数理解析研究所講究録
979 巻 1997 年 42-4942

作用素不等式と -normal C*-環について(On operator inequalities and a-normal -algebras)$\sigma$$C^{*}$

東北大学大学院理学研究科 斎藤和之 (Kazuyuki SAIT\^O)

be a Hilbert space and letLetbe the Banach -algebra of all bounded linear
operators on .
A -algebra $A$ is a -subalgebra ofwhich is closed under the norm topology.
Gelfand, Naimark and Kaplansky gave an elegant axiomatic characterization of -algebras
in the following form:
Let $A$ be any Banach algebra with an involution. If the norm on $A$ satisfies that$\mathcal{H}$$\mathcal{L}(\mathcal{H})$$\star$$\mathcal{H}$$C^{*}$$\mathcal{L}(\mathcal{H})$$\star$$C^{*}$$||_{X^{*}}x||=||x||2$

for each $x\in A$ , then there exist a Hilbert space
and a -isomorphism from $A$ onto a-algebra on .
A von Neumann algebra $A$ is a -subalgebra ofwhich is closed under the weak
operator topology and contains the identity operator
on . Then, the following theorem
is well-known:$\mathcal{H}$$C^{*}$$\star$$\mathcal{H}$$\star$$\mathcal{L}(\mathcal{H})$$\mathcal{H}$$1_{\mathcal{H}}$

Theorem A Let
be any increasing net of hermitian operators in $A$ majorized byan hermitian operator $B\in A$ . Then there exists a strong limit $A\in A$ ofin such away that$\{A_{\alpha}\}$$\{A_{\alpha}\}$$A= \sup_{\alpha}A_{\alpha}$in the ordered spaceof all hermitian elements in $A$ .
Moreover, for any given $A\in A$ , put’ wheretion onto the subspace
and we have that
$A_{h}$$P=P_{\overline{rang(A^{*})}}e$$P_{\mathcal{M}}$is the orthogonal projec-$\mathcal{M}$$((A^{*}A) \frac{1}{2}+\frac{1}{n}1_{\mathcal{H}})^{-1}(A^{*}A)\frac{1}{2}\nearrow P(narrow\infty)$

and for each ,$n$$((A^{*}A)^{\frac{1}{2}}+ \frac{1}{n}1_{\mathcal{H}})^{-1}(A^{*}A)\frac{1}{2}\in A$.

Hence, it follows that $P\in A$ and $PA^{*}=A^{*}$ . This $P$ can be characterized as follows:$P$
is the smallest projection $E\in A$ that satisfies $EA^{*}=A^{*}$ .
It follows also that for each $A\in A$ , denoting the right annihilator of $\{A\}$ by,$\mathcal{R}a(\{A\})$.

$\mathcal{R}a(\{A\})=(1_{\mathcal{H}}-P)A$

In his book [3],. Berberian says the following: “Von Neumann algebras areblessed with an excess of structures-algebraic, geometric, topological-so much,$\mathrm{S}.\mathrm{K}$43

that one can easily obscure, through proof by overkill, what makes a particulartheorem work.”
In connection with this philosohpy, along lines with an abstact treatment of the theoryof von Neumann algebras not directly concerned with the representation of the algebras-algebras, which is now called. Rickart introduced a concept ofon Hilbert spaces,-algebras as follows:
Rickart$Bp^{*}$$\mathrm{C}.\mathrm{E}$$C^{*}$

Definition(see [3] for details) A Rickart
each $x\in A$ ,$\mathcal{R}a(\{x\})$

with

$g$

$C^{*}$-algebra is a$C^{*}$-algebra $A$ such that, for: $(=\{y\in A|_{X}y=0\})=gA$

a projection in $A$ .

Remark Such a projection$g$is uniquely determined. It follows also that$\mathcal{L}a(\{x\}):(=\{y\in A|yx=0\})=\mathcal{R}a(\{x^{*}\})^{*}=(hA)^{*}=Ah$for a suitable projection $h\in A$ .
By the preceding argument just before the Definition, it follows that every von Neumann
algebra is a Rikart -algebra, however, the converse is not true in general as the following
examples show:
be the(1) Let X be a perfect locally compact separable metric space and let$X$$B(X)$
is a Rickart
algebra of all complex-valued bounded Borel functions on . Then,-algebra which is not a von Neumann algebra.
$A$
be the Borel envelope of $A$ ,. Let be a unital non-atomic -algebra and let-algebra which is not a von Neumann algebra.
then $B(A)$ is a Rickart$A$be the regular a-completion of $A$ . Thenbe a unital -algebra and let(3) Letis a singular Rickart -algebra.
$C^{*}$$C^{*}-$$\mathcal{B}(X)$$C^{*}$$C^{*}$$(2)’$$\mathcal{B}(A)$$C^{*}$$C^{*}$$\hat{A}$$\hat{A}$$C^{*}$-algebras have very nice order properties, in particular, they have plenty of
Rickartprojections in such a way that the set Proj $(A)$ of all projections in a Rickart
algebra $A$ is a a-complete lattice with respect to the natural order of projections(see Lemma 4).
$C^{*}$$C^{*}-$-algebra $A$ is said to be monotone (resp. monotone ) complete, if everyA
increasing net (resp. sequence) of elements in the ordered space of all hermitian elements
of $A$ has a supremum in . It is straightforward to verify that every monotone complete
to be true,is-algebra. For type I $AW^{*}$ -algebras, the-algebra is
however, for a general $AW^{*}$ -algebras, this question is still open, although an impressive
who showed that
attack on the problem was made by E. Christensen andproperly infinite $AW^{*}$ -algebras are monotone a-complete [4].
$C^{*}$$\sigma-$

$A$

$A$

$C^{*}$$\mathrm{c}\mathrm{o}\dot{\mathrm{n}}\mathrm{V}\mathrm{e}\Gamma \mathrm{s}\mathrm{e}$$\mathrm{k}\mathrm{n}\mathrm{o}\mathrm{V}^{\ulcorner}\mathrm{n}$$\mathrm{a}\mathrm{n}.AW^{*}$$-$$\dot{\mathrm{G}}.\mathrm{K}.\mathrm{P}\mathrm{e}\mathrm{d}\mathrm{e}\Gamma \mathrm{S}\mathrm{e}\mathrm{n}$44

A parallel question in Rickart-algebra theory is whether or not every Rickart
algebra is nmotone a-complete.
A-algebra $A$ is said to be -normal (resp. normal) if every increasing sequence(resp. net) of projections in $A$ has a supremum in .
P. Ara and D. Goldstein recently showed in [2] that every Rickart -algebra is -normal.
Their proof uses an ingenious, deep analysis of the structure of Rickart-algebras andregular rings, together with a result by Christensen and Pedersen, which says that every
properly infinite Rickart -algebra is monotone a-complete.
$C^{*}$$C^{*}$$C^{*}-$$\sigma$$A$$C^{*}$$\sigma$$C^{*}$$C^{*}$In this talk, I would like to give you a proof of a theorem (see, below, Theorem) which
is a nominally more general result on the one hand and a simple alternative proof of the anormality of Rickart -algebras on the other (see, for details, [8]). In fact, in the course of
my proof, under the assumption that Rickart -algebras are
Lemma 6), I use the following simple properties on -algebras only:$C^{*}$$C^{*}$$UMF-\mathrm{a}\mathrm{l}\mathrm{g}\mathrm{e}\mathrm{b}\mathrm{r}\mathrm{a}\mathrm{s}(\mathrm{s}\mathrm{e}\mathrm{e}$$C^{*}$

Let $A$ be a unital -algebra. Then $||x||^{2}=||x^{*}x||$ for any $x\in A$ . If $a\leq b$ inthen $x^{*}ax\leq x^{*}bx$ for every $x\in A$ .
$C^{*}$$A$,

The main part of this talk will be published in the Bulletin of the London MathematicalSociety.

Throughout this note, all -algebras are unital. Let $A$ be a -algebra and let Proj $(A)$be the ordered subset (of ) of all projections in $A$ .
A -algebra $A$ is said to be a $UMF$-algebra (uniform majorization-factorization$b\in A$ , whenever $a^{*}a\leq b^{*}b$ , there exists $c\in A$ such thatalgebra), if, for any pair$||c||\leq 1$ and $a=cb$ (see for details, [1]).
Note that Rickart-algebras (and so von
Neumann algebras) are UMF-algebras(
Lemma 6). Concerning another non-trivial
ones, for examples, we note that a corona-algebra $M(A)/A$ is a $UMF$-algebra forany separable non-unital, whereis the multiplier algebra of $A$ . In
particular, the Calkin algebra
andare UMF-algebras.
$C^{*}$$C^{*}$$A$$C^{*}$$a,$$C^{*}$$\mathrm{s}\mathrm{e}\mathrm{e}$$C^{*}$$C^{*}- \mathrm{a}\mathrm{l}\mathrm{g}\mathrm{e}\mathrm{b}\mathrm{r}\mathrm{a}.4’$$\mathcal{L}(\mathcal{H})/\mathcal{K}(\mathcal{H})$$\mathcal{M}(A)$$\ell^{\infty}/c_{0}$Theorem Let $A$ be a UMF-C*-algebra. Let
be an increasing sequence of elements$(A)$$Proj(A)$inand letbe in Proj
. Suppose that whenever $x\in A$ satisfies $xe_{n}=0$ for
all $n,$ $xe_{0}=0$ . Then, for any $a\in A,$.
for all implies that$\{e_{n}\}$$e_{0}$$e_{n}\leq a$$e_{0}\leq a$

A Proof of Theorem.
Suppose that $a\in A$ satisfies thatfor all . Then, for every positive real number
for all . Thus we have that$e_{n}\leq a$$\epsilon,$$n$$a+\epsilon\geq e_{n}$

$n$

$n$

$(a+ \epsilon)-\frac{1}{2}e_{n}(a+\epsilon)-\frac{1}{2}\leq 1$

45

for all$n$

and hence we get that$||(a+ \epsilon)^{-}\frac{1}{2}en||=||(a+\epsilon)^{-}\frac{1}{2}en(a+\epsilon)-\frac{1}{2}||^{\frac{1}{2}}\leq 1$

for all . Define$n$$x(k)=n1 \sum_{=}^{k}\frac{1}{2^{n}}(a+\epsilon)-\frac{1}{2}e_{n}$for $k=1,2,$ , and it follows that $||x(k)-x(l)||arrow 0(k, larrow\infty)$ . Since $A$ is complete by
the norm topology, we can find $x\in A$ such that $||x-x(k)||arrow 0(karrow\infty)$ . Formally, we
write$\cdots$$x= \sum_{n=1}\frac{1}{2^{n}}(a+\epsilon)-\frac{1}{2}e_{n}\infty$.

By the same way, define$y(k)= \sum_{1n=}k\frac{1}{2^{n}}e_{n}$for $k=1,2,$$\cdots$, and we can get an element $y\in A$ such that$y= \sum_{n=1}^{\infty}\frac{1}{2^{n}}e_{n}$and $||y(k)-y||arrow 0(karrow\infty)$ . Since$e_{k}\geq e_{n}$for each $n\leq k$ , it follows that$x(k)=(a+ \epsilon)^{-}\frac{1}{2}e_{k}y(k)$for each and so we have that $x(k)^{*}X(k)\leq y(k)^{*}y(k)$ for each . If goes to infinity, thenit follows that $x^{*}x\leq y^{*}y$ . Since $A$ is a $UMF$-algebra, one can find $c\in A$ with $||c||\leq 1$such that $x=w$ , that is,
$k$

$k$

$k$

.

$\sum_{n=1}^{\infty}\frac{1}{2^{n}}(a+\epsilon)^{-}\frac{1}{2}e_{n}=\sum_{1n=}\frac{1}{2^{n}}Ce\infty n$

Multiplying by$e_{1}$

on the right in the above equality, we get thatce 1,$(_{n=} \sum_{1}^{\infty}\frac{1}{2^{n}})(a+\epsilon)^{-\frac{1}{2}}e_{1}=(\sum_{n=1}^{\infty}\frac{1}{2^{n}}\mathrm{I}$that is,$(a+\epsilon)^{-\frac{1}{2}}e_{1}=ce_{1}$and so we have.

$\sum_{n=2}^{\infty}\frac{1}{2^{n}}(a+\epsilon)^{-}\frac{1}{2}e_{n}=\sum_{2n=}\frac{1}{2^{n}}Ce\infty n$on the right in the above equality, we have2.
Again multiplying by
ourHence,
by.
all
for
By repeating these arguments, it follows that$||c||\leq
1$, we have that
. Sinceassumption, we get that$(a+\epsilon)^{-\frac{1}{2}}e_{2}=ce$$e_{2}$$(a+\epsilon)^{-\frac{1}{2}}e_{n}=ce_{n}$$(a+\epsilon)^{-\frac{1}{2}}e_{0}=ce_{0}$,$(a+ \epsilon)-\frac{1}{2}e_{0}(a+\epsilon)^{-}\frac{1}{2}=ce0^{c^{*}\leq 1}$

$n$

46

and so$e_{0}\leq a+\epsilon$

for all$\epsilon$

follows, that is, we get that

Corollary ([2]) Let $A$ be$a$Rickart$C^{*}$$e_{0}\leq a$. This completes the proof.

-algebra. Then, $A$ is a-normal.

Before going into the proof of Corollary, we shall make a survey of results on Rickart-algebras for the sake of completeness ([1], [2] and [3]).

$C^{*}$

Lemma 1 (See [3]) Let $A$ be$a$Rickart$C^{*}$-algebra. Then $A$ is unital.

In fact, by our definition, the right annihilator $\mathcal{R}a(\{0\})=eA$ for some $e\in ProjA$ . Since$\mathcal{R}a(\{0\})=A$ , it follows that $A=eA$ and sois the unit for $A$ because $A$ is-algebra.
From now onward, we shall denote this by 1.
$e$$\mathrm{a}*$$e$Lemma 2 (See [3]) Let $A$ be Rickart -algebra and $x\in A$ . Then, there are a unique$f\in A$ such thatpair of projections(1) $xe=x$ ,(2) for any $y\in A,$ $xy=0$ if, and only if, $ey=0$ ,(3) $fx=x$ ,and(4) for any $y\in A,$ $yx=0$ if, and only if, $yf=0$ .
$C^{*}$$a$$e,$

In fact, since $\mathcal{R}a(\{x\})=(1-e)A$ and the left annihilator $\mathcal{L}a(\{x\})=A(1-f)$ , Lemma
2 follows easily.
From now onward, we shall denote by $RP(x)$ and by $LP(x)$ respectively.
$f$

$e$

Lemma 3 (See [3]) Let $A$ be Rickart-algebra and suppose that a family of$A$
projections in
has a supremum in Proj $(A)$ . Then, for any $x\in A$ , whenever $xe_{i}=0$
for all , we have $xe=0$ .
$C^{*}$$a$$\{e_{i}\}$

$e$

$i$

In fact, if $xe_{i}=0$ for all , then $e_{i}\leq 1-RP(x)$ for all and so we have $e\leq 1-RP(x)$by our definition, that is, $xe=0$ follows.
$i$

$i$

of projections in $A$ , put $ef=f+RP(e(1-f))$ and put $e\wedge f=$$e-LP(e(1-f))$ . Then by these operations, Proj $(A)$ becomes a lattice with respect tothe natural order.
For any pair

$e,$

$f$

Lemma 4 (See [3]) Let $A$ be
complete.

$a$

Rickart$C^{*}$-algebra.

Then, the lattice Proj $(A)$ is$\sigma-$47

First of all, we shall show that if every orthogonal sequence of projections in $A$ has asupremum in Proj $(A)$ , then every sequence of projections has a supremum in Proj $(A)$ .
be any given sequence of projections. Put $f_{1}=e_{1}$ and put
In fact, let$\{e_{n}\}$$f_{n}=e_{1}e_{2}\cdots e_{n}-(e_{1}\vee e_{2}\cdots\vee e_{n-1})(n\geq 2)$.

. Note that
. We shall show thatis orthogonal. Let. Then note that $f_{0}\geq e_{\vee}e_{2}\cdots\vee e_{n+1}-(e_{1}$. Suppose that $f_{0}\geq e_{i}i=1,2,$. Hence, $f_{0}\geq e_{1}\vee e_{2}\cdots\vee e_{n+1}$ follows. So
and that
we are done.
be any orthogonal family of projections in $A$ . Define as before
Next letThen$f_{0}=_{n=1}^{\infty}e_{n}$$f_{0}=_{n=1}^{\infty}f_{n}$$\{f_{n}\}$$\cdots,$ $n$$f_{0}\geq e_{1}$$f_{0}\geq e_{1}\vee e_{2}\vee\cdots\vee e_{n}$$e_{2}\vee\cdots\vee e_{n})$$\{p_{n}\}$$x_{n}= \sum_{k=1}^{n}\frac{1}{2^{k}}p_{k}(n=1,2, \cdots)$

and we have $||x_{m}-x_{n}||arrow 0$ as $m,$ $narrow\infty$ as before and so there is $x_{0}\in A$ such thatas $narrow\infty$ . Let $e=RP(x_{0})$ . If $f\in Proj(A)$ satisfies that $p_{n}\leq f$ for all ,then $x_{n}f=x_{n}$ for all and so we have that $xf=x$ , that is, $e\leq f$ follows. It remains tofor all . For any fixed index $m,$ $p_{m}x_{n}=2^{-m}pm$ for all $n\geq m$ . Therefore,show that$e_{m}x_{0}=2^{-m}p_{m}$ for all $m$ , that is, $p_{m}=2^{m}e_{m}x_{0}$ for all $m$ . Since $x_{0}e=x_{0}$ , it follows that$p_{m}e=p_{m}$ for all $m$ . So we are done.
$n$$||x_{n}-x\mathrm{o}||arrow 0$$n$$p_{n}\leq e$$n$if, for each $x\in B$ , there exists a
A -algebra has polar decom.position property
which satisfies properties (1) and (2) in Lenuna and there
unique projection$w\inB$such that $x=w|x|$ and $w^{*}w=RP(|x|)$ .
exists a partial isometry$C^{*}$$(\mathrm{P}\mathrm{D})$$\mathcal{B}$$RP(x)\in \mathcal{B}$Lemma 5 ([6]) Let
over have$\mathcal{M}_{2}(B)$Take any

Then

$B$

$a,$

$B$

$(\mathrm{P}\mathrm{D})$$b\in B$$2c$

be a unital -algebra such that
. Then is a $UMF- C^{*}-$ algebra.

with$C^{*}$and the$2\cross 2$

matrix algebra$\mathcal{B}$$a^{*}a\leq b^{*}b$

. Let

$x=$

$x\in \mathcal{M}_{\mathit{2}}(\mathcal{B})$$B$.

. Let $x=w|x|$ be a polar decomposition of

$x$

in

$\mathcal{M}_{2}(B)$. Note that$x^{*}x==$and hence it follows that$|x|=$ .

48

Put

$w=$

.

Then

$w^{*}w===RP(|x|)$.

Hence we have that$w_{2}^{*}w_{2}+w^{*}w44=0$

and so $w_{2}=w_{4}=0$ follows, which impleis that$w=$ .

Since

$==$

$w_{1}^{*}w_{1}+w_{3}^{*}w_{3}$is a projection, we get that $||w||\leq 1$ . Hence it follows that,that is, $a=w_{1}|b|=w_{1}u^{*}b$ , where $b=u|b|$ is a polar decomposition of in . Put $c=w_{1}u^{*}$
and we get the desired factorization $a=cb$ and $||c||\leq 1$ . Hence is a UMF-C*-algebra.
$b$$\mathcal{B}$$\mathcal{B}$By an ingenious, deep analysis of the structure of Rickart -algebras and regular rings,
Ara and Goldstein showed in [1] that for every Rickart -algebra $A,$
is a Rickart-algebra withfor each . Hence we have the following:$C^{*}$$C^{*}$$C^{*}$$(\mathrm{P}\mathrm{D})$$\mathcal{M}_{n}(A)$$n$Lemma 6 ([1]) Every Rickart$C^{*}$-algebra is a UMF-C*-algebra.

A Proof of Corollary
In fact, letbe any increasing sequence in Proj $(A)$ . Then it has a supremum
in Proj $(A)$ in such a way that for any $x\in A,$ $e_{n}x=0$ for all implies that $xe_{0}=0$ by
Lemma 3 and Lemma 4. Since $A$ is a $UMF$-algebra by Lemma 6, is just the supremum
ofin by the theorem.
$\{e_{n}\}$$e_{0}$$n$$e_{0}$$\{e_{n}\}$$A$

Remark. A parallel problem in $AW^{*}$ -algebra theory was considered by.M. Wrightand the author [10], [7] and [9] (see also [5]). The above corollary gives us also a simple
alternative proof of the normality of a-finite $AW^{*}$ -algebras (see for details, [10], [5], [7] and[9] .
$\mathrm{J}.\mathrm{D}$

$)$

49

References[1] P. Ara and D. Goldstein, A solution of the matrix problem for Rickart -algebras,
Math. Nachr., 164(1993), 25a-270.
[2] P. Ara and D. Goldstein, Rickart -algebras are -normal, Archiv Math., 65(1995),$C^{*}$$C^{*}$$\sigma$

505-510.
[3] S.K. Berberian, Baer*-rings, Springer, Berlin and New York, 1972.
-algebras are monotone[4] E. Christensen and G.K. Pedersen, Properly infinite
sequentially complete, Bull. London Math. Soc., $16(1984)$ , 407-410.
$AW^{*}$[5] M. Hamana, Regular embeddings of -algebras in monotone complete -algebras,
J. Math. Soc. Japan, 33(1981), 159183.
[6] G.K. Pedersen, Three quavers on unitary elements in -algebras, Pacific J. Math.,
137(1989), 169-179.
[7] K. Sait\^o, On normal $AW^{*}$ -algebras, T\^ohoku Math. J., 33(1981), 567-572.
[8] K. Sait\^o, On -normal -algebras, To appear in the Bull. London Math. Soc..
[9] K. Sait\^o and J. D.M. Wright, All $AW^{*}$ -factors are normal, J. London Math. Soc.,$C^{*}$$C^{*}$$C^{*}$$\sigma$$C^{*}$

44(1991), 143154.
[10] J.D.M. Wright, Normal $AW^{*}$ -algebras, Proc. Roy. Soc. Edinburgh A 85(1980),

137-141.