数理解析研究所講究録 1243 巻 2002 年 57-68

57

Non-commutative Korovkin-type theorems.
Mitsuru Uchiyama
内山充

Department of Mathematics, Pukuoka Univ. of Education
福岡教育大学数学科

Munakata, Fukuoka , 811-4192, Japan

1. Introduction
Let $A$ be aC’-algebra with an identity 1. Apositive linear map on $A$
is called aSchwarz map if it satisfies $\Phi(a)^{*}\Phi(a)\leq\Phi(a^{*}a)$ for every $a\in A$ .
It is well-known that if $A$ is commutative then every contractive positive
linear map is aSchwarz map. Robertson [11] has proved that, for asequenceof Schwarz maps, the set { $a\in A$ : $\Phi_{n}(x)arrow x(narrow\infty)$ for,is a -subalgebra. As acorollary he also stated that for$x=a$ ,of contractive positive linear maps on the commutative
asequenceC’-algebra $C(X)$ of continuous complex valued functions on acompact
Hausdorff space $X$ , the set $\{u\in C(X) : \Phi_{n}(u)arrow u, \Phi_{n}(|u|^{2})arrow|u|^{2}\}$-subalgebra. By identifying $C_{f}(X)$ with the subalgebra of $C(X)$ ,is a
the Stone-Weierstrass theorem shows that this contains the Korovkin
theorem.
Let us recall that if is aC’-subalgebra of $C(X)$ and if for any point$x\in X$ there is a $f\in B$ such that $f(x)\neq 0$ and if
separates $X$ , then$\Phi$$\{\Phi_{n}\}$$a^{*}a$$aa^{*}\}$$C^{*}$$\{\Phi_{n}\}$$C^{*}$$B$$B$$B=C(X)$ .

of
Limaye and Namboodiri[7] have shown that for asequence
Schwarz maps and a -homomorphism$, the set { $a\in A$ :-closed)subalgebra,is aclosed (not$\{\Phi_{n}\}$$\Phi_{n}(a)arrow$$*$$\Phi(a)$$\Phi_{n}(a^{*}a)arrow\Phi(a^{*}a)\}$$\mathrm{n}\mathrm{e}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{l}\mathrm{y}*$58
and that { $a\in A:\Phi_{n}(x)arrow\Phi(x)$ for $x=a$ ,,}, the intersectionof this subalgebra and its adjoint, is aC’-subalgebra. By the Kadison
theorem acontractive positive linear map
satisfies $\{\Phi(a)^{*}, \Phi(a)\}\leq$for all $a\in A$, where , is the Jordan product, i.e., $\{x,y\}=$$a^{*}a$$aa^{*}$$\Phi$$\Phi(\{a^{*}, a\})$

$\{$

$\}$

$xy+yx$ .

Limaye and Namboodiri [8] have shown that, for asequenceof$\{a\in
A:\Phi_{n}(a)arrow$
positive linear maps and -homomorphism , the set, $\Phi_{n}(\{a^{*},a\})arrow\Phi(\{a^{*}, a\})\}$ is -closed, norm closed subspace whichis also closed with respect to the Jordan product.
$\{\Phi_{n}\}$$\Phi$$\mathrm{a}*$$\Phi(a)$$\mathrm{a}*$Acontinuous real valued function $f(t)$ on
called an operator
monotone function if $f(a)\geq f(b)$ whenever $a\geq b\geq 0$ , $a,b\in A$. Thisfunction is characterized as follows: is an operator function on
if
and only if has an analytic extension $f(z)$ to the upper half plane suchthat $Imf(z)>0$ for $Imz>0$ . Therefore if is an operator function,then so are
and $f(1/t)^{-1}$ .
and $\log(t+1)$ are operator
monotone functions. It is well-known that an operator monotone function
is increasing and concave.
$[0, \infty)\mathrm{i}\mathrm{s}$$[0, \infty)$

$f$

$f$

$f$

$f(\sqrt{t})^{2}$$t^{\mathrm{p}}(0<p\leq 1)$

The aim of this paper is to give estimates of the norms related toschwarz maps and to extend Korovkin-type theorems by using operator
monotone functions. These estimates seem to be very useful for studying
Korovkin-type theorems in anon-commutative -algebra;for instace wegive aquite simple proofs for many results given above.
2. generalized Schwarz maps.
$C^{*}$$\mathrm{w}\mathrm{i}\mathrm{u}$Let $A$ be aC’-algebra with aunit 1. Alinear map is called aSchwarz
map if $\Phi(a)^{*}\Phi(a)\leq\Phi(a^{*}a)$ for every $a\in A$ , and apositive linear map
with $\Psi(1)\leq 1$ was called aJordan-Schwarz map in [3], since it satisfies$\{\Psi(a)’, \Psi(a)\}\leq\Psi(\{a^{*}, a\})$ as we mentioned in the previous section. To
investigate two cases given above all at once and to extend them, weconsider the following binary operation in $A$ :$\Phi$$\Psi$$\circ$(i) $(\alpha x+\beta y)\circ z=\alpha(x\circ z)+\beta(y\circ z)$(ii) (xo$y)^{*}=y^{*}\mathrm{o}x^{*};$$(\alpha,\beta\in \mathrm{C},x,y, z\in A)$

;

59
(iii)

;

$x^{*}\circ x\geq 0$(iv) there is areal number $M$ such that $||x\mathrm{o}y||\leq M||x||||y||$ ;(v) $(x\circ y)\circ z=x\circ(y\circ z)$ ,or(v)’ $(x\mathrm{o}y)=(y\circ x)$ and $a\circ a=a^{2}$ for $a=a^{*}$ .

One may regard this binary operation as the ordinary product or the
Jordan produt.
-closed and norm-closed subspace of $A$ which is
Beckhoff [3] called
also closed with respect to the Jordan product a -subalgebra of $A$ .
$\mathrm{a}*$$J^{*}$

We call alinear subspace, $y\in B$ , and -subalgebra if$B\subseteq A$$x$$0^{*}$

a -subalgebra if $x\circ y\in B$ , whenever$\circ$is a -subalgebra

$B$

$0$

$\mathrm{a}\mathrm{n}\mathrm{d}*$-closed.

If a -subalgebra is complete, that is norm-closed, then it is called acomplete o’-subalgebra.
$0^{*}$

Definition. Alinear map
map w.r.t. if satisfies$0$$\Phi$: $Aarrow A$ is called ageneralized Schwarz$\Phi$$\Phi(x^{*})=\Phi(x)^{*}$

and$\Phi(x^{*})\circ\mathrm{x})\leq\Phi(x^{*}\circ x)$

for every $x\in A$.

We remark that ageneralized Schwarz map is not necessarily positive(that is positive means $\Phi(a)\geq 0$ whenever $a\geq 0$).
$\Phi$$\Phi$

is called aw.r.t.
Definition. Ageneralized Schwarz map
homomorphism w.r.t. if $\Phi(x)^{*}0\Phi(x)=\Phi(x^{*}\circ x)$ for every $x\in A$.
$\Phi$

$0$

$*-$

$0$

Let us note that if 4is
tion$\mathrm{a}*$-homomorphism w.r.t , then by apolariza$\circ$$4x^{*} \circ y=\sum_{n=0}^{3}i^{n}(i^{n}x+y)^{*}\circ(i^{n}x+y)$,

we deduce $(x)o\Phi (y)=\Phi (x ) for every $x,y\in A$ . It is clear that if-homomorphism w.r.t. isis the original product in $A$ , thenhomomorphism in the ordinal sense, and that if is the Jordan product$\circ y$$\mathrm{o}$$0$$\mathrm{a}*$$\circ$$\mathrm{a}*-$

60
then

-homomorphism w.r.t. is aC’-homomorphism in the ordinalsense. Abounded linear functional \langle 6 of A is called astate if$ is positive$\mathrm{a}*$and $(1)$\circ$$\ovalbox{\tt\small REJECT}$

1.

Theorem 2.1. Let
For $x,y\in A$ set

be a generalized Schwarz map w.r.t.

$\Phi$$X:=\Phi(x^{*}\mathrm{o}x)-\Phi(x)^{*}0\Phi(x)\geq 0$$\mathrm{Y}:=\Phi(y^{*}\mathrm{o}y)-\Phi(y)^{*}\circ\Phi(y)\geq 0$$Z:=\Phi(x^{*}\mathrm{o}y)-\Phi(x)^{*}0\Phi(y)$Then we have.

$\circ$on $A$ .

,

,

(1)

$|\phi(Z)|\leq\phi(X)^{1}\pi\phi(\mathrm{Y})\#$

for every state$\phi$$\in A’$.

$Fk\hslash her$

we have(2)$\frac{1}{2}||Z||\leq||X||^{\}}||\mathrm{Y}||\}$

Proof. For every complex number , we have$\alpha$,$0\leq\Phi((x+\alpha y)^{*}\circ(x+\alpha y))-\Phi(x+\alpha y)^{*}\circ\Phi(x+\alpha y)=X+\alpha Z+\overline{\alpha}\overline{Z}+|\alpha|^{2}\mathrm{Y}$ffom which it follows thatfor every state $\phi\in A’$ . Thus we can
is a state of $A$} is the numerical radius$0\leq\phi(X)+2Re\alpha\phi(Z)+|\alpha|^{2}\phi(\mathrm{Y})$easily get (1). Since $\sup$ { $\phi(Z)$ :$w(Z)$ , from (1) we obtain $w(Z)\leq w(X)\tau w(\mathrm{Y})\pi 11$ .
It is well-known that $\frac{1}{2}||a||\leq w(a)\leq||a||$ for every $a\in A$ .
Thus we obtain (2).
$\phi$$\square$Rom the inequality (2) we can easily prove results mentioned in thefirst section.
Proposition 2.2. Let
be a sequence of generalized Schwarz maps
on $A$ with, and $\Phi a*$ -homomorphism $w.r.t$.
on$A$ with $||\Phi||\leq 1$$\inA$. Then the set $D:=\{x$: $||\Phi_{n}(x)-\Phi(x)||arrow$.
,as} $u$ a complete -subalgebra.
$\{\Phi_{n}\}$$w.\mathrm{f}\mathrm{o}\mathrm{r}$

$0$

$0$

$||\Phi_{n}||\leq 1$$||\Phi_{n}(x^{*}\mathrm{o}x)-\Phi(x.

\mathrm{o}x)||arrow 0$$0$$narrow\infty$$\circ$61

0Proof. Suppose x cE D. From the definition of$\ovalbox{\tt\small REJECT}$$||\mathrm{O}_{n}(\mathrm{r})’\circ \mathrm{O}_{n}(\mathrm{r})$$\circ$it follows that$-\mathrm{O}(\mathrm{r})’\circ+(\mathrm{r}\ovalbox{\tt\small REJECT}|$$M||^{\ovalbox{\tt\small REJECT}}!_{n}\ovalbox{\tt\small REJECT}(\mathrm{r})’-\ovalbox{\tt\small REJECT}!\ovalbox{\tt\small REJECT}(\mathrm{r})’||1\mathrm{F}^{\ovalbox{\tt\small REJECT}}(x\ovalbox{\tt\small REJECT}|+M|\mathrm{F}^{\ovalbox{\tt\small REJECT}}n(\mathrm{r})’||||^{\ovalbox{\tt\small REJECT}}!_{n}^{\ovalbox{\tt\small REJECT}}(\mathrm{r})-\ovalbox{\tt\small REJECT}!\ovalbox{\tt\small REJECT}(\mathrm{r}\ovalbox{\tt\small REJECT}|\mathrm{e}0$$\ovalbox{\tt\small REJECT}$.

This and$\Phi_{n}(x^{*}\circ x)arrow\Phi(x^{*}\mathrm{o}x)=\Phi(x)^{*}\circ\Phi(x)$imply$||\Phi_{n}(x^{*}\mathrm{o}x)-\Phi_{n}(x)^{*}\circ\Phi_{n}(x)||arrow 0(narrow\infty)$.

Thus for every $y\in A$ , in virtue of (2) we get$||\Phi_{n}(x^{*}\circ y)-\Phi_{n}(x)^{*}\circ\Phi_{n}(y)||arrow 0(narrow\infty)$,

which implies that$\Phi_{n}(x^{*}\mathrm{o}y)arrow\Phi(x)^{*}0\Phi(y)$$if$ $x\in D$

and $\Phi_{n}(y)arrow\Phi(y)$ .

Prom this one can see that $x\circ y\in D$ if $x,y\in D$ . Sincebounded,is complete.

$\{\Phi_{n}\}$is uniformly$\square$$D$

Corollary 2.3. Under the above condition the set
o’-subalgebra.

$D\cap D^{*}$is acomplete

Remark. Since every bounded linear functional on $A$ is alinear combiof $A$ weakly converges
nation of at most four states of $A$ , asequence
to if and only if $\phi(a_{n})arrow\phi(a)$ for every state . By using (1) we cansee that$\{a_{n}\}$$\phi$$a$$D_{1}:=\{x\in A:\Phi_{n}(x) arrow\Phi(x)(w), \Phi_{n}(x^{*}\circ x)arrow\Phi(x^{*}\circ x)(w)\}$is acomplete -subalgebra, and hence that
subalgebra.
$0$$D_{1}\cap D_{1}^{*}$is acomplete$0^{*}-$

Proposition 2.2, Corollary 2.3 and Remark were proved in [7] [8] [11]when is the original product or the Jordan product in $A$ , but the above
proof seems to be simple.
We denote the -subalgebra of $A$ generated by asubset of $A$ by$J^{*}(S, 0)$ or simply by $J^{*}(S)$ . We define the Korovkin closure $K\sigma rA(S)$$\circ$$\circ^{*}$

$S$

62
of asubset $S\subseteq A$ as follows :KorA(S) is the set of all $x\in A$ suchthat for every sequenceof positive generalized Schwarz maps w.r.t.
with, $\Phi_{n}xarrow x$ wheneverfor every $a\in S$ . Herethe convergence means convergence in the norm topology. Prom thisdefinition the next follows :$\{\Phi_{n}\}$$||\Phi_{n}||\leq 1$$\mathrm{o}$$\Phi_{n}aarrow a$

Lemma 2.4. $Kor_{A}(S)$S$\subseteq Kor_{A}(T)$$\subseteq Kor_{A}(T)$.

Corollary 2.5. For a subset$J^{*}(S)\subseteq Kor_{A}(S_{1})$$S\subseteq A$

if S

.

$\subseteq T$

if$Kor_{A}(S)\subseteq Kor_{A}(T)$, we have,where $S_{1}:=S\cup\{x^{*}\mathrm{o}x$ : x$\in S\}\cup\{x\mathrm{o}x^{*}$:x$\in S\}$.

(3)

Proof, Fix asequenceof positive generalized Schwarz maps w.r.t.
withsuch thatfor every. We have only toshow
for every $t\in J^{*}(S)$ . By Corollary 2.3, the set { $x\in A$ :,,is a -subalgebra.
Since it contains , it contains $J^{*}(S)$ too. Thus we haveforevery $t\in J^{*}(S)$ .
$\{\Phi_{n}\}$$||\Phi_{n}||\leq 1$$\circ$$\Phi_{n}(t)arrow t$$t\in S_{1}$$\Phi_{n}(t)arrow t$$\Phi_{n}(x)$$arrow x$$\Phi_{n}(x^{*}\mathrm{o}x)arrow x^{*}\mathrm{o}x$$\Phi_{n}(x\mathrm{o}x^{*})arrow x\circ x^{*}\}$$\circ^{*}$$S$$\Phi_{n}(t)arrow t$$\square$

Theorem 2.6. Let be an operator monotone function on$f(0)=0$ and $\lim_{\mapsto\infty}f(x)=\infty$ . Set $g=f^{-1}$ . Then for a subset
we have$f$$J^{*}(S)\subseteq KorA(S_{2})$$[0, \infty)$
$S$

with

of

$A$

,where $S_{2}:=S\cup\{g(x^{*}\mathrm{o}x) : x\in S\}\cup\{g(x\mathrm{o}x^{*}) : x\in S\}$ .
(4)

Proof, Let
be asequence of positive generalized Schwarz maps$\circwith$w.r.t.
such that $\Phi_{n}(t)arrow t$ for every $t\in S_{2}$ . It was
shown in [4] [5] that$\{\Phi_{n}\}$$||\Phi_{n}||\leq 1$$\Phi_{n}(f(a))\leq f(\Phi_{n}(a))$

for every$a\geq 0$,(5)which implies$0\leq\Phi_{n}(x^{*}\mathrm{o}x)-\Phi_{n}(x)^{*}0\Phi_{n}(x)\leq f(\Phi_{n}(g(x^{*}\mathrm{o}x)))-\Phi_{n}(x)^{*}\circ\Phi_{n}(x)$63oor), it follows thatfor every r.
e x’ ox. Thus the right side of the above inequality converges to 0,
from which it follows that$\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{m}+.(g(x’$$\mathrm{r}))\ovalbox{\tt\small REJECT}$$f(+_{n}(g(\mathrm{r}’\circ$$g(\mathrm{x}’$$\mathrm{r})))$$\lim\Phi_{n}(x^{*}\circ x)=\lim\Phi_{n}(x)^{*}\circ\Phi_{n}(x)=x^{*}\circ x^{*}$.

.
Similarly we can getfor every in which was given inThus we have shown that
$S_{1}\subseteq
KorA(S_{2})$ . By (3) and Lemma
Corollary 2.5, that is, we have shown
2.4 we have $J^{*}(S)\subseteq Kor_{A}(S_{1})\subseteq Kor_{A}(S_{2})$ . Consequently we get (4).
$\mathrm{h}.\mathrm{m}$$\Phi_{n}(x\mathrm{o}x^{*})=x\mathrm{o}x^{*}$$t$$\Phi_{n}(t)arrow t$$S_{1}$$\square$

Theorem 2.7. Let be a function given in Theorem 2.6. For a finite
set $S=\{s_{1}, \ldots, s_{n}\}$ , we have$g$$J^{*}(S)\subseteq KorA(Sx)$above$S_{3}=S \cup\{g(\sum_{\dot{|}=1}^{n}(s_{\dot{0}}^{*}\mathrm{o}s:+s:\mathrm{o}s_{\dot{l}}^{*}))\}$. Let us take an arbitrary sequence$Pro\mathrm{o}/$Schwarz maps w.r.t$t\in S_{3}$$\leq$

with$||\Phi_{n}||\leq 1$(6)of positive generalized
such that $\{\Phi_{n}(t)\}arrow t$ for every$\Phi_{n}$.

For each0$\circ$.

$i$$\Phi_{n}(s_{i}^{*}\mathrm{o}s:)-\Phi_{n}(s:)^{*}\circ\Phi_{n}(s:)\leq\sum_{j=1}^{n}\{\Phi_{n}(s_{j}^{*}\mathrm{o}s_{j})-\Phi_{n}(s_{j})^{*}\circ\Phi_{n}(s_{j})\}$$\leq$$\Phi_{n}(\sum_{j}(s_{j}^{*}\mathrm{o}s_{j}+s_{j}\mathrm{o}s_{j}^{*}))-\sum_{j}\{\Phi_{n}(s_{j})^{*}0\Phi_{n}(s_{j})+\Phi_{n}(s_{j})\circ\Phi_{n}(s_{j})^{*}\}$.

$\leq$$f( \Phi_{n}(g(\sum_{j}(s_{j}^{*}\circ s_{j}+s_{j}\circ s_{j}^{*}))))-\sum_{j}\{\Phi_{n}(s_{\dot{f}})^{*}\circ\Phi_{n}(s_{j})+\Phi_{n}(s_{j})\circ\Phi_{n}(s_{j})^{*}\}$.
converges toSince the right side converges to 0,. Thus we have
converges to
Similarly we can see that
$S_{\dot{l}}^{*}\circ S:$$\Phi_{n}(s_{\dot{1}}^{*}\circ s:)$$\Phi_{n}(s_{i}\circ s^{*}.\cdot)$$s_{i}\circ s_{\dot{l}}^{*}$

shown that $S_{1}:=S\cup\{s^{*}\circ s:s\in S\}\cup\{s\mathrm{o}s^{*} : s\in S\}\subseteq KorA(S_{3})$ .
By (3) and Lemma 2.4, we get (6).

$\square$

64
Theorem 2.8. Under the same assumption as Theorem 2.6, we have$J^{*}(S)\subseteq KorA(S\cup\{g(x.\circ x+x\circ x^{*}) :Proof By substituting for$x$orem 2.7, we get0$s_{\dot{1}}$x\in S\})$.

(7)in the inequalities of the proof of The-$\leq$$\Phi_{l},(x.

\mathrm{o}x)-\Phi_{n}(x)^{*}0\Phi_{n}(x)$$\leq$$f(\Phi_{n}(g(x^{*}\mathrm{o}x+x\mathrm{o}x.)))-\{\Phi_{n}(x).0\Phi_{n}(x)+\Phi_{n}(x)0\Phi_{n}(x)^{*}\}$

Thus in the same fashion as Theorem 2.6 we can get (7).

.
$\square$In the above three theorems we needed conditions $/(0)=0$ ,in order that $f^{-1}=g$ is defined onthat (5) is valid for every
positive map. However, when we consider the case of $1\in S$ , we can loose
the condition $f(0)=0$ .
$f(\infty)=\mathrm{o}\mathrm{o}$$[0, \infty)\mathrm{a}\mathrm{n}\mathrm{d}$Theorem 2.9. Suppose $1\in S\subseteq A$ . Let be an operator monotone
such that $f(0)\leq 0,f(\infty)=\infty$ . Set $g=f^{-1}$ .
function defined on
Then we have$J^{*}(S)\subseteq KorA(S_{2})$ , where $S_{2}=S\cup\{g(x^{*}\mathrm{o}x)|x\in S\}\cup\{g(x\mathrm{o}x^{*})|x\in S\}$.
$f$$[0, \infty)$

Proof Let us take an arbitrary sequence

Schwarz maps w.r.t withBy (5) we get$\circ$$||\Phi_{n}||\leq 1$$\{\Phi_{n}\}$such that$\Phi_{n}(f(a)-f(0)1)\leq f(\phi_{n}(a))-f(0)1$of positive generalized
for every.

$\Phi_{n}(t)arrow t$

for every$t\in S_{2}$$a\geq 0$,

and hence$\Phi_{n}(a)=\Phi_{n}(f(g(a)))\leq f(\Phi_{n}(g(a)))-f(0)(1-\Phi_{n}(1))$.

From this, for every $x\in S$ we deduce0$\leq$$\Phi_{n}(x^{*}\mathrm{o}x)-\Phi_{n}(x)^{*}0\Phi_{n}(x)$$\leq$$f(\Phi_{n}(g(x^{*}\mathrm{o}x)))-f(0)(1-\Phi_{n}(1))-\Phi_{n}(x)^{*}\circ\Phi_{n}(x)$.

65
Since the bigger side in the above converges to 0, we obtain thatx
. By (3) we get
Similarly we can$)$\mathrm{O}_{n}(\mathrm{r}’\circ$$-+\mathrm{J}’\circ\.$$\mathrm{g}\mathrm{e}\mathrm{t}+.(\mathrm{r}\circ \mathrm{r}’)\ovalbox{\tt\small REJECT}$$\mathrm{J}’(S)\ovalbox{\tt\small REJECT}_{\ovalbox{\tt\small REJECT}}K\mathrm{c}\mathrm{v}_{A}(S_{2})$$\circ x’$.

[1

In the same fashion as the above proof, we can easily extend Theorem
2.7 and Corollary 2.8 to the case of $1\in S$ as follows:
Theorem 2.10. Let $S=\{s_{1}, \ldots, s_{n}\}$ be a subset of $A$ and include
such that
1. Let be an operator monotone function defined on$f(0)\leq 0$ . $f(\infty)=\infty$ . Set $g=f^{-1}$ . Then we have $J^{*}(S)\subseteq K\sigma rA(S_{3})$ ,.
where$[0, \infty)$$f$$S_{3}=S \cup\{g(\sum_{\dot{|}=1}^{n}(s_{\dot{1}}^{*}\mathrm{o}s:+s:\mathrm{o}s_{\dot{l}}^{*}))\}$

Corollary 2.11. Under the same assumption as Theorem 2.9, we have

: x $\in S\})$ .

$J^{*}(S)\subseteq K\sigma rA(S\cup\{g(x^{*}\circ x+x\circ x^{*})$

Remark. In the above theorems we studied not the universal KOrovkin closures (the definiton is given below) but the Korovkin closures,. To get the same
instead ofthat is, the case whereisconclusions for 4as theorems, we would have to assume that-homomorphism in the ordinary sense behomomorphism w.r.t.
ca se of $\Phi(g(a))=g(\Phi(a))$ ;we thought it is abit complicated assumption. If abinary operation is the ordinary product or the Jordan prod-homomorphismuct,-homomorphism in the ordinary sense is
w.r.t.
too. Now we consider this case. Let us define the universal
as follows : $Kor_{A}^{u}(S)$ isof asubset
Korovkin closure-homomorphism and for evthe set of all $x\in A$ such that forwithery sequenceof positive generalized Schwarz maps w.r.t.
$a\inS$. Whenfor every,
whenever $n
is the ordinary product or the Jordan product, it is not difficult to seethat we can substitute $Kor_{A}^{u}(S)$ for KorA in the above theorems.
$\Phi_{n}arrow\Phi$$\Phi_{n}arrow 1$$\Phi$$*-$$\circ \mathrm{a}\mathrm{n}\mathrm{d}*$$\circ$$\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}*$$\mathrm{a}*$$0$$K\sigma r_{A}^{u}(S)$$S\subseteq A$$\Phi$$\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{y}*$$\{\Phi_{n}\}$$||\Phi_{n}||\leq 1$$\circ$$(\mathrm{a})arrow\Phi(a)$$\Phi_{n}xarrow\Phi x$$\circ$At the end of this section we consider the case where is the ordinary
product, and we extend the Robertson’s theorem in avisible form :$\circ$Theorem 2.12. Let$\{\Phi_{n}\}$be a sequence of Schwarz maps and$\Phi a$66
-homomorphism, and let f be an operator monotone function on [0, oo)
with0,oo. Set g. Then the set C {a c;- A$*$$f(0)\ovalbox{\tt\small REJECT}$$+.(\mathrm{r})\ovalbox{\tt\small REJECT}$$f(oo)\ovalbox{\tt\small REJECT}$$+(\mathrm{z})f_{orovalbox{\tt\small REJECT}}a$

Proof Thatand that$\Phi_{n}(a)$$\Phi_{n}(g(a^{*}a))$$\ovalbox{\tt\small REJECT}$, $g(a’ a)$ ,$f^{-1}$$g(\mathrm{o}\mathrm{o}’)\}$$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}$

is a C’-subalgebra.

converges to
impliesconverges to $\Phi(g(a^{*}a))$ implies$\Phi(a)$$\Phi_{n}(a)^{*}\Phi_{n}(a)arrow\Phi(a)^{*}\Phi(a)$$f(\Phi_{n}(g(a^{*}a)))arrow f(\Phi(g(a^{*}a)))=\Phi(a^{*}a)=\Phi(a)^{*}\Phi(a)$.

Thus we have $f(\Phi_{n}(g(a^{*}a)))-\Phi_{n}(a).\Phi_{n}(a)arrow 0$. Prom (5) it follows that$0\leq\Phi_{n}(a^{*}a)-\Phi_{n}(a)^{*}\Phi_{n}(a)\leq f(\Phi_{n}(g(a^{*}a)))-\Phi_{n}(a)^{*}\Phi_{n}(a)$.

Hence we get. Similary we can get.
Thus, where $D$ is given in Proposition 2.2. Conversely, since$D\cap D^{*}$
is a -subalgebra(Corollary 2.3),. Consequently
is aC’-subalgebra.
Cl$\Phi_{n}(a^{*}a)arrow\Phi(a^{*}a)$$\Phi_{n}(aa^{*})arrow\Phi(aa^{*})$$C\subseteq D\cap D^{*}$$C^{*}$$D\cap D^{*}\subseteq C$$C$

3. Korovkin sets in $C(X)$ .
Let X be acompact Hausdorff space and $C(X)$ aC’-algebra of all complex valued continuous functions . Though we treat only complex algebras, the results which will be gotten for complex algebras in this section
hold for real algebras too. Since apositive linear map on $C(X)$ satisfies$|\Phi(fg)|^{2}\leq\Phi(|f|^{2})\Phi(|g|^{2})$ ,is aSchwarz map with respect to the ordinary product if $\Phi(1)\leq 1$ . Asubset S of $C(X)$ is called aKorovkin
set if $K_{C(X)}(S)=C(X)$ . Here $KC(x)$ is the set of every x $\in C(X)$
which satisfies that $\Phi_{n}(x)arrow x$ for every sequence of Schwarz maps
(i.e,$0\leq\Phi_{n}, \Phi_{n}(1)\leq 1)$ such that $\Phi_{n}(s)arrow s$ for aU s $\in S$. $C^{*}(S)$stands for the -aubalgebra generated by S.
$\Phi$$\Phi$$C^{*}$

Theorem 3.1. Let be an operator monotone function defined onsuch that $f(0)\leq 0$ , $f(\infty)=\infty$ , and set $g=f^{-1}$ . Then for
subset of $C(X)$$f$$[0, \infty)$

$a$

$S$

$C^{*}(S)\subseteq KC(x)(S \cup\{g(|u|^{2}) :u\in S\})$

if $f(0)=0$ , or $1\in S$ .

,

67
Proof. This follows from Theorems 2.6, 2.9.

$\square$

Theorem 32. Let be an operator monotone function defined onwith $f(0)\leq 0$ , $f(\infty)=\infty$ , and set $g=f^{-1}$ . If a finite subset$S\cup$$S=\{u_{1}, \ldots, u_{m}\}\subseteq C(X)$ separates strongly the points of $X$ , then$\{g(|u_{1}|^{2}+\cdots+|u_{m}|^{2})\}$ is a Korovki
set if $f(0)=0$ , or $1\in S$ .
$f$$[0, \infty)$$.n$

Proof. By Theorems 2.7, 2.10, we have
if $f(0)=0$ or $1\in S$ . Prom the Stone-Weierstrass theorem$C^{*}(S)\subseteq Kc(x)(S\cup\{g(|u_{1}|^{2}+$$\ldots+|u_{m}|^{2})\})$$C’(S)$ $=C(X)$ follows.

$\square$In [9], the above theorem was shown in the case where $g(t)=t$ . The
forms of Korovkin sets given above include many Korovkin sets in Appendix of [1].
$C$

参考文献
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Korovkin theorem
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maps on C’-algebras. Math. Slovaca 36, 91-99 (1986)$\ovalbox{\tt\small REJECT}$[9] Nishishiraho, T. :Convergence of positive linear approximation pr0cesses. T\^ohoku Math. J. 40, 617-632 (1983)[10] Priestley, W. M. :Anoncommutative Korovkin theorem. J. Approx.
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$C^{*}$