal{D}(\pi_{p})=\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}$span of$\{\Pi_{p}(\chi+N_{p})\xi;x\in \mathfrak{R}_{p}$, and$\pi_{p}(a)(\sum_{k}\Pi_{p}(x_{k}+N_{p})\xi_{k}\mathrm{I}=\sum_{k}\Pi_{p}(aX_{k}+N_{p})\xi_{k}$for $a\in A,$$\{x_{k}\}\subset \mathfrak{R}_{p}$$\xi\in \mathcal{H}_{\Pi_{p}}\}$(finite sums)and$\{\xi_{k}\}\subset \mathcal{H}_{\Pi_{p}}$

.

Since
$(\Pi_{p}(ax+N_{p})\xi|\Pi(Py+N_{p})\eta \mathrm{I}=(\xi|\Pi_{p}((ax+N)^{*}(\mathcal{Y}^{+}N_{p})p)\eta)$$=(\xi|\Pi_{p}(x^{*}ay+N_{p}*)\eta)$$=(\xi|\Pi_{P}(X^{*}+Np)\Pi_{p}(a^{*}y+N_{p})\eta \mathrm{I}$$=(\Pi_{p}(X+N_{p}\mathrm{I}^{\xi}|\Pi p(^{*}a_{\mathcal{Y}}+N)P\eta)$

for each $a\in A,$$x,$linear operator on$*$$\mathcal{D}(\pi_{p})$-representation ofof$\mathcal{D}(\pi_{p})$indefinition of$\mathcal{H}_{\Pi_{\rho}}$$\pi_{p}$$y\in \mathfrak{R}_{p}$

$A$

and

$\xi,$$\eta\in \mathcal{H}_{\Pi_{p}}$, it follows that $\pi_{p}(a)$ is a welL-definedfor each $a\in A$ , so that it is easily shown thaton the Hilbert space) with domain

we have$\mathcal{D}(\pi_{p})$$\mathcal{H}_{\pi_{p}}\equiv[\mathcal{D}(\pi_{p})]=\overline{\mathcal{D}(\pi_{p})}^{\Downarrow\#}$. Take an arbitrary$\pi_{p}(b)=\pi_{p}^{\mathrm{o}}(b)|\mathcal{D}(\pi)P$’$\Pi_{p}\in \mathrm{F}\mathrm{R}\mathrm{e}\mathrm{p}(A_{p})$and$x\in \mathfrak{R}_{p}$with$||\xi||\leq 1$. By the.
$||\overline{\pi_{p}(x)}||\geq p(x)$$y= \frac{X}{p(x)}\in \mathfrak{R}_{p}$, we have,$||\Pi_{p}(\mathrm{y}+N_{p})\xi||\leq||\Pi_{p}(\mathcal{Y}^{+}N_{p})||||\xi||=p(\mathcal{Y})||\xi||\leq 1$

and so(the closure$b\in \mathcal{D}(p)$. It is sufficient to show thatIf $p(x)=0$ , then it is obvious. Suppose $p(x)\neq 0$ . We put$\xi\in \mathcal{H}_{\Pi_{p}}$is aand hence$||\overline{\pi_{p}(b)}||\leq||\Pi_{p}(b+N\mathrm{I}p||\leq||b+N_{p}||_{p}=p(b)$Suppose$\pi_{p}$.

. For each56$|| \overline{\pi_{p}(_{\mathcal{Y}})}||=||\overline{\pi_{p}(\mathcal{Y})*}||\geq\sup\{||\pi(py)’\Pi p(\mathcal{Y}^{+N}p)\xi||;\xi\in \mathcal{H}_{\Pi_{p}}\mathrm{s}.\mathrm{t}.||\xi||\leq 1\}$$= \sup\{||\Pi_{p}(^{*}\mathcal{Y}y+N_{p})\xi||;\xi\in \mathcal{H}_{\Pi_{p}}\mathrm{S}.\mathrm{t}.||\xi||\leq 1\}$$=||\Pi_{p}(y^{*}y+N_{p})||$$=p(y^{*}y)=p(\mathrm{y})^{2}=1$

Hence, we have$||\overline{\pi_{p}(x)}||\geq p(x)$:.

. This completes the proof.

We have the following diagram:$9\dagger_{\Pi_{\rho}})$$\cap$$\downarrow$$\pi_{p}$$A-\pi_{p}(A)$ ( -algebra in$O^{*}$$\mathcal{H}_{\pi_{\rho}}\subset \mathcal{H}_{\Pi_{\rho}}$).

of $A$ defined above by an unbounded
Remark: The -representation
is non-zero if and only ifof Rep-seminorm on $A$ and an element. In what follows, we discuss several situations keeping this in mind.
$*$$\pi_{p}$$C^{*}$$(A_{p})$$\Pi_{p}$$p$$A\mathfrak{R}_{p}\not\subset N_{p}$orm on $A$ . We denote by Rep $(A,p)$ ,be an unbounded
FRep $(A,p)$ and FNRep $(A,p)$ the sets of all *-representations of $A$ constructed
as above by $(A,p)$ , that is,
Let$C^{*}- \mathrm{s}\mathrm{e}\mathrm{m}\dot{\mathrm{m}}$$p$

Rep $(A,p)=\{\pi_{p}$ ;$\Pi_{p}\in \mathrm{R}\mathrm{e}\mathrm{p}(A_{p})\},$

FRep $(A,p)=\{\pi_{p}$ ;.

$\Pi_{p}\in \mathrm{F}\mathrm{R}\mathrm{e}\mathrm{p}(A_{P})\}$

FNRep $(A,p)=\{\pi_{p}$ ;,$\Pi_{p}\in \mathrm{F}\mathrm{N}\mathrm{R}\mathrm{e}\mathrm{p}(A_{p})\}$.

57

on $A$ is said to be-seminorm
Definition 2.3. An unbounded
is total inwith respect to the seminorm . An
nondegenerate if; and is said-seminorm on $A$ is said to be finite if
unboundedinsuch thatto be uniforndy semifmite if there exists a net
andfor eacharid is said to befor each and$m^{*}$$\mathcal{D}(q)^{2}$$q$$\mathcal{D}(q)$$q$$m^{\mathrm{c}}$$\mathcal{D}(q)=\mathfrak{R}_{q}$$q$$\{u_{\alpha}\}$$q(u_{\alpha})\leq 1$

semifinite if$C^{*}$$\mathfrak{R}_{q}$-seminomis dense in

on

$p$

$\mathcal{D}(q)$$u_{\alpha}=u_{\alpha}*$$\mathfrak{R}_{q}$$x\in \mathcal{D}(q);$$\lim_{\alpha}q(\chi u_{\alpha}-X)=0$$\alpha$$q$$q$with respect to the seminorm$q$. An unboundedis said to be weakly semifinite if FRep $(A,p)$. An elementofis said to$A$$\equiv\{\pi_{p}\in \mathrm{F}\mathrm{R}\mathrm{e}\mathrm{p}(A,p);\mathcal{H}_{\pi_{p}}=\mathcal{H}_{\Pi_{\rho}}\}\neq\emptyset$$\mathrm{R}\mathrm{e}\mathrm{p}^{\mathrm{W}\mathrm{B}}(A,p)$$\pi_{p}$be a well-behaved *-representation of $A$ in Rep $(A,p)$ .

Definition 2.4.
; and$\mathrm{A}^{*}$$[\pi(A)\mathcal{D}(\pi)]=\mathcal{H}_{\pi}$$\pi$

of $A$ is said to be nondegenerate if-representation
is said to be strongly nondegenerate if there exists aleft$\pi$ideal 7 of $A$ contained in the bounded partsuch that$[\overline{\pi(3)}\mathcal{H}]\pi=\mathcal{H}_{\pi}$$A_{\mathrm{b}}^{\pi}\equiv\{x\in A;\overline{\pi(x)}\in \mathfrak{B}(\mathcal{H}_{\pi})\}$

of

$\pi$

.

Proposition 2.5. Let$p$be an unbounded$C^{*}$-seminorm on $A$ . Then thefollowing statements hold:(1)and.ever.y$\mathrm{R}\mathrm{e}\mathrm{p}^{\mathrm{W}\mathrm{B}}(A,p)\subset \mathrm{F}\mathrm{N}\mathrm{R}\mathrm{e}\mathrm{p}(A,p)$$\pi_{p}\in{\rm Re}_{\mathrm{P}^{\mathrm{w}\mathrm{B}}}(A,p)$satisfies thefollowing conditions (i), (ii) and (i\"u):
(i)

(ii)

(iii)

$[\overline{\pi_{p}(\mathfrak{R}_{P}\mathrm{I}}\mathcal{H}_{\pi p}]=\mathcal{H}_{\pi_{\rho}}$$||\overline{\pi_{p}(_{X})}||=p(X),$, and$\pi_{p}$$\forall_{X\in \mathcal{D}(p)}$$\pi_{p}(A)_{w}’=\overline{\pi_{P}(\mathcal{D}(p))}’$

andis strongly nondegenerate.

.
$\pi_{p}(A)’w\mathcal{D}(\pi_{p})\subset \mathcal{D}(\pi_{p})$Conversely suppose
Then there exists an elementof(2) Suppose
is semifinite. Then$\pi_{p}\in \mathrm{F}\mathrm{R}\mathrm{e}\mathrm{p}(A,p)$$\pi_{p}^{WB}$$\mathcal{D}(p)$with respect to(3) Suppose

$p$

$p$

satisfies conditions (i) and (\"u) above.
which is a representation of .
andis$\mathrm{R}\mathrm{e}\mathrm{p}^{\mathrm{W}\mathrm{B}}(A,p)$$p$total in.

, and so$\pi_{p}$$\mathrm{R}\mathrm{e}\mathrm{p}^{\mathrm{W}\mathrm{B}}(\mathcal{A},p)=\mathrm{F}\mathrm{N}\mathrm{R}\mathrm{e}\mathrm{p}(A,p)$$p$$\mathfrak{R}_{p}^{2}$

is nondegenerate.

is uniformly semifinite. Then

$A_{\mathrm{b}}^{\pi_{\rho}}=A_{\mathrm{b}}^{p}\equiv\{a\in A;k_{a}\exists>0\mathrm{s}.\mathrm{t}.p(aX)\leq k_{a}p(x),$$|| \overline{\pi_{p}(b)}||=\sup\{p(b\chi);\chi\in \mathfrak{R}_{p}$and $p(x)\leq 1\},$$\forall_{\chi}\in \mathfrak{R}_{p}\}$$\forall_{b}\in A_{\mathrm{b}}^{p}$,

58

for each(4)$\pi_{p}\in \mathrm{F}\mathrm{R}\mathrm{e}\mathrm{p}(A_{P},)$$p$.

is finite if and only if

3. UNBOUNDED$C^{*}$$\mathcal{D}(p)$is a left ideal of $A$ .

-SEMDfORMS DEFINED BY -REPRESENTATIONS
$*$

) of
In Section 2 we constructed a family Rep $(fl,P)$ (resp.
-seminorm
representation of $A$ from an (resp. weakly semifinite) unbounded
on $A$
orm
on $A$ . Conversely we shall construct an unboundedof $A$
of $A$ and the natural representationfrom a -representation
Let
be a
and
constucted from , and investigate the relation$*-$$\mathrm{R}\mathrm{e}\mathrm{p}^{\mathrm{W}\mathrm{B}}(A,p)$$C^{*}$$C^{*}- \mathrm{s}\mathrm{e}\mathrm{m}\dot{\mathrm{m}}$$r_{\pi}$

$p$

$*$

$\pi_{r_{n}}^{N}$

$\pi$

$\pi$

$*_{-}$$\pi$$\pi_{r_{\pi}}^{N}$$r_{\pi}$

representation of$A$

on a Hilbert space$\mathcal{H}_{\pi}$$\mathrm{A}^{\pi}=\{x\in A;\overline{\pi(X)}\in \mathfrak{B}(\mathcal{H}_{\pi})\}$

and$\pi_{\mathrm{b}}(x)=\overline{\pi(\chi)},$$x\in A_{\mathrm{b}}^{\pi}$.

onis a bounded *-representation of-subalgebra of $A$ and. We now define-algebra generated bythe. We denote by

Then$\mathcal{H}_{\pi}$. We put$A_{\mathrm{b}}^{\pi}$is$A_{\mathrm{b}}^{\pi}$$\mathrm{a}^{*}$$\pi_{\mathrm{b}}$$C^{*}$$C^{*}(\pi)$an unbounded$C^{*}$-seminorm$\mathcal{D}(r_{\pi})=A_{\mathrm{b}}^{\pi}$$r_{\pi}$$\pi_{\mathrm{b}}(A_{\mathrm{b}}^{\pi})$

on

and

$A$

as follows;$r_{\pi}(x)=||\pi_{\mathrm{b}}(x)||,$$x\in \mathcal{D}(\gamma_{\pi})$.

Then we put$\Pi(x+N_{r\hslash})=\pi_{\mathrm{b}}(_{X}),$

Since$||\Pi(X+N_{r_{\pi}})||=r_{\pi}(x)=||x+N_{\Gamma_{\hslash}}||_{r_{\hslash}}$for eachofextended to a faithful *-representationof $A$ defined by-representation
representation of $A$ induced by . Since$\Pi_{r_{\pi}}^{N}$$*$$\Pi_{r_{l}}^{N}$$\pi_{r_{n}}^{N}$$\pi$$x\in A_{\mathrm{b}}^{\pi}$.

$x\in A_{\mathrm{b}}^{\pi}$$A_{r_{\pi}}$, it follows thaton the Hilbert space

$\Pi$

canbe

$\mathcal{H}_{\pi}$. The

as above is called the naturalis a, it follows that$g\{_{\Pi_{r_{\pi}}^{N}}=\mathcal{H}_{\pi}$$\mathcal{H}_{\pi_{r_{\pi}}^{N}}$. We simply note the above method of the conshuction ofclosed subspace ofby the following diagram:$\mathcal{H}_{\pi}$$\pi_{r_{\pi}}^{N}$

59

$\pi$

completion

unbounded-seminorm($C^{*}$

1$C^{-}$-algebra)$\Pi_{r_{\pi}}^{N}$$C^{*}(\pi)=\overline{\pi(\mathrm{b}A\pi)\mathrm{b}}\mathrm{N}\mathrm{N}$$\pi_{r_{\pi}}^{N}$$C^{*}$We have the following results for the relation of

and$\pi_{r_{\pi}}^{N}$$\mathcal{H}_{\pi}$:is -representation of $A$ on a Hilbert space
Proposition 3.1. Suppose. Then the following statements hold:$\pi$$\mathcal{H}_{\pi}$$\pi$-algebra on(1)$\pi_{r_{\pi}}^{N}\subset\tilde{\pi}$$\pi$

true, then.

is nondegenerate. Thenis strongly nondegenerate if and only if(2) Suppose(3)$\mathrm{a}^{*}$$\pi_{r_{\pi}}^{N}$$\pi_{r_{\pi}}^{N}\in \mathrm{F}\mathrm{N}{\rm Re}_{\mathrm{P}}(A,r_{\pi})$$\pi_{\mathrm{b}}$is strongly nondegenerate withsemifinite.
(4) Suppose there exists a net$s- \lim_{\alpha}\pi(a\mathcal{U}_{\alpha})=\pi(a)$$\{u_{\alpha}\}$infor each $a\in A$ . Then$\mathfrak{R}_{r_{\pi}}$$\pi_{r_{n}}^{N}\in \mathrm{R}\mathrm{e}\mathrm{p}^{WB}(A,r_{\pi})$$A_{\mathrm{b}}^{\pi_{r_{l}}^{N}}=A_{\mathrm{b}}^{\pi}$such that$\pi_{r_{\pi}}^{N}=\tilde{\pi}$.

By Proposition 3.1 we have the following diagram:
$p$

.

,

and

$r_{\pi}$. If this is

is weakly$s- \lim_{\alpha}\pi(u_{\alpha})=I$

and

60

We here investigate the relations of unbounded-seminorms
andandthe -representation
andWe first define an order relation among$C^{*}$$p$$r_{\pi_{p}}$$*$$\pi_{p}$$\pi_{r_{\pi_{\rho}}}$unbounded seminorms as follows:

Definition 3.2. Let
and be unbounded seminorms on $A$ . We say that(oris an extention ofis a restriction of ) if
and$q(x)=p(X)$ for each, and then denote by
.
$p$

$p$

$q$

$q$

$q$

$p$

$x\in \mathcal{D}(q)$$q\subset p$

the set of all unbounded $C$ -seminorms on $A$ . Thenis an ordered set with the order . For any
we putWe denote by$\mathrm{C}^{*}\mathrm{N}(A)$$\mathcal{D}(q)\subset \mathcal{D}(p)$$\mathrm{C}^{\cdot}\mathrm{N}(A)$$p\in \mathrm{C}^{*}\mathrm{N}(A)$$\subset$$\mathrm{C}^{*}\mathrm{N}(p)=\{q\in \mathrm{c}\cdot \mathrm{N}(A);p\subset q\}$.

Then it follows from Zom’s lemma thathas a maximal element. We showthat if
is weakly semifmite thenis a maximal element of.
$\mathrm{C}^{*}\mathrm{N}(p)$$p$$\mathrm{C}^{*}\mathrm{N}(p)$$r_{\pi_{\rho}}$Lemma 3.3. Let. Then, for any
such that.

$p$$p\subset r$

and

$r$

be unbounded-seminorms on $A$ . Suppose
there exists an elementof Rep $(A,r)$$C^{*}$$\pi_{p}\in \mathrm{R}\mathrm{e}\mathrm{p}(A,p)$$\pi_{r}$$\pi_{p}\subset\pi_{r}$Propo ition 3. 4. Suppose
seminorm on $A$ and$\mathrm{s}$$p$$\pi_{p}\in \mathrm{R}\mathrm{e}\mathrm{p}^{\mathrm{W}\mathrm{B}}(A,p)$and$r_{\pi_{p}}=r_{\pi_{\rho}’}$for each$\pi_{p},$is a weakly semifmite unbounded. Thenis a maximal element of$C^{*}-$$\mathrm{C}^{*}\mathrm{N}(p)$$r_{\pi_{p}}$$\pi_{p}’\in \mathrm{R}\mathrm{e}\mathrm{p}^{\mathrm{W}\mathrm{B}}(A,p)$.

By Proposition 3.1, (3) and Proposition 3.4 we have the following
Corollary 3.5. Suppose$A$ . Thenis maximal.

$\pi$

is a strongly nondegenerate *-representation of$r_{\pi}$

For the relation of -representation$*$

Proposition 3. 6. Suppose

seminorm on

$A$

and

$p$

$\pi_{p}$and$\pi_{r_{\pi_{P}}}^{N}$

we have the following

is a weakly semifmite unbounded$\pi_{p}\in{\rm Re}_{\mathrm{P}^{\mathrm{w}\mathrm{B}}}(A,p)$. Then$\sim$$\pi_{p}\subset\pi_{r_{\pi_{p}}}^{N}$and$\pi_{r_{n_{\rho}}}^{N}=\pi_{p}$.

$C^{*}-$

61

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Soc., 15(1965), 399-421.

3. S. J. Bhatt, A.-Inoue and H. Ogi, On-spectral algebras, to appear in Rend.
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