YOKOHAMA MATHEMATICAL
JOURNAL VOL. 34, 1986
ON LOCALLY
$W^{*}$
-ALGEBRAS
By
MARIA FRAGOULOPOULOU
(Received March 25, 1985; Revised October 1, 1985)
We consider in this paper a more general class than that of the classical
algebras; namely the class of locally
-algebras defined as inverse limits of
algebras (Definition 1.1). Among the examples of this sort of algebras we work out, is
the locally
-algebra $L(H),$ $H$ a ’‘locally Hilbert space” [8; Section 5]. As it is
apparent from what follows, $L(H)$ represents, in effect, the most general case of a
locally
-algebra. That is, defining the respective -weak (operator) topology on
$L(H)$ , we prove that every locally
-algebra $E$ equipped with the inverse limit
(Proposition
topology
1.3) coincides (within an isomorphism of topological
algebras) with a -weakly closed -subalgebra of some $L(H),$ $H$ a locally Hilbert
space (Theorem 2.1). In this respect, one gets that the center $Z(E)$ of a locally
algebra $E$, is a $\sigma- closed*$ -subalgebra of
, hence also a locally
-algebra (cf.
Corollary 2.2 and Example 1.4.1, Proposition). On the other hand, one always
obtains that every locally
-algebra admits an ”Arens-Michael-type decomposition” consisting of
-aIgebras (Proposition 1.3). An application of the above
gives information concerning the inner derivations of a locally
-algebra $E$. More
$E$
precisely, one has that each inner derivation
, of is an inverse limit of
inner derivations
, corresponding to the
-algebra factors
of $E$.
$E$
Thus, the set
of all inner derivations of , becomes a complete locally convex
spaoe (Theorem 3.1), so that if
, is a defining family of seminorms for
,
one gets that for every
$W^{*}-$
$W^{*}$
$W^{*}-$
$C^{*}$
$W^{*}$
$\sigma$
$W^{*}$
$\sigma$
$*$
$\sigma$
$W^{*}-$
$W^{*}$
$(E, \sigma)$
$W^{*}$
$W^{*}$
$W^{*}$
$\delta_{x},$
$\delta_{x_{a}},$
$\alpha\in A$
$x=(x_{\alpha})\in E$
$W^{*}$
$\overline{E}_{\alpha}^{\sigma_{\alpha}},$
$\alpha\in A$
$\delta_{0}(E)$
$(q_{\alpha}),$
$\alpha\in A$
$\delta_{0}(E)$
$\delta_{x}\in\delta_{O}(E)$
$q_{\alpha}(\delta_{x})=2\inf\{p_{\alpha}(x-z):z\in Z(E)\}$
,
$\alpha\in A$
,
, is a family of (submultiplicative)
where
-seminorms defining the
$E$
topology of . The latter extends in our case a previous result of L. Zsid\’o [16]
conceming an estimation of the norm of an (inner) derivation acting on an abstract
-algebra. Similar estimations referred to (inner) derivations of the
-algebra
of all bounded linear operators on a Hilbert space $H$, or of a
-algebra
acting on a separable Hilbert space, have been also given earlier by J. G. Stampfli [14]
and P. Gajendragadkar [5] respectively. Furthermore we use Zsid6’s technique of
Theorems 1 and 2 in [16] in order to estimate the above numbers
, by
means of a certain map
, at the cost however of some particular
restriction on the locally
-algebra $E$ (Theorem 3.5). A further application of the
latter leads to a new information even for the normed case, according to which
$(p_{\alpha}),$
$\alpha\in A$
$C^{*}$
$W^{*}$
$C^{*}$
$\mathscr{L}(H)$
$W^{*}$
$q_{\alpha}(\delta_{x}),$
$\Phi:E\rightarrow Z(E)$
$W^{*}$
$\alpha\in A$
M. FRAGOULOPOULOU
36
[ ; Theor. 3] is, in effect, a relatively open
Zsido’s continuous map
(Theorem 4.3).
section of a continuous open linear surjection
$16$
$v:\delta_{0}(E)\rightarrow E$
$u:E\rightarrow\delta_{0}(E)$
1. The topological algebras considered throughout are all over the field $C$ of
complex numbers and have an identity element.
-algebra $E$ endowed with a
By an lmc (locally $m$ -convex)*-algebra, we mean
(a directed index set), of -preserving
topology defined by a family
submultiplicative seminorms. A complete lmc -algebra $E$ is called a locally
$x\in E$ . Given an lmc
, for any
algebra [8] if, in addition,
the completion of the normed algebra
and denote by
algebra $E$, set
$x\in E$ and
(cf.
, with norm defined by
, within a topological algebraic
[1], [11]). Then, if $E$ is complete one gets
$a^{*}$
$*$
$\alpha\in A$
$(p_{\alpha}),$
$*$
$C^{*}-$
$*-$
$\alpha\in A,$
$p_{\alpha}(x)^{2}=p_{\alpha}(x^{*}x)$
$\tilde{E}_{\alpha}$
$N_{\alpha}\equiv ker(p_{\alpha})$
$\dot{p}_{\alpha}(x_{\alpha})=p_{\alpha}(x),$
$\dot{p}_{\alpha}$
$E_{\alpha}\equiv E/N_{\alpha}$
$\alpha\in A$
$x_{\alpha}=x+N_{\alpha}\in E_{\alpha},$
$E=\varliminf_{\alpha}\tilde{E}_{\alpha}$
isomorphism (ibid.), the latter expression being called an Arens-Michael decomis always
, if $E$ is a locally $C^{*}- algebra,$
position of $E$ [ ; Def. III, 3.1]. In
(cf. [13; Folg. 5.4]), a
for every
-algebra, i.e.,
complete, hence a
$E$ has no unit (cf. [4; Prop. 2.1, (ii)]), $E$ given thus
fact being actually true even if
as an inverse limit of -algebras.
-algebras, where the
, be an inverse system of
Now, let
with respect to the
,
continuous
are considered
in
connecting maps
into
. The canonical map of the inverse limit
uniform (norm) topology of
$E_{\alpha}$
$10$
$P_{\sim}^{articular}$
$C^{*}$
$\alpha\in A$
$E_{\alpha}=E_{\alpha}$
$C^{*}$
$(F_{\alpha},f_{\alpha\beta}),$
$f_{\alpha\beta},$
$W^{*}$
$\alpha\in A$
$A$
$\alpha\leq\beta$
$F_{\alpha}’ s$
$\varliminf_{\alpha}F_{\alpha}$
$F_{\alpha}$
, will be
denoted by
. In this regard, we now set the next.
Mallios). An algebra is said to be a locally
$f_{\alpha},$
$\alpha\in A$
Deflnition 1.1 (A.
is given as an inverse limit of W*-algebras, i.e.,
$E$
$W^{*}$
-algebra, if it
, is a
, where each
$W^{*}$
$F_{\alpha},$
$E=\varliminf_{\alpha}F_{\alpha}$
$\alpha\in A$
-algebra.
-algebra equipped with
-algebra $E$ is a locally
Scholium 1.2. Every locally
-algebra
the inverse limit topology , induced on it by the uniform topology of its
is given exactly
. In addition, the Arens-Michael decomposition of
factors
.
by the *-subalgebras
of
,
-norm defining the uniform topology of
denotes the
In fact, if
one gets by [10; Lemma III, 3.2] that
$C^{*}$
$W^{*}$
$W^{*}$
$\tau$
$F_{\alpha},$
$(E, \tau)$
$\alpha\in A$
$f_{\alpha}(E)$
$F_{\alpha}’ s,$
$\alpha\in A$
$C^{*}$
$\Vert\cdot\Vert_{\alpha}$
(1.1)
$F_{\alpha},$
$\varliminf_{\alpha}F_{\alpha}=(E, \tau)=\varliminf_{a}f_{\alpha}(E)=\varliminf_{\alpha}\overline{f_{\alpha}(E)}$
where
“–,,
(1.2)
means
$\Vert\cdot\Vert_{\alpha}$
$\alpha\in A$
,
-closure. Now the relation
$p_{\alpha}=\Vert\cdot\Vert_{\alpha}\circ f_{\alpha}$
,
$\alpha\in A$
,
-seminorm on $E$, and it is clear by (1.1)
defines a ( -preserving submultiplicative)
-algebra. Now,
, and makes $E$ into a locally
that is defined by the family
, one has by (1.2) that
, of
considering the Arens-Michael factor
$*$
$\tau$
$C^{*}$
$(p_{\alpha}),$
$C^{*}$
$\alpha\in A$
$E_{\alpha},$
$\alpha\in A$
$\dot{p}_{\alpha}(x_{\alpha})=p_{\alpha}(x)=\Vert f_{\alpha}(x)\Vert_{\alpha}$
,
$(E, \tau)$
$x\in E$
,
$\alpha\in A$
,
ON LOCALLY
$W^{*}$
-ALGEBRAS
consequently the map
(1.3)
$E_{\alpha}\equiv E/N_{\alpha}\rightarrow f_{\alpha}(E):x_{\alpha}-\rightarrow f_{\alpha}(x)$
,
$x\in E$
,
$\alpha\in A$
,
(cf. discussion before
is a topological algebraic isomorphism, and since
. Thus, we finally have
Definition 1.1), one also gets
$E_{\alpha}=\tilde{E}_{\alpha},$
$\alpha\in A$
$f_{\alpha}(E)=\overline{f_{\alpha}(E)}^{-}\alpha\in A$
(1.4)
$\varliminf_{a}(F_{\alpha}, \Vert\cdot\Vert_{\alpha})=(E, \tau)=\varliminf_{\alpha}(E_{\alpha}\cong f_{\alpha}(E),\dot{p}_{\alpha})$
,
within isomorphisms of topological algebras.
,
Note that, because of (1.3), the connecting maps of the inverse system
. Thus, from now on, we agree to
, on
coincide with the restrictions of
, for the connecting maps, respectively, canonical
keep the symbols
, as well.
maps of the inverse system
, there is a Banach
, for each
Now, given a locally
-algebra
$(E_{\alpha}),$
$f_{\alpha\beta},$
$f_{\alpha\beta},$
$\alpha\leq\beta$
$\alpha\in A$
$f_{\beta}(E)$
$\alpha\in A$
$\alpha\leq\beta,f_{\alpha},$
$\alpha\in A$
$(E_{\alpha}),$
$\alpha\in A$
$W^{*}$
$E=\varliminf_{\alpha}F_{\alpha}$
the
[ ; Defs. 1.1.2, 1.1.3]. We denote by
space
the dual of which is
-algebra
the
and by
on
weak -topology
. In this respect, we now have the next.
endowed with
$F_{\alpha},$
$M_{*}^{\alpha}$
$\alpha\in A$
$12$
$\sigma_{\alpha}$
$*$
$F_{\alpha}\cong(M_{*}^{\alpha})^{*}$
$\sigma((M_{*}^{\alpha})^{*}, M_{*}^{\alpha})$
$W^{*}$
$(M_{*}^{\alpha})_{\sigma_{\alpha}}^{*}$
$F_{\alpha}$
$\alpha\in A$
$\sigma_{\alpha},$
Proposition 1.3.
Let
$E=\varliminf F_{\alpha}$
be a locally
$W^{*}$
-algebra, in such a way that the
$\alpha$
, are weakly
$E$
,
than the
Then,
coarser
topology
is
with
inverse
limit
endowed
the
-continuous.
admits an Arens-topology (cf. Scholium 1.2). In addition,
inverse limit lmc
-algebras, in the sense that
Michael-type decomposition consisting of
connecting maps
$f_{\alpha\beta},$
$\alpha\leq\beta$
in
$A$
,
of the respective inverse system
$(F_{\alpha}),$
$\alpha\in A$
$*$
$\sigma$
$C^{*}$
$(E, \sigma)$
$\tau$
$W^{*}$
(1.5)
$(E, \sigma)=\varliminf_{\alpha}\overline{E}_{\alpha^{\alpha}}^{\sigma}$
within a topological algebraic isomorphism, where
,
’
$-\sigma_{\alpha}$
means
$\sigma_{\alpha}$
-closure in
$F_{\alpha},$
.
one
$\alpha\in A$
,
Proof. By the above comments and the weak -continuity of
, is a projective system of topological vector spaces,
concludes that
in such a way that
$*$
$f_{\alpha\beta},$
$((M_{*}^{\alpha})_{\sigma_{\alpha}}^{*},f_{\alpha\beta}),$
$\alpha\leq\beta$
$\alpha\leq\beta$
(1.6)
$E=\varliminf_{\alpha}(M_{*}^{\alpha})_{\sigma_{\alpha}}^{*}$
,
within an isomorphism of vector spaces. So that by (1.6) $E$ is obviously equipped with
-topology
, which is coarser than the lmc
the inverse limit topology
$C^{*}$
$\sigma=\varliminf\sigma_{\alpha}$
$\tau$
$\alpha$
defined by the family
diagram
$(p_{\alpha}),$
$\alpha\in A$
(cf. (1.2)),
as this follows by the next commutative
$id_{E}(E,\sigma)(E,\tau)\downarrow\rightarrow(F_{\alpha}, \sigma_{\alpha})=(M_{*}^{\alpha})_{\sigma_{\alpha}}^{*}\rightarrow(F\alpha_{I^{id_{F_{\alpha}}}}||\cdot\Vert_{\alpha})f_{\alpha}f_{\alpha}$
38
M. FRAGOULOPOULOU
and the fact that
on for every
.
Now, denote by
the -closure of into
(cf. Scholium 1.2). Then,
-subalgebra of the
[ ; Def.
-algebra as a
-algebra
is a
1.1.4]. On the other hand, since
are weakly *-continuous, they are uniquely
;
, in such a way that
extended to
, is a projective
$\sigma_{\alpha}\leq\Vert\cdot\Vert_{\alpha}$
$\alpha\in A$
$F_{\alpha}$
$\overline{E}_{\alpha^{\alpha}}^{\sigma}$
$E_{\alpha}$
$F_{\alpha},$
$\sigma_{\alpha}$
$W^{*}$
$\overline{E}_{\alpha^{\alpha}}^{\sigma}$
$f_{\alpha\beta},$
$f_{\alpha\beta}$
$\overline{E}_{\beta}^{\sigma\rho}\rightarrow\overline{E}_{\alpha^{\alpha}}^{\sigma},$
system too. Thus,
$\alpha\in A$
$W^{*}$
$\sigma_{\alpha}- closed^{*}$
$F_{\alpha},$
$\alpha\in A,$
$12$
$\alpha\leq\beta$
$\alpha\leq\beta$
$(\overline{E}_{\alpha}^{\sigma_{\alpha}},f_{\alpha\beta}),$
$\alpha\in A$
one gets
$E=\varliminf_{\alpha}E_{\alpha}\subset\varliminf_{\alpha}\overline{E}_{\alpha^{\alpha}}^{\sigma}\subset\rightarrow\varliminf_{\alpha}F_{\alpha}=E$
,
which implies
$(E, \sigma)=\varliminf_{\alpha}\overline{E}_{\alpha^{\alpha}}^{\sigma}$
,
within an isomorphism of topological algebras.
$\square $
Motivated by the above when we speak in the sequel about the -topology of
-algebra, we shall always mean that the connecting maps of the
a locally
respective inverse system are weakly *-continuous.
$\sigma$
$W^{*}$
1.4. Examples of locally
1.
Proof.
.
-algebras.
The first example of a locally
Proposition.
is also a locally
$(E, \tau)$
$W^{*}$
Every
-algebra.
$\sigma- closed^{*}$
$W^{*}$
-algebra is given by the next.
-subalgebra
$G$
of a locally
$W^{*}$
-algebra
$E=\varliminf F_{\alpha}$
$W^{*}$
Sinoe
$\sigma\leq\tau$
on
$E,$
,
$q$
$G$
is also -closed, hence a locally
$C^{*}$
$\tau$
-subalgebra of
Thus, (cf. [1], [11])
$(G, \tau|_{G})=\varliminf_{\alpha}G_{\alpha}$
,
I
within a topological algebraic isomorphism, where
(
),
(cf.
1.1).
algebra
discussion before Definition
In particular,
is -closed one gets by [10; Lemma III, 3.2]
(1.3)), so that, since
$G_{\alpha}\equiv G/ker$
$p_{\alpha}$
$\alpha\in A$
$G$
$G_{\alpha}\cong f_{\alpha}(G),$
$G$
a
$C^{*}-$
$\alpha\in A$
(cf.
, is
$\sigma$
(1.7)
$(G, \sigma|_{G})=\varliminf_{\alpha}\overline{G}_{\alpha}^{\sigma_{\alpha}}$
,
within an isomorphism of topological algebras, where each
-subalgebra of the
-algebra
.
$W^{*}$
$\sigma_{\alpha}- closed^{*}$
$\overline{E}_{\alpha}^{\sigma_{\alpha}},$
$\alpha\in A$
$\overline{G}_{\alpha^{\alpha}}^{\sigma}$
is a
$W^{*}$
-algebra as a
$\square $
$n\in N$ (natural numbers), be
2. Let
a descending sequence of
-algebras
with non-trivial intersection (cf. [3]). Let also that uniform, respectively, weak
$n\in N$,
topologies on
form an ascending sequence (for any $n\leq m$ in $N$,
$(E_{n}),$
$W^{*}$
$*-$
$E_{n},$
$\Vert\cdot\Vert_{n}|_{E_{m}}\leq\Vert\cdot\Vert_{m}$
,
as well as
$\sigma_{n}|_{E_{m}}\leq\sigma_{m}$
). Then,
$E\equiv\bigcap_{n}E_{n}$
,
ON LOCALLY
is a locally
$W^{*}$
$W^{*}$
-ALGEBRAS
-algebra. It is easily seen that
$\varliminf_{n}E_{n}=\bigcap_{n}E_{n}$
,
within an algebraic isomorphism. Moreover, the canonical injections
:
$j_{nm}$
$E_{m}\subset-E_{n}$
,
are norm, respectively, weakly*-continuous, so that
limit topologies
a locally
$W^{*}$
$\tau=\varliminf_{n}\Vert\cdot\Vert_{n},$
$\sigma=\varliminf_{n}\sigma_{n}$
in
$n\leq m$
, where
$\sigma\leq\tau$
$N$
is endowed with the inverse
$\varliminf_{n}E_{n}$
since
,
$\sigma_{n}\leq\Vert\cdot\Vert_{n},$
$n\in N$
. Thus,
$E$
is
-algebra by Definition 1.1. In particular (cf. also Proposition 1.3),
$\varliminf_{n}(E_{n}, \sigma_{n})=(E, \sigma)\cong\varliminf_{n}\overline{F}_{n}^{\sigma_{n}}$
,
where
$F_{n}\equiv(E, \Vert\cdot\Vert_{n|E})$
is a
a
$W^{*}$
,
$n\in N$
,
-algebra corresponding to the Arens-Michael decomposition of
$n\in N$.
-subalgebra of
$C^{*}$
3.
$\langle$
,
$(E, \tau)$
, and
$\overline{F}_{n^{n}}^{\sigma}$
$E_{n},$
Let
$\rangle_{\lambda}=\langle, \rangle_{\mu}$
$(H_{\lambda}),$
on
and
a directed family of Hilbert spaces, with
in . Then,
endowed with the respective
for any
$\lambda\in\Lambda$
$H_{\lambda}$
, be
$H_{\lambda}\subset H_{\mu}$
$\Lambda$
$\lambda\leq\mu$
$H=\rightarrow\lim_{\lambda}H_{\lambda}$
locally convex inductive limit topology, is called a locally Hilbert space (cf. [8; Def.
5.2]). Thus, if
$L(H)=\{T\in \mathscr{L}(H)$
: for every
$\lambda\leq\mu|_{\Lambda},$
, and
$T_{\mu}\circ i_{\mu\lambda}=i_{\mu\lambda}\circ T_{\lambda}$
. where
the canonical injection of
$T_{\lambda}=\tau|_{H_{\lambda}}$
into
},
$L(H)$ is, in fact, a locally
-algebra (cf. [8; Prop. 5.1]). In particular, $L(H)$ is a locally
-algebra, in such a way that the connecting maps of the inverse system of
algebras corresponding to $L(H)$ , are weakly -continuous. The topology of $L(H)$ is
,
defined by a family of ( -preserving submultiplicative)
-seminorms
$\in \mathscr{L}(H_{\lambda}),$
$\lambda\in\Lambda$
$i_{\mu\lambda}$
$H_{\lambda}$
$H_{\mu}$
$C^{*}$
$W^{*}$
$W^{*}-$
$*$
$*$
$C^{*}$
$(p_{\lambda}),$
$\lambda\in\Lambda$
(ibid). Now, considering the
,
such that
-algebra
corresponding to the Arens-Michael decomposition of $L(H)$ , we conclude that the
$p_{\lambda}(T)=\Vert T_{\lambda}\Vert,$
$C^{*}$
$\lambda\in\Lambda$
$L(H)/N_{\lambda},$
$\lambda\in\Lambda$
map
$L(H)_{\lambda}\equiv L(H)/N_{\lambda}\rightarrow \mathscr{L}(H_{\lambda}):T+N_{\lambda}-\rightarrow T_{\lambda}=\tau|_{H_{\lambda}}$
,
$\lambda\in\Lambda$
,
is an isomorphism of topological algebras, so that
(1.8)
$L(H=\rightarrow\lim_{\lambda}H_{\lambda})=\varliminf_{\lambda}\mathscr{L}(H_{\lambda})$
,
within a topological algebraic isomorphism, where each
, is a
algebra; hence, by Definition 1.1 $L(H)$ is a locally
-algebra.
We shall now show that the connecting maps
in , of the inverse system
,
, are weakly *-continuous. For each
$\mathscr{L}(H_{\lambda}),$
$\lambda\in\Lambda$
$W^{*}-$
$W^{*}$
$f_{\lambda\mu},$
$(L(H)_{\lambda}\cong \mathscr{L}(H_{\lambda})),$
$\lambda\in\Lambda$
$\lambda\leq\mu$
$\Lambda$
$\lambda\in\Lambda \mathscr{L}(H_{\lambda})=(\mathscr{L}_{*}(H_{\lambda}))^{*}$
40
M. FRAGOULOPOULOU
the dual of the compact operators of
.
Moreover, the weak -topology
, is the -weak (operator)
, defined by the following family of seminorms
topology of
with
$\mathscr{L}(H_{\lambda}),$
$\mathscr{L}_{*}(H_{\lambda})=\mathscr{L}\mathscr{C}(H_{\lambda})^{*}$
$*$
$\lambda\in\Lambda$
$\sigma(\mathscr{L}(H_{\lambda}), \mathscr{L}_{*}(H_{\lambda})),$
$\lambda\in\Lambda$
$\sigma$
$\mathscr{L}(H_{\lambda})$
(1.9)
$p_{\langle\xi_{\lambda}^{*}),\langle\eta_{\lambda}^{n})}(T_{\lambda})=|\sum_{n=1}^{\infty}\langle T_{\lambda}\xi_{\lambda}^{n}, \eta_{\lambda}^{n}\rangle_{\lambda}|$
,
$T_{\lambda}\in \mathscr{L}(H_{\lambda})$
,
$\lambda\in\Lambda$
,
,
in
with
,
for any sequences
[ ; p. 67],
. Thus, if
is a net in
such that
with respect to
, then since
, one also gets that
and ,
respect
to
in
with
.
By the above and Proposition 1.3 we now have that $L(H)$ is endowed with the
inverse limit topology
$(\xi^{n})$
$15$
$\lambda\in\Lambda$
$(\eta_{A}^{n})$
$\sum_{n=1}^{\infty}\Vert\xi_{\lambda}^{n}\Vert^{2}<\infty$
$H_{\lambda}$
$(T_{\mu}^{\alpha})$
$\sigma(\mathscr{L}(H_{\mu}), \mathscr{L}_{*}(H_{\mu}))$
$\mathscr{L}(H_{\mu})$
$T_{\lambda}^{\alpha}=T_{\mu}^{\alpha}$
I
(1.10)
$H_{\lambda}$
The topology
$\sigma$
on
$L(H)$
with
,
$\sigma=\varliminf_{\lambda}\sigma_{\lambda}$
$T_{\mu}^{\alpha}\rightarrow 0$
$\rangle_{\lambda}=\langle, \rangle_{\mu}|_{H_{\lambda}}$
$\langle$
$\sigma(\mathscr{L}(H_{\lambda}), \mathscr{L}_{*}(H_{\lambda})),$
$T_{\lambda}^{\alpha}\rightarrow 0$
$\sum_{n=1}^{\infty}\Vert\eta_{\lambda}^{n}\Vert^{2}<\infty$
$\Lambda$
$\lambda\leq\mu$
$\sigma_{\lambda}\equiv\sigma(\mathscr{L}(H_{\lambda}), \mathscr{L}_{*}(H_{\lambda}))$
.
will be called, in the sequel, -weak (operator) topology.
$\sigma$
2. In this Section we shall show that every locally
-algebra equipped with
(Proposition
1.3)
(within
topology
the inverse limit
is identified
a topological algebraic isomorphism) with a -weakly closed *-subalgebra of some $L(H),$ $H$ a locally
Hilbert space (Theorem 2.1). The latter constitutes in our case, an analogue of the
respective classical situation for
-algebras (cf., for instance, [15; Theor. III’, 3.5]).
$W^{*}$
$E$
$\sigma$
$\sigma$
$W^{*}$
-algebra $E$ endowed with the inverse limit
Theorem 2.1. Every locally
topology coincides, within an isomorphism of topological algebras, with a -weakly
closed *-subalgebra of some $L(H),$ $H$ a locally Hilbert space.
$W^{*}$
$\sigma$
$\sigma$
Proof.
$E=\varliminf_{\backslash }F_{\alpha}$
,
where each
$F_{\alpha},$
$\alpha\in A$
, is
a
$W^{*}$
-algebra, hence [15; Theor.
$\alpha$
III, 3.5] there is a faithful representation
(2.1)
$H_{\alpha},$
$\alpha\in A$
$\varphi_{\alpha}$
,
:
$F_{\alpha}\rightarrow \mathscr{L}(H_{\alpha})$
,
$\alpha\in A$
,
a Hilbert space, bicontinuous with respect to the topologies
$\sigma_{\alpha}\equiv\sigma(F_{\alpha}, M_{*}^{\alpha})$
,
, respectively (cf. comments before Proposition
of
1.3 and (1.10)). Now, set (cf. also [8; Theor. 5.1])
$\sigma(\mathscr{L}(H_{\alpha}), \mathscr{L}_{*}(H_{\alpha}))$
$F_{\alpha},$
$\mathscr{L}(H_{\alpha}),$
$\alpha\in A$
$\ovalbox{\tt\small REJECT}_{\lambda}=\bigoplus_{\alpha\leq\lambda}H_{\alpha}$
,
where $\oplus means$ orthogonal direct sum. Then, for any
,
, so that
on
$\langle$
$\rangle_{\mu}$
$\lambda\leq\mu\ovalbox{\tt\small REJECT}_{\lambda}\subset\ovalbox{\tt\small REJECT}_{\mu}$
$\ovalbox{\tt\small REJECT}_{\lambda}$
$H=\varliminf_{\lambda}\ovalbox{\tt\small REJECT}_{\lambda}$
,
is a locally Hilbert space (cf. Example 1.4.3). In this regard, define
$\varphi:E\rightarrow L(H)\cong\lim_{\overline{\lambda}}\mathscr{L}(\ovalbox{\tt\small REJECT}_{\lambda}):x\mapsto\varphi(x):\varphi(x)|_{x_{\lambda}}=\varphi(x)_{\lambda}$
,
and
$\langle$
,
$\rangle_{\lambda}=$
ON LOCALLY
$W^{*}$
-ALGEBRAS
with
$\varphi(x)_{\lambda}(\xi_{\lambda})=\bigoplus_{\alpha\leq\lambda}\varphi_{\alpha}(x_{\alpha}X\xi_{\alpha})$
for every
,
in
$\xi_{\lambda}=(\xi_{\alpha})_{\alpha\leq\lambda}$
$\ovalbox{\tt\small REJECT}_{\lambda}$
.
It is easily seen that is a 1-1 algebraic morphism. On the other hand, if
canonical map of $L(H)$ onto
is continuous if, and only if, each
$E$
is a net in with
we have to show that
continuous. Thus, if
$\varphi$
$\mathscr{L}(\ovalbox{\tt\small REJECT}_{\lambda}),$
$\varphi$
$(x_{\delta})$
$ x_{\delta}\rightarrow 0\sigma$
$\pi_{\lambda}(\varphi(x_{\delta}))=$
$\lambda$
$(\xi_{\lambda}^{n}),$
$(\eta_{\lambda}^{n})$
in
$\ovalbox{\tt\small REJECT}_{\lambda}$
,
is the
is
$\pi_{\lambda}\circ\varphi$
’
for all . According to the definition of
$\varphi(x_{\delta})_{\lambda}\rightarrow 0\sigma_{\lambda}$
$\pi_{\lambda}$
$\sigma_{\lambda}$
(cf. (1.9)), for any
sequences
such that
(2.1)
$\sum_{n=1}^{\infty}\Vert\xi_{\lambda}^{n}\Vert^{2}<\infty$
,
$\sum_{n=1}^{\infty}\Vert\eta_{\lambda}^{n}\Vert^{2}<\infty$
,
we must show
(2.2)
$|\sum_{n=1}^{\infty}\langle\varphi(x_{\delta})_{\lambda}\xi_{\lambda}^{n}, \eta_{\lambda}^{n}\rangle_{\lambda}|\rightarrow 0$
.
But,
(2.3)
where
$|\sum_{n=1}^{\infty}\langle\varphi(x_{\delta})_{\lambda}\xi_{\lambda}^{n}, \eta_{\lambda}^{n}\rangle_{\lambda}|=|\sum_{n=1}^{\infty}\sum_{\alpha\leq\lambda}\langle\varphi_{\alpha}(x_{\alpha}^{\delta})\xi_{\lambda,\alpha}^{n}, \eta_{\lambda.\alpha}^{n}\rangle_{\alpha}|$
$(\xi_{\lambda,\alpha}^{n}),$
$(\eta_{\lambda,\alpha}^{n})$
are sequences in
(2.4)
$\alpha\leq\lambda$
$\sum_{n=1}^{\infty}\Vert\xi_{\lambda.\alpha}^{n}\Vert^{2}<\infty$
On the other hand,
$\alpha\in A$
$\varphi_{\alpha},$
$x_{\delta}\rightarrow 0\sigma\Leftrightarrow x_{\alpha}^{\delta}\rightarrow 0\sigma_{\alpha}$
, such that
$\sum_{n=1}^{\infty}\Vert\eta_{\lambda,\alpha}^{n}\Vert^{2}<\infty$
.
for all , which by the weak *-continuity of
$\alpha$
, yields that
(2.5)
for all
$H_{\alpha},$
,
$\varphi_{\alpha}(x_{\alpha}^{\delta})\rightarrow 0$
$\alpha\in A$
.
But, (2.5)
,
with respect to
$\sigma(\mathscr{L}(H_{\alpha}), \mathscr{L}_{*}(H_{\alpha}))$
means that for any sequences as in
$|\sum_{n=1}^{\infty}\langle\varphi_{\alpha}(x_{\alpha}^{\delta})\xi_{\lambda,\alpha}^{n}, \eta_{\lambda,\alpha}^{n}\rangle_{\alpha}|\rightarrow 0$
,
,
(2.4)
$\alpha\leq\lambda$
.
Thus, (2.2) follows now from the fact that the right-hand side of (2.3) is less than or
equal to
.
Conversely, let
be a net in $\varphi(E)\subset L(H)$ , with
$\sum_{\alpha\leq\lambda}|\sum_{n=1}^{\infty}<\varphi_{\alpha}(x_{\alpha}^{\lambda})\xi_{\lambda,\alpha}^{n},$
$\eta_{\lambda,\alpha}^{n}>_{\alpha}|$
$(\varphi(x_{\delta}))$
(2.6)
where
$\varphi(x_{\delta})\rightarrow 0\sigma$
$\sigma$
is now given by (1.10). We must show that
$(f_{\alpha}\circ\varphi^{-1})(\varphi(x_{\delta}))=x_{\alpha}^{\delta}\rightarrow 0$
$\sigma_{\alpha}$
or equivalently
,
for all
$\alpha\in A$
,
42
M. FRAGOULOPOULOU
(2.7)
$\varphi_{\alpha}(x_{\alpha}^{\delta})\rightarrow 0$
By (2.6) we have that
sequences
for all sequences
$(\xi_{\lambda}^{n}),$
$(\eta_{\lambda}^{n})$
,
$\sigma(\mathscr{L}(H_{\alpha}), \mathscr{L}_{*}(H_{\alpha}))$
,
$\alpha\in A$
for all , which means that (2.2) is true for all
$\ovalbox{\tt\small REJECT}_{\lambda}$
$(\eta_{\alpha}^{n})$
$(\xi_{\alpha}^{n}),$
fulfilling (2.1). Moreover, (2.7) will have been proved, if
, one gets
in
with
,
$\sum_{n=1}^{\infty}\Vert\xi_{\alpha}^{n}\Vert^{2}<\infty,$
$H_{\alpha}$
$|\sum_{n=1}^{\infty}\langle\varphi_{\alpha}(x_{\alpha}^{\delta})\xi_{\alpha}^{n}, \xi_{\alpha}^{n}\rangle_{\alpha}|\rightarrow 0$
,
$\sum_{n=1}^{\infty}\Vert\eta_{\alpha}^{\hslash}\Vert^{2}<\infty$
$\alpha\in A$
.
, so that the sequences
is imbedded to some
But each
Thus, (cf. also (2.3))
(2.1).
which satisfy
define sequences
$H_{\alpha}$
$\ovalbox{\tt\small REJECT}_{\lambda},$
$(\xi_{\lambda}^{n}),$
$\alpha\leq\lambda$
$(\xi_{\alpha}^{n}),$
$(\eta_{\alpha}^{n})$
as before,
$(\eta_{\lambda}^{n})$
$|\sum_{n=1}^{\infty}\langle\varphi_{\alpha}(x_{\alpha}^{\delta})\xi_{\alpha}^{n}, \eta_{\alpha}^{n}\rangle_{\alpha}|=|\sum_{n=1}^{\infty}\langle\varphi(x_{\delta})_{\lambda}\xi_{\lambda}^{n}, \eta_{\lambda}^{n}\rangle_{\lambda}|\rightarrow 0$
$\alpha\leq\lambda$
.
$\lambda$
$\varphi(x_{\delta})_{\lambda}\rightarrow 0\sigma_{\lambda}$
in
(2.8)
with respect to
,
, which proves (2.8).
We now show that
$\varphi(E)$
is -weakly closed in
$\sigma$
$L(H)$
.
Let
,
$\varphi(x_{\delta})$
be a net in
$\varphi(E)$
with
$\varphi(x_{\delta})\rightarrow T\in L(H)\sigma$
Then,
$\varphi(x_{\delta})_{\lambda}\rightarrow T_{\lambda}\sigma_{\lambda}$
the notation
$\sigma_{\alpha}$
for
’
for all
$\lambda$
(cf. (1.10)),
$\sigma(\mathscr{L}(H_{\alpha}), \mathscr{L}_{*}(H_{\alpha})),$
$\varphi_{\alpha}(x_{\alpha}^{\delta})\rightarrow T_{\alpha}\sigma_{\alpha}$
where
$T_{\alpha}\in\varphi_{\alpha}(F_{\alpha}),$
$\alpha\in A$
where, in particular,
, since
$f_{\alpha\beta}(y_{\beta})=y_{\alpha}$
$\alpha\in A$
for all
,
is
, for any
$\varphi_{\alpha}(F_{\alpha})$
so that applying the argument of (2.3) and
, too, we conclude that
$\sigma_{\alpha}$
$\alpha\in A$
,
-closed. Thus,
in . On the other hand,
$T_{\alpha}=\varphi_{\alpha}(y_{\alpha}),$
$y_{\alpha}\in F_{\alpha},$
$\alpha\in A$
,
$A$
$\alpha\leq\beta$
$y_{\alpha}=\varphi_{\alpha}^{-1}(\varphi_{\alpha}(y_{\alpha}))=\varphi_{\alpha}^{-1}(\lim_{\delta^{\alpha}}^{\sigma}\varphi_{\alpha}(x_{\alpha}^{\delta}))=\lim_{\delta^{\alpha}}^{\sigma}x_{\alpha}^{\delta}$
,
for every
$\alpha\in A$
.
Hence,
$\lim\delta\delta\sigma x=y\in E$
and
$T=\lim_{\delta}^{\sigma}\varphi(x_{\delta})=\varphi(\lim_{\delta}^{\sigma}x_{\delta})=\varphi(y)\in\varphi(E)$
Corollary 2.2. The center
algebra. In particular,
(2.9)
$Z(E)$
of a locally
$W^{*}$
-algebra
$E$
$(E_{\alpha}),$
$\alpha\in A$
$\square $
is also a locally
$(Z(E), \sigma|_{Z\langle E)})=Z((E, \sigma)=\varliminf_{\alpha}\overline{E}_{\alpha^{\alpha}}^{\sigma})=\varliminf_{\alpha}Z(\overline{E}_{\alpha^{\alpha}}^{\sigma})$
within a topological algebraic isomorphism, where
Michael decomposition of $E$.
.
$W^{*}-$
,
, corresponds to the Arens-
Proof. It is easily seen that multiplication of $L(H),$ $H$ a locally Hilbert space, is
separately continuous with respect to the -weak topology. Thus, by the preceding
theorem, one also gets that multiplication of $E$ is separately continuous with respect
,
to . A consequence of the latter is that $Z(E)$ is a $\sigma- closed*$ -subalgebra of
-algebra according to Example 1.4.1, Proposition. Moreover,
therefore a locally
$\sigma$
$(E, \sigma)$
$\sigma$
$W^{*}$
ON LOCALLY
$W^{*}$
-ALGEBRAS
43
by (1.7) (cf. also Proposition 1.3)
(2.10)
$(Z(E), \sigma|_{Z\langle E)})=\varliminf_{\alpha}\overline{Z(E)_{\alpha}}^{\sigma_{\alpha}}$
,
within an isomorphism of topological algebras, where
Michael decomposition of $Z(E)$ , as a locally
-subalgebra of
$\alpha\in A$
,
is the Arens-
$(E, \tau)$
.
Now,
$(Z(E)_{\alpha}),$
$C^{*}$
(2.11)
$Z(E)_{\alpha}=Z(E_{\alpha})$
,
$\alpha\in A$
,
within the topological algebraic isomorphism
$Z(E)_{\alpha}\rightarrow Z(E_{\alpha});z+ker(p_{\alpha}|_{Z\langle E)})\vdash\rightarrow z+N_{\alpha}$
,
$\alpha\in A$
,
while,
(2.12)
$Z(\overline{E}_{\alpha^{\alpha}}^{\sigma})=Z(E_{\alpha})=\overline{Z(E_{\alpha})}^{\sigma_{\alpha}}$
,
$\alpha\in A$
,
as this follows by the separate continuity of the multiplication of
(cf.,
(2.11), (2.12).
$\alpha\in A$
$\sigma_{\alpha},$
with respect to
for instance, [15; Theor. III, 3.5]). Thus, (2.9) follows now by (2.10),
$\overline{E}_{\alpha}^{\sigma_{\alpha}}$
$\square $
2.3. The connecting maps of the inverse system
(cf.
are those of
restricted on
. Thus, according to Section 1
(cf., in particular, proof of Propositlon 1.3 and discussion before it), when no
confusion is likely to result, we shall keep the symbols
, of the
connecting, respectively, canonical maps of the projective system
, for both
of the projective systems
and
.
Remark 2.4. If $L(H),$ $H$ a locally Hilbert space, is the localIy
-algebra of
Example 1.4.3, one also defines on $L(H)$ the respective of the classical weak
$(Z(\overline{E}_{\alpha}^{\sigma_{\alpha}})=Z(E_{\alpha})),$
(2.12)),
$(\overline{E}_{\alpha}^{\sigma_{\alpha}})$
$Z(\overline{E}_{\alpha}^{\sigma_{\alpha}}),$
$f_{\alpha\beta},$
$\alpha\leq\beta,$
$f_{\alpha},$
$(F_{\alpha}),$
$\alpha\in A$
$(\overline{E}_{\alpha}^{\sigma_{\alpha}}),$
$\alpha\in A$
$\alpha\in A$
$(Z(\overline{E}_{\alpha}^{\sigma_{\alpha}})=Z(E_{\alpha})),$
$\alpha\in A$
$\alpha\in A$
$\alpha\in A$
$W^{*}$
(operator) topology. In fact, if
$T=(T_{\lambda})\in L(H=\lim H_{\lambda})\cong\lim \mathscr{L}(H_{\lambda})$
$\overline{\lambda}$
, there is
$\eta\in H$
$\lambda\in\Lambda$
such that
$\xi,$
(2.13)
$\eta\in H_{\lambda}$
,
the seminorms
, define a locally convex topology on
called weak (operator) topology and is denoted by .
Furthermore, for any
, one defines
$\xi,$
$\eta\in H$
$lt$
$\xi,$
$\omega_{\xi,\eta}$
where
:
$\xi$
. Thus, if
$p_{\xi,\eta}(T)=|\langle T_{\lambda}\xi, \eta\rangle_{\lambda}|$
$(p_{\xi,\eta}),$
(cf. (1.8)) and ,
$\overline{\lambda}$
$L(H)$ ,
which is
$w$
$\eta\in H$
$L(H)\rightarrow C:T^{\llcorner}\rightarrow\omega_{\xi,\eta}(T)=\langle T_{\lambda}\xi, \eta\rangle_{\lambda}$
,
is that index in
with
. Then,
(topological dual of
$(L(H), w))$ , for any
. Thus, if
is the linear subspace of $L(H)$ generated
by
, the pair $(L(H), L\mathscr{F}(H))$ forms a dual system, in such a way that one
particularly obtains
$\lambda$
$\Lambda$
$\xi,$
$\xi,$
$\omega_{\xi,\eta},$
$\eta\in H$
$\eta\in H_{\lambda}$
$\omega_{\xi,\eta}\in L(H)^{\prime}$
$L\mathscr{F}(H)$
$\eta\in H$
(2.14)
On the other hand, if
and
$\xi,$
$w=\sigma(L(H), L\mathscr{F}(H))$
$w_{\lambda}$
.
denotes the weak (operator) topology on
,
, one has (cf.
linear subspace of
$\mathscr{L}\mathscr{F}(H_{\lambda})therespectivetoL\mathscr{F}(H)$
$\mathscr{L}(H_{\lambda}),$
$\mathscr{L}(H_{\lambda})^{\prime},$
$\lambda\in\Lambda$
$\lambda\in\Lambda$
M. FRAGOULOPOULOU
44
also [15; p. 68])
(2.15)
$\lambda\in\Lambda$
.
, of the inverse system
In particular, the connecting maps
continuous,
so that one finally gets
, are weakly
$f_{\lambda\mu},$
$\lambda\in\Lambda$
,
$w_{\lambda}=\sigma(\mathscr{L}(H_{\lambda}), \mathscr{L}\mathscr{F}(H_{\lambda}))$
(2.16)
$\lambda\leq\mu$
$w=\lim w_{\lambda}=\lim\sigma(\mathscr{L}(H_{\lambda}), \mathscr{L}\mathscr{F}(H_{\lambda}))=\sigma(L\langle H),$ $L\mathscr{F}(H))$
$(\mathscr{L}(H_{\lambda}))$
,
.
$\overline{\lambda}$
$\overline{\lambda}$
Moreover (cf. [15; p. 68] as well as (1.10)),
(2.17)
$w_{\lambda}\leq\sigma_{\lambda}\equiv\sigma(\mathscr{L}(H_{\lambda}), \mathscr{L}_{*}(H_{\lambda}))$
,
$\lambda\in\Lambda$
,
consequently,
(2.18)
$w\leq\sigma=\varliminf_{\lambda}\sigma_{\lambda}$
,
with the -weak topology on $L(H)$ . Thus, a consequenoe of (2.18) and Example
1.4.1, Proposition is that every -closed*-subalgebra of $L(H),$ $H$ a locally Hilbert
-algebra.
space, is a locally
$\sigma$
$\sigma$
$w$
$W^{*}$
3. In Sections 3 and 4 we apply the theory developed above, in order to get
-algebra. Some of our
some information about the inner derivations of a locally
previous results of L.
-algebras,
locally
of
general
case
more
results extend in the
-algebra.
Zsid\’o [16] concerning the norm of a derivation of an abstract
$E$ is
such that $\delta(xy)=$
a linear map
A derivation of an algebra
$\delta(x)y+x\delta(y)$ for any
for some
in $E$. A derivation of $E$ is called inner, if
$E$
in , with $\delta_{x}(y)=xy-yx,$ in .
, is a
-algebra. Since each
be a locally
Now, let
$W^{*}$
$W^{*}$
$W^{*}$
$\delta:E\rightarrow E$
$\delta$
$x,$ $y$
$E$
$\delta=\delta_{x}$
$x$
$\alpha\in A$
$W^{*}-$
$y$
$W^{*}$
$F_{\alpha},$
$E=\varliminf_{\alpha}F_{\alpha}$
, is norm-continuous and inner [12]. Thus, if $x=$
algebra, every derivation of
, respectively defined
, are the inner derivations of
and
, one gets
by
$F_{\alpha},$
$(x_{\alpha})\in E$
$x,$
$\delta_{x},$
$\delta_{x_{\alpha}},$
$\alpha\in A$
$E,$
$\alpha\in A$
$F_{\alpha},$
$\alpha\in A$
$\alpha\in A$
$x_{\alpha},$
$f_{\alpha\beta}\circ\delta_{x\rho}=\delta_{x_{a}}\circ f_{\alpha\beta}$
for any
linear map
$\alpha\leq\beta$
in
$A$
and
$x_{\beta}\in F_{\beta}$
with
$f_{\alpha\beta}(x_{\beta})=x_{\alpha}$
$\delta=\varliminf_{\alpha}\delta_{x_{\alpha}}$
:
,
. Hence, there is a unique continuous
$E\rightarrow E$
,
$x=(x_{\alpha})\in E$ .
, and
such that
is the set of all inner derivations of $E$ and
In this respect, if
, one has the following.
the Banach space of all (inner) derivations of
$f_{\alpha}\circ\delta=\delta_{x_{\alpha}}\circ f_{\alpha},$
$\alpha\in A$
$\delta=\delta_{x},$
$\delta(F_{\alpha}),$
$\delta_{0}(E)$
$F_{\alpha},$
Theorem 3.1. For every locally
convex space, in such a way that
$W^{*}$
$\alpha\in A$
,
$\alpha\in A$
-algebra
$E,$
$\delta_{0}(E)$
is a complete locally
ON LOCALLY
$W^{*}$
,
$\delta_{0}(E)=\varliminf_{\alpha}\delta(\overline{E}_{\alpha^{\alpha}}^{\sigma})$
where
$(E_{\alpha}),$
$\alpha\in A$
45
-ALGEBRAS
, corresponds to the Arens-Michael decomposition
of .
$E$
Proof. Each
, is complete [13; Folg. 5.4] (cf. also discussion
in , are onto,
before Definition 1.1) therefore the connecting maps :
which yields that
$\alpha\in A$
$E_{\alpha}\equiv E/N_{\alpha},$
$E_{\beta}\rightarrow E_{\alpha},$
$f_{\alpha\beta}$
$f_{\alpha\beta}(Z(E_{\beta}))\subset Z(E_{\alpha})$
Hence, according to (2.12)
Proposition 1.3)
,
one also gets
$f_{\alpha\beta}(Z(\overline{E}_{\beta}^{\sigma\rho}))\subset Z(\overline{E}_{\alpha}^{\sigma_{\alpha}})$
Thus, for any
(3.1)
:
$g_{\alpha\beta}$
Now, every
in
$\alpha\leq\beta$
$A$
$\overline{E}_{\alpha^{\alpha}}^{\sigma},$
$\alpha\in A$
$\delta_{x_{\alpha}}\in\delta(\overline{E}_{\alpha}^{\sigma_{\alpha}}),$
, for all
,
in
$\alpha\leq\beta$
.
2.3 and comments after
$A$
.
we may define the continuous morphisms
,
,
is a
$W^{*}$
$x_{\alpha}\in Z(\overline{E}_{\alpha^{\alpha}}^{\sigma}),$
(3.3)
$u_{\alpha}$
:
z_{\alpha}\in Z(\overline{E}_{\alpha}^{\sigma_{\alpha}})\}$
,
$\alpha\in A$
,
(cf. [16; p. 148, Corol.]). Thus, sinoe
, it follows that the map
$\alpha\in A$
$x_{\alpha}\in\overline{E}_{\alpha^{\alpha}}^{\sigma},$
$\alpha\in A$
.
-algebra (cf. Proposition 1.3), so that
$\Vert\delta_{x_{\alpha}}\Vert=2\inf\{\Vert x_{\alpha}-z_{\alpha}\Vert_{\alpha} :
$\delta_{x_{\alpha}}=0$
(cf. besides
$A$
$A$
$\overline{E}_{\beta}^{\sigma\rho}/Z(\overline{E}_{\beta}^{\sigma\rho})\rightarrow\overline{E}_{\alpha}^{\sigma_{\alpha}}/Z(\overline{E}_{\alpha}^{\sigma_{\alpha}}):\dot{x}_{\beta}\equiv x_{\beta}+Z(\overline{E}_{\beta}^{\sigma\rho})-\rightarrow f_{\alpha\beta}(x_{\beta})\wedge$
(3.2)
for any
in
$\alpha\leq\beta$
$\alpha\leq\beta$
$\overline{E}_{\alpha^{\alpha}}^{\sigma}/Z(\overline{E}_{\alpha}^{\sigma_{\alpha}})\rightarrow\delta(\overline{E}_{\alpha^{\alpha}}^{\sigma}):\dot{x}_{\alpha}$
– $\delta_{x_{a}}$
,
moreover
,
$\alpha\in A$
is a topological vector spaoe isomorphism. A consequence of (3.1), (3.3) is now that
in , is a projective system of Banach spaces, where
the pair
$(\delta(\overline{E}_{\alpha}^{\sigma_{\alpha}}), h_{\alpha\beta}),$
$h_{\alpha\beta}$
:
$A$
$\alpha\leq\beta$
$\delta(\overline{E}_{\alpha}^{\sigma p})\rightarrow\delta(\overline{E}_{\alpha^{\alpha}}^{\sigma}):\delta_{x_{\beta}}$
}
$-\delta_{x_{\alpha}}$
with
,
$x_{\alpha}=f_{\alpha\beta}(x_{\beta})$
,
Hence, taking also into account the discussion before Theorem 3.1
$\delta_{0}(E)=\varliminf_{\alpha}\delta(\overline{E}_{\alpha}^{\sigma_{\alpha}})$
$\alpha\leq\beta$
.
we obtain
,
becomes a complete locally convex space, for which a defining family of
seminorms is given by
so that
$\delta_{0}(E)$
(3.4)
$ q_{\alpha}(\delta_{x})=\Vert\delta_{x_{\alpha}}\Vert$
for any
$\delta_{x}\in\delta_{0}(E),$
$x=(x_{\alpha})\in E$
.
,
$\alpha\in A$
,
$\square $
An interesting result in this direction would be, of course, that every derivation
-algebra is inner.
of a locally
The mext theorem provides in our case an analogue of a known result concerning
-algebra, see [16; p. 148, Corol.] as well as [5; Theor.
the norm of a derivation of a
$W^{*}$
$W^{*}$
1].
Theorem 3.2.
Let
$E$
be a locally
$W^{*}$
-algebra. Then,
46
M. FRAGOULOPOULOU
(3.5)
$q_{\alpha}(\delta_{x})=2\inf\{p_{\alpha}(x-z):z\in Z(E)\}$
for any
$x=(x_{\alpha})\in E$
$\delta_{x}\in\delta_{0}(E),$
$\alpha\in A$
,
.
By Proposition 1.3
Proof.
,
$E\cong\varliminf_{\alpha}\overline{E}_{\alpha}^{\sigma_{\alpha}}$
,
so that if
$x=(x_{\alpha})$
(cf. also 2.3). Thus, by Theorem 3.1,
with
Moreover, by (2.12), (2.11)
. Hence, according to (3.2), (3.4), (1.2) we conclude that
every
in
$E,$
$x_{\alpha}=f_{\alpha}(x)$
for
$\alpha\in A$
.
$Z(\overline{E}_{\alpha}^{\sigma_{\alpha}})=Z(E_{\alpha})\cong Z(E)_{\alpha}=Z(E)/ker(p_{\alpha}|_{Z(E)})=f_{\alpha}(Z(E))$
,
$\alpha\in A$
$\delta_{x}=(\delta_{f_{\alpha}\{x)})$
$\delta_{f_{\alpha}\langle x)}\in\delta(\overline{E}_{\alpha}^{\sigma_{\alpha}}),$
$\alpha\in A$
$q_{\alpha}(\delta_{x})=\Vert\delta_{f_{\alpha}\langle x)}\Vert=2\inf\{\Vert f_{\alpha}(x)-z_{\alpha}\Vert_{\alpha} :
z_{\alpha}\in Z(\overline{E}_{\alpha}^{\sigma_{\alpha}})\}$
$=2\inf\{\Vert f_{\alpha}(x)-f_{\alpha}(z)\Vert_{\alpha} :
$=2\inf\{p_{\alpha}(x-z):z\in Z(E)\}$
for any
$\delta.\in\delta_{0}(E),$
$x\in E$
.
z\in Z(E)\}$
,
$\alpha\in A$
$\square $
As a corollary to the preceding theorem we now get Stampfli’s result for $L(H),$ $H$
a locally Hilbert space, referred to the norm of a derivation acting on the -algebra
$H$ a Hilbert spaoe (cf. [14; Theor. 4]).
$C^{*}$
$\mathscr{L}(H),$
Corollary 3.3. Let $H$ be a locally Hilbert space and $L(H)$ the locally
in $L(H)$ . Then,
of Example 1.4.3. Let also in $\delta_{0}(L(H)),$
(3.6)
$q_{\lambda}(\delta_{T})=2\inf\{p_{\lambda}(T-zid_{H}):z\in C\}$
for all
$\lambda\in\Lambda$
$W^{*}$
-algebra
$T=(T_{\lambda})$
$\delta_{T}$
,
.
, of
Proof. Working out the Example 1.4.3, we saw that each factor
$L(H)$
the Arens-Michael decomposition of
coincides (algebraically-topologically)
, a Hilbert space. Moreover, by (2.11)
with the
-algebra
, where
,
, with
$L(H)_{\lambda},$
$W^{*}$
$\mathscr{L}(H_{\lambda}),$
$H_{\lambda},$
$\lambda\in\Lambda$
$\lambda\in\Lambda$
$\lambda\in\Lambda$
$Z(L(H))\cong\varliminf_{\lambda}Z(\mathscr{L}(H_{\lambda}))$
$Z(\mathscr{L}(H_{\lambda}))=\{z_{\lambda}id_{H_{\lambda}} :
z_{\lambda}\in C\}\equiv C_{\lambda}$
. Hence, (cf. [2; p. 77, Exemple 2)]),
all
so that the assertion now follows by Theorem 3.2.
$C_{\lambda}\cong C$
, for
$\lambda\in\Lambda$
$Z(L(H))=\{zid_{H} :
z\in C\}$ ,
$\square $
-algebra. Then, $\delta_{0}(E)=E/Z(E)$ within an
Let $E$ be a locally
isomorphism of locally convex spaces or, equivalently, the sequence
Corollary 3.4.
$W^{*}$
$0$
–
$Z(E)\rightarrow E\rightarrow^{u}\delta_{0}(E)\rightarrow 0$
,
is topologically exact.
The map $u:E\rightarrow\delta_{0}(E):x-u(x)=\delta_{x}$ is alinear surjection with $ker(u)=$
(cf. (3.5)), therefore is also
, for any $x\in E,$
$Z(E)$ . Moreover,
linear bijection
continuous. Thus, taking the induced from
fact,
is,
, this
in
an isomorphism of locally convex
spaces by (3.5). Hence, the continuous linear surjection is a ”topological homomorphism” [7; p. 106, Def. 2], which is equivalent to the fact that is moreover open [7; .
Proof.
$q_{\alpha}(u(x))\leq 2p_{\alpha}(x)$
$\alpha\in A$
$u$
$\overline{u}:E/Z(E)\rightarrow$
$u$
$\delta_{0}(E):\dot{x}\equiv x+Z(E)-\overline{u}(\dot{x})=\delta_{\chi}$
$u$
$u$
$p$
ON LOCALLY
106, Theor. 1].
$W^{*}$
-ALGEBRAS
47
$\square $
L. Zsid\’o realized the norm of a derivation acting on an abstract
-algebra $E$
by means of a certain particular map $\Phi:E\rightarrow Z(E)$ [ ; Theorems 1, 2]. Using Zsid\’o’s
technique we shall also try to estimate the numbers
, (cf. (3.5)) via of a
corresponding to our case map , at the cost however of some extra restriction for the
locally
-algebra involved (cf. Theorem 3.5). Thus, let $E$ be a locally
-algebra.
Then (Proposition 1.3),
$W^{*}$
$16$
$q_{\alpha}(\delta_{x}),$
$\alpha\in A$
$\Phi$
$W^{*}$
$W^{*}$
$E\cong\varliminf_{\alpha}\overline{E}_{\alpha}^{\sigma_{\alpha}}$
.
Denote by
,
$\mathfrak{M}_{\alpha}\equiv \mathfrak{M}(Z(\overline{E}_{\alpha}^{\sigma_{\alpha}}))$
the spectrum (Gel’fand space) of
$\alpha\in A$
$Z(\overline{E}_{\alpha}^{\sigma_{\alpha}}),$
,
$Z(\overline{E}_{\alpha}^{\sigma_{\alpha}})\cong \mathscr{C}(\mathfrak{M}_{\alpha})$
by the Gel’fand-Naimark theorem. Thus, if
(3.7)
$M_{\alpha}=\{\sum_{i=1}^{n}m_{\alpha}^{i}y_{\alpha}^{i}$
:
$m_{\alpha}^{i}\in ker(h_{\alpha}),$
$n$
,
$\alpha\in A$
, where
$\alpha\in A$
,
is a positive integer and
,
$y_{\alpha}^{i}\in\overline{E}_{\alpha^{\alpha}}^{\sigma}\}^{-}\equiv[ker(h_{\alpha})]$
$h_{\alpha}\in \mathfrak{M}_{\alpha}$
$\alpha\in A$
set
,
where “–,, means norm-closure in
. Then,
is the smallest closed 2-sided
ideal of
containing
, so that it is primitive by [6; Theor. 4.7].
Therefore, every
, has a faithful irreducible representation
in some Hilbert
spaoe
. In this regard, L. Zsid\’o shows in [16; Theor. 1] the existenoe of a
unique norm continuous map
$\overline{E}_{\alpha^{\alpha}}^{\sigma},$
$\overline{E}_{\alpha}^{\sigma_{a}}$
$ker(h_{\alpha}),$
$\alpha\in A$
$M_{\alpha}$
$\alpha\in A$
$\overline{E}_{\alpha}^{\sigma_{\alpha}}/M_{\alpha}$
$\psi_{\alpha}$
$H_{\alpha},$
$\alpha\in A$
(3.8)
$\Phi_{\alpha}$
:
$\overline{E}_{\alpha}^{\sigma_{\alpha}}-Z(\overline{E}_{\alpha}^{\sigma_{\alpha}})\cong \mathscr{C}(\mathfrak{M}_{\alpha})$
,
,
$\alpha\in A$
defined by the relation
(3.9)
$\Phi_{\alpha}(x_{\alpha})(h_{\alpha})=oenter$
of the operator
$\psi_{\alpha}(x_{\alpha}+M_{\alpha})$
for any
, where by the center of an operator
the unique complex number
for which
$x_{\alpha}\in\overline{E}_{\alpha}^{\sigma_{\alpha}},$
$h_{\alpha}\in \mathfrak{M}_{\alpha},$
$\alpha\in A$
,
$T$
in
$\mathscr{L}(H_{\alpha})$
is meant
$z_{T}$
(3.10)
$\Vert T-z_{T}\Vert=\inf\{\Vert T-zid_{H}\Vert :
z\in C\}$
(cf. [14]). By means of , L. Zsid\’o estimates the norm of a derivation in
(cf.
[16; Theor. 2]). Namely, for any
and
, the map (3.8) has the
$\Phi_{\alpha}$
$\overline{E}_{\alpha}^{\sigma_{\alpha}},$
$x_{\alpha}\in\overline{E}_{\alpha}^{\sigma_{\alpha}}$
$z_{\alpha}\in Z(\overline{E}_{\alpha}^{\sigma_{\alpha}}),$
$\alpha\in A$
$\alpha\in A$
following properties:
(3.11)
(3.12)
(3.13)
$\Phi_{\alpha}(x_{\alpha}+z_{\alpha})=\Phi_{\alpha}(x_{\alpha})+z_{\alpha}$
,
$\Phi_{\alpha}(x_{\alpha}z_{\alpha})=\Phi_{\alpha}(x_{\alpha})z_{\alpha}$
$\Vert x_{\alpha}-\Phi_{\alpha}(x_{\alpha})\Vert_{\alpha}=\inf\{\Vert x_{\alpha}-z_{\alpha}\Vert_{\alpha} ; z_{\alpha}\in Z(\overline{E}_{\alpha}^{\sigma_{\alpha}})\}$
$\Vert\delta_{x_{\alpha}}\Vert=2\Vert x_{\alpha}-\Phi_{\alpha}(x_{\alpha})\Vert_{\alpha}$
,
$\delta_{x_{\alpha}}\in\delta(\overline{E}_{\alpha}^{\sigma_{\alpha}})$
,
,
.
Following the preceding notation, as well as the notation of 2.3, we consider the
M. FRAGOULOPOULOU
48
primitive ideal
. Thus, for any
$M_{\beta}\equiv[ker(h_{\beta})]$
$\alpha\in A$
(3.14)
:
$\lambda_{\alpha\beta}$
$\alpha\leq\beta$
in
in
$A$
$\overline{E}_{\beta}^{\sigma\rho}$
$\pi_{\alpha}$
$h_{\beta}=h_{\alpha}\circ f_{\alpha\beta}$
in
$\mathfrak{M}_{\beta},$
$\alpha\leq\beta,$
in
$h_{\alpha}$
$\mathfrak{M}_{\alpha}$
,
we can define the map
$\overline{E}_{\beta}^{\sigma\rho}/M_{\beta}\rightarrow\overline{E}_{\alpha}^{\sigma_{\alpha}}/M_{\alpha}$
In this respect, if
(cf. (3.7)), with
:
.
one has
$x_{\beta}+M_{\beta}|\rightarrow\lambda_{\alpha\beta}(x_{\beta}+M_{\beta})=f_{\alpha\beta}(x_{\beta})+M_{\alpha}$
denotes the quotient map of
$\lambda_{\alpha\beta}\circ\pi_{\beta}=\pi_{\alpha}\circ f_{\alpha\beta}$
$\overline{E}_{\alpha^{\alpha}}^{\sigma}$
for any
,
onto
in
$\alpha\leq\beta$
$\alpha\in A$
$\overline{E}_{\alpha}^{\sigma_{\alpha}}/M_{\alpha},$
$A$
,
,
so that (3.14) is continuous. On the other hand, the faithful irreducible representation
, is in fact, an isometric representation, since
of the primitive algebra
, is 1-1, this also
, is a
-algebra. Besides, supposing that
becomes an isometry. Hence, considering the diagram
$\overline{E}_{\alpha}^{\sigma_{\alpha}}/M_{\alpha},$
$\alpha\in A$
$\psi_{\alpha}$
$\alpha\in A$
$\overline{E}_{\alpha^{\alpha}}^{\sigma}/M_{\alpha},$
$C^{*}$
$\lambda_{\alpha\beta},$
$\alpha\leq\beta$
$\psi_{\beta}^{-1}|\downarrow\psi_{\alpha}\overline{E}_{\beta}^{\sigma\rho}/M_{\beta}\overline{E}_{\alpha^{\alpha}}^{\sigma}/M_{\alpha}\underline{\lambda_{\alpha\beta}}$
(3.15)
$\alpha\leq\beta$
in
$A$
${\rm Im}(\psi_{\beta})\subset \mathscr{L}(H_{\beta}){\rm Im}(\psi_{\alpha})\subset \mathscr{L}(H_{\alpha})\overline{\psi_{\alpha}\circ\lambda_{\alpha\beta}\circ\psi_{\beta}^{-1}}$
, becomes 1-1 for any
is an isometry too. Note that
, bijections,
:
with connecting maps
-algebra
henoe norm and weakly *-bicontinuous (for the latter see [12; Corol. 4.1.23]).
In this concern, one now gets the next theorem, which is an analogue in our case
of [16; Theorems 1, 2].
-algebra, in such a way that the map (3.14) is
Theorem 3.5. Let $E$ be a locally
, with
1-1. Then, there is a unique -continuous (cf. Scholium 1.2) map
the properties:
i) $\Phi(x+z)=\Phi(x)+z,$ $\Phi(xz)=\Phi(x)z$ , for any $x\in E,$ $z\in Z(E)$ .
.
ii) $p_{\alpha}(x-\Phi(x))=\inf\{p_{\alpha}(x-z):z\in Z(E)\}$ , for any $x\in E,$
$(x\in E),$
.
, for any
iii)
the map
locally
$\psi_{\alpha}\circ\lambda_{\alpha\beta}\circ\psi_{\beta}^{-1}$
$\lambda_{\alpha\beta},$
$W^{*}$
$\alpha\leq\beta$
$F_{\beta}\rightarrow F_{\alpha},$
$f_{\alpha\beta}$
$ E=\lim_{\alpha}F_{\alpha}\leftarrow$
$\alpha\leq\beta$
’
$W^{*}$
$\Phi:E\rightarrow Z(E)$
$\tau$
$\alpha\in A$
$\alpha\in A$
$\delta_{x}\in\delta_{0}(E),$
$q_{\alpha}(\delta_{x})=2p_{\alpha}(x-\Phi(x))$
Proof. We shall first show that the maps
system. Thus, we consider the diagram
$(\Phi_{\alpha}),$
$\alpha\in A$
, (cf. (3.8)) form
an inverse
$\overline{E}_{\beta}^{\sigma\rho}Z(\overline{E}_{\beta}^{\sigma_{\beta}})\cong \mathscr{C}(\mathfrak{M}_{\beta})\underline{\Phi_{\beta}}$
(3.16)
$\downarrow{}^{t}(^{t}f_{\alpha\beta})$
$ f_{\alpha\beta}\downarrow$
$\alpha\leq\beta$
in
$A$
$\overline{E}_{\alpha^{\alpha}}^{\sigma}\rightarrow Z(\overline{E}_{\alpha}^{\sigma_{a}})\cong \mathscr{C}(\mathfrak{M}_{\alpha})\Phi_{\alpha}$
denotes the transpose of
, one has (cf. also (3.9), (3.10))
where
$A$
${}^{t}f_{\alpha\beta}$
$f_{\alpha\beta},$
$\alpha\leq\beta$
. Then, for any
$x_{\beta}\in\overline{E}_{\beta}^{\sigma\rho},$
$h_{\alpha}\in \mathfrak{M}_{\alpha},$
$\alpha\leq\beta$
in
ON LOCALLY
$W^{*}$
-ALGEBRAS
49
$(({}^{t}({}^{t}f_{\alpha\beta})\circ\Phi_{\beta})(x_{\beta}))(h_{\alpha})=(\phi_{\beta}(x_{\beta})\circ {}^{t}f_{\alpha\beta})(h_{\alpha})=\phi_{\beta}(x_{\beta})(h_{\alpha}\circ f_{\alpha\beta})$
.
$=\phi_{\beta}(x_{\beta})(h_{\beta})=z_{\psi_{\beta}\langle x\rho+M_{\beta})}$
But, sinoe
$\lambda_{\alpha\beta},$
$\alpha\leq\beta$
, is 1-1, the
map
$\psi_{\alpha}\circ\lambda_{\alpha\beta}\circ\psi_{\beta}^{-1}$
of diagram (3.15) is an isometry, so
that
$z_{\psi_{\beta}\{x\rho+M_{\beta})}=z_{\langle\psi_{\alpha}\circ\lambda_{\alpha\beta}\circ\psi\rho^{-1})\langle\psi_{\beta}\langle x\rho+M_{\beta}))}$
$=z_{\psi_{\alpha}\langle\lambda_{\alpha\beta}\langle xp+M_{\beta}))}=z_{\psi_{\alpha}\langle f_{\alpha\beta}(x\rho)+M_{\alpha})}$
$=(\Phi_{\alpha}\circ f_{\alpha\beta})(x_{\beta})(h_{\alpha})$
,
$\alpha\leq\beta$
.
a unique
Thus, the diagram (3.16) is commutative, which yields the existenoe of
continuous map (cf. also Proposition 1.3 and Corollary 2.2)
(3.17)
$\Phi=\varliminf_{a}\Phi_{\alpha}$
:
$E\cong\varliminf_{\alpha}\overline{E}_{\alpha}^{\sigma_{\alpha}}\rightarrow Z(E)\cong\varliminf_{\alpha}Z(\overline{E}_{\alpha^{\alpha}}^{\sigma})$
$\tau-$
,
such that
$f_{\alpha}\circ\Phi=\Phi_{\alpha}\circ f_{\alpha}$
,
for all
$\alpha\in A$
.
Now, the definition of and (3.11) imply i). On the other hand, (cf. also (1.2)
and (3.12)), for every $x=(x_{\alpha})\in E$
$\Phi$
$p_{\alpha}(x-\Phi(x))=\Vert x_{\alpha}-\Phi_{\alpha}(x_{\alpha})\Vert_{\alpha}\leq\Vert x_{\alpha}-z_{\alpha}\Vert_{\alpha}$
for all
$z_{\alpha}=f_{\alpha}(z)\in Z(\overline{E}_{\alpha^{a}}^{\sigma})\cong f_{\alpha}(Z(E)),$
$\alpha\in A$
,
$\alpha\in A$
,
(cf. proof of Theorem 3.2). Thus,
$p_{\alpha}(x-\Phi(x))\leq\inf\{p_{\alpha}(x-z):z\in Z(E)\}\leq p_{\alpha}(x-\Phi(x))$
,
for any $x\in E,$
.
Concerning iii), this isaconsequenoe of (3.4), (3.13) and (1.2).
$\alpha\in A$
$\square $
4. If $(E, (p))$ is a locally convex spaoe and $Z$ a closed linear subsapce of ,
denote by the quotient map of onto $E/Z$ and by the seminorm of $E/Z$ derived by
. In this respect, using the properties of the map (3.17) of Theorem 3.5, we are led to
$E$
$E$
$\pi$
$r$
$p$
the following general statement.
Theorem 4.1. Let $(E, (p))$ be a locally convex space and $Z$ a closed linear
subspace of E. Then,
is a continuous map with the properties $\Psi(x+z)=$
$\Psi(x)+z,$ $x\in E,$ $z\in Z$, and $p(x-\Psi(x))=\inf\{p(x-z):z\in Z\},$ $x\in E$,
for every if,
and only if,
admits a continuous relatively open section , in the sense that
$\pi\circ s=id_{E/Z}$ and $p(s(\dot{x}))=r(\dot{x}),\dot{x}\equiv x+Z,$ $x\in E$,
for any .
$\psi:E\rightarrow Z$
$p$
$\pi$
$s$
$p,$
$r$
$\Psi(x)=x-y+\Psi(y)$ , so that one defines
where
and $p(x-\Psi(x))=r(\dot{x}),$ $x\in E$, for
$x\in E$, for any
all
. Conversely, if is a section of with
, we first
$\pi(x-s(\dot{x}))=0$
$\Psi:E\rightarrow
Z:x->\Psi(x)=x-s(\dot{x})$
hence,
,
have that
. Then,
one defines
$p(\Psi(x))\leq 2p(x),$ $x\in E$, for every
, so that
is continuous, having besides the re-
Proof.
For any
$x\in E$
and
$s:E/Z\rightarrow E:\dot{x}\mapsto s(\dot{x})=x-\Psi(x)$
$p,$
$r$
$y\in\dot{x},$
,
$s$
$\pi\circ s=id_{E/Z}$
$\pi$
$p$
$p(s(\dot{x}))=r(\dot{x}),$
$\Psi$
$p,$
$r$
M. FRAGOULOPOULOU
50
quired by Theorem 4.1 properties.
$\square $
A direct consequence of Theorems 4.1, 3.5 is now the next.
Corollary 4.2.
Then,
$\pi:E\rightarrow E/Z(E)$
, where
$x\in E$
$\Phi$
be a locally
-algebra for which the map (3.14) is 1-1.
has a continuous relatively open section , with $s(\dot{x})=x-\Phi(x)$ ,
Let
$E$
$W^{*}$
$s$
is the map (3.17)
$\square $
Now, according to Corollary 3.4, for every locally
$W^{*}$
-algebra
$E$
,
one defines
the ”topological homomorphism”
(4.1)
$u:E\rightarrow\delta_{0}(E):x\mapsto u(x)=\delta_{x}$
,
in such a way that the induced linear bijection
(4.2)
$\overline{u}:E/Z(E)\rightarrow\delta_{0}(E):\dot{x}-\rightarrow\overline{u}(\dot{x})=u(x)$
,
becomes an isomorphism of locally convex spaces.
The next theorem gives an extension and strengthening as well of a previous
result of L. Zsid\’o [16; Theor. 3]. More precisely, Theorem 4.3 below, gives a new
information, even in the norm case, ensuring that Zsid\’o’s continuous map of
Theorem 3 in [16] (cf., for instance, the map of the next theorem) is, in addition, a
relatively open section of the topological homomorphism (4.1). That is, one has.
$v$
Theorem 4.3. Let $E$ be a locally
-algebra for which the map (3.14) is 1-1. Let
also be the map (3.17) and $v:\delta_{0}(E)\rightarrow E:\delta_{x}-v(\delta_{x})=x-\Phi(x)$ . Then, the continuous
open linear surjection (cf. (4.1)) has as a continuous relatively open section. In
particular,
$W^{*}$
$\Phi$
$u$
$v$
$2p.(v(\delta.))=q.(\delta.)$
for any
$\delta_{x}\in\delta_{0}(E),$
Proof.
If
$(x\in E),$
$\delta_{x},$
.
$\alpha\in A$
$\delta_{y}\in\delta_{0}(E),$
,
$(x, y\in E)$ ,
with
$\delta_{x}=\delta_{y}$
,
one gets that
$x-y\in Z(E)$ ,
consequently (cf. Theorem 3.5, )) $\Phi(x)=\Phi(x-y+y)=x-y+\Phi(y)$ , which yields that
, for every
,
is well defined. On the other hand,
$x\in E$; moreover,
4.2,
, with
as in Corollary
respectively (4.2). Hence, is,
in effect, a continuous relatively open section of , having besides the property (cf.
$(x\in E)$ .
Theorem 3.5) $q.(\delta.)=2p.(v(\delta.))$ , for any
$i$
$v$
$\delta_{x}\in\delta_{0}(E)$
$(u\circ v)(\delta_{x})=\delta_{x}-\delta_{\Phi\langle x)}=\delta_{x}$
$v=s\circ\overline{u}^{-1}$
$s,\overline{u}$
$v$
$u$
$\alpha\in A,$
$\delta_{x}\in\delta_{0}(E),$
$\square $
Acknowledgements. The initial motive to this paper was the subject matter of
my M. Sc. Thesis for the supervision of which I am grateful to Dr. A. L. Brown
(Univ. of Newcastle upon Tyne). I also thankfully acknowledge the financial support
to this work of the Greek Ministry of Coordination. However, to put the whole stage
within this framework, it was the idea of Professor A. Mallios (Univ. of Athens), as a
result of which the paper took the present form. I wish to express him my thanks for
advice, several penetrating discussions and his perseverance in finishing this study.
ON LOCALLY
$W^{*}$
-ALGEBRAS
References
[1]
[2]
[3]
R. Arens: A generalization of normed rings, Pacific J. Math., 2 (1952), 455-471.
N. Bourbaki: Th\’eorie des Ensemples, Chapitre 3, Hermann, Paris, 1967.
C. R. Chen: On the intersections and the unions of Banach algebras, Tamgang J. Math., 9 (1978),
21-27.
[4]
M. Fragoulopoulou: Kadison’s transitivity for locally
$C^{*}$
-algebras, J. Math. Anal. Appl., 108 (1985),
422-429.
[5]
P. Gajendragadkar: Norm
of a derivation on a von Neumann algebra, Trans. Amer. Math. Soc., 170
165-170.
H. Halpem: Irreducible module homomorphisms of a von Neumann algebra into its center, Trans.
Amer. Math. Soc., 140 (1969), $19\succ 221$ .
J. Horv\’ath: Topological Vector Spaces and Distributions I, Addison-Wesley Publishing Company,
1966.
A. Inoue: Locally
-algebras, Mem. Faculty Sci. Kyushu Univ., 25 (1971), 197-235.
G. K\"othe: Topological Vector Spaces I, Springer-Verlag, Berlin Heidelberg New York, 1969.
A. Mallios: Topological Algebras: Selected Topics, North Holland, Amsterdam, 1986.
E. A. Michael: Locally multiplicatively-convex topological algebras, Mem. Amer. Math. Soc., No. 11,
1952.
S. Sakai:
-Algebras and
-Algebras, Springer-Verlag, Berlin Heidelberg New York, 1971.
K. Schm\"udgen: Uber LMC*-Algebren, Math. Nachr., 68 (1975), 167-182.
J. G. Stampfli: The norm of a derivation, Pacific J. Math., 33 (1970), 737-747.
M. Takesaki: Theory of Operator Algebras I, Springer-Verlag, New York Heidelberg Berlin, 1979.
L. Zsid\’o: The norm of a derivation in a
-algebra, Proc. Amer. Math. Soc., 38 (1973), 147-150.
(1972),
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
$C^{*}$
$C^{*}$
$W^{*}$
$W^{*}$
Mathematical Institute
University of Athens
57, Solonos Street
10679 Athens
Greece