Small bound isomorphisms of the domain of a closed ∗ derivation

Vol. 24, No. 5 (2000) 315–326 S0161171200002532 © Hindawi Publishing Corp.

SMALL BOUND ISOMORPHISMS OF THE DOMAIN

OF A CLOSED

∗

-DERIVATION

TOSHIKO MATSUMOTO and SEIJI WATANABE

(Received 31 December 1998)

Abstract.The domainᏰ(δ)of a closed∗-derivationδinC(K)(K: a compact Hausdorff space) is a generalization of the spaceC(1)_{[0,1]of differentiable functions on[0,1]. In} this paper, a problem proposed by Jarosz (1985) is studied in the context of derivations instead ofC(1)_[0,1].

Keywords and phrases. Closed∗-derivation, linear isomorphism.

2000 Mathematics Subject Classification. Primary 46E15, 46J10, 46J15, 47B38.

LetK1andK2be two compact Hausdorff spaces.C(Ki)denotes a space of all com-plex valued continuous functions onKi (i=1,2). Let T be a surjective linear isom-etry fromC(K1) to C(K2). Then the Banach-Stone theorem states that there exist a homeomorphism τ from K2 to K1 and a functionw in C(K2)with |w(y)| = 1 (y∈K2)such that

T f (y)=w(y)fτ(y) forf∈C(K1), y∈K2. (1) That is, the existence of a surjective linear isometry betweenC(K1)andC(K2)implies thatK1andK2are homeomorphic. Amir [1] and Cambern [2] extended this theorem from this viewpoint as follows.

Theorem1(see [1, 2]). If there is a surjective linear isomorphismT:C(K1)→C(K2) such thatTT−1_{<2, thenK}

1andK2are homeomorphic.

LetX be a compact subset of the real line RandC(1)_{(X)be the space of} continu-ously differentiable functions onXwith theΣ-norm defined byfΣ=supx∈X|f (x)|+ supx∈X|f (x)|.

In [4], Jarosz proposed the following question: “Is there a positive ε such that for any compact subsetsX,Y of the real lineRand any linear isomorphismT:C(1)_(X)→ C(1)_{(Y ),TT}−1_{< εimplies thatXandY are homeomorphic?”}

In [5], Jun and Lee obtained some partial answers for this question.

Theorem2(see [5]). LetXandY be compact subset ofRandX⊂[a,b]andY⊂ [c,d]. IfT is a linear isomorphism betweenC1_(X)andC1_{(Y )which satisfies}

(i) iff (t)≡0, then(T f ) ≡0, (ii) f g ≤ T f T g ≤(1+ε)2_{f g,} (iii) f ≤ T f ≤(1+ε)f,

Theorem3[5]. LetX andY be compact subsets ofRandX⊂ni=1[ai,bi] (ai< bi< ai+1)andmaxi{|bi−ai|}< kandY⊂mj=1[cj,dj] (cj< dj< cj+1)andmaxi{|dj− cj|}< k. IfT is a linear map fromC1_(X)ontoC1_{(Y )which satisfies}

(i) f (t)≡0if and only if(T f ) ≡0, (ii) f ≤ T f ≤(1+ε)f,

(iii) k < (4−√10)/6andε <6k2_−8k+1, thenXandY are homeomorphic.

In this paper, we consider this problem from another viewpoint. To the end, we recall a closed∗-derivation.

LetKbe a compact Hausdorff space andC(K)denotes the space of all complex val-ued continuous functions onKwith the supremum norm·∞. A closed∗-derivation δinC(K)is a linear mapping inC(K)satisfying the following conditions:

(1) The domainᏰ(δ)ofδis a norm dense subalgebra ofC(K). (2)δ(f g)=δ(f )g+f δ(g)(f ,g∈Ᏸ(δ)).

(3) Iffn∈Ᏸ(δ), fn→f, andδ(fn)→gimpliesf∈Ᏸ(δ)andδ(f )=g (i.e.,δis closed as a linear operator).

(4)f∈Ᏸ(δ)impliesf∗_{∈Ᏸ(δ)andδ(f}∗_{)=δ(f )}∗_{, wheref}∗ _{means the complex}

conjugate off.

The differentiation d/dt on the space C(1)_{([0,1]) of continuously differentiable} functions on [0,1] is a typical example of closed ∗-derivations. For any closed ∗-derivation δ in C(K), we may regard the domain Ᏸ(δ) of δ as a generalization of the Banach spaceC(1)_{([0,1]). Moreover, ifᏰ(δ)=C(K), δis bounded and hence} δ≡0.

Properties of the domains of closed ∗-derivations have been studied by many authors.

We summarize useful properties of closed∗-derivations which is used later fre-quently without references.

Property4[7]. Forf (=f∗_{)∈Ᏸ(δ) and h∈C}(1)_([−f

∞,f∞]), h(f )(=h◦ f )∈Ᏸ(δ)andδ(h(f ))=h(f )δ(f ), whereh means the derivative ofh.

Property5[7]. If f ∈ Ᏸ(δ) is a constant in a neighborhood of x ∈ K, then δ(f )(x)=0.

Property6[7]. LetJ1andJ2be disjoint closed subsets ofK. Then there is a func-tionf∈Ᏸ(δ)such that

f=0 onJ1, f=1 onJ2, (0≤f≤1). (2) Now, for any fixed pointx∈K, we define a linear functionalηx◦δonᏰ(δ)by

ηx◦δ(f ):=δ(f )(x) f∈Ᏸ(δ). (3)

LetK(δ)be the set ofx∈Ksuch thatηx◦δ≠0, i.e.,

K(δ)=x∈K:ηx◦δ≠0=x∈K:∃f∈Ᏸ(δ)such thatδ(f )(x)≠0. (4)

Throughout this paper, the norminᏰ(δ)is given by

f:= f∞+δ(f )∞ f∈Ᏸ(δ). (5)

Then we note that forx0∈K(δ), the norm of a linear functionalηx0◦δis 1 (see [6]). In [6], we obtained the following result.

Theorem7. LetKibe a compact Hausdorff space and letδibe a closed∗-derivation inC(Ki) (i=1,2). LetTbe a surjective linear isometry betweenᏰ(δ1)andᏰ(δ2). Then, there exist a homeomorphismτfromK2toK1, w1∈ker(δ2)and a continuous function w2onK2(δ2)such thatτ(K2(δ2))=K1(δ1), |w1(y)| =1for ally∈K2, |w2(y)| =1 for ally∈K2(δ2),

(T f )(y)=w1(y)fτ(y) forf∈Ᏸδ1, y∈K2,

δ2(T f )(y)=w2(y)δ1(f )τ(y) forf∈Ᏸδ1, y∈K2δ2. (6) In this paper, we consider Jarosz’s problem in the same context as this theorem. We use the following notation, for a Banach spaceB,B∗_{denotes the conjugate space}

ofB.B1andB1∗denote the closed unit balls ofBandB∗, respectively.Tdenotes the unit circle{z∈C:|z| =1}in the complex plane.

We shall prove the following theorem.

Theorem8. Let Ki be a compact Hausdorff space satisfying the first countable axiom, and letδi be a closed ∗-derivation in C(Ki) (i=1,2). If there exist a linear isomorphismT ofᏰ(δ1)ontoᏰ(δ2)withTT−1<2andT,T−1are bounded un-der the uniform norm, thenK1(δ1)andK2(δ2)are homeomorphic. Moreover, if the range᏾(δi)contains 1(i=1,2), thenK1andK2are homeomorphic.

The proof of this theorem is done along the line in [3].

LetKbe a compact Hausdorff space satisfying the first countable axiom and letδ be a closed∗-derivation inC(K).

The following two lemmas will be used in the rest of the paper.

Lemma9. Forx0∈K(δ), an open neighborhoodU ofx0andε (0< ε <1), there exists a functionf∈Ᏸ(δ)such that

f ≤1, f∞≤ε, fx0=0,

f=δ(f )=0 onK\U, 1>δ(f )x0>1−ε. (7)

Proof. We take an open neighborhoodVofx0such thatV⊂Uand take a function g∈Ᏸ(δ)such that

0≤g≤1, gx0=1, g=0 onK\V. (8) Then,g=δ(g)=0 onK\U. Since x0∈K(δ), there is a functiongε(=gε∗)∈Ᏸ(δ) such that

_{gε<1, 1−ε=ηx}

Forcε:=min{(1−δ(gε)∞)/(1+δ(g)∞),ε}, there is a functionh∈C1(0[−gε∞,

gε∞])such that

h∞≤cε, hgεx0=0, hgεx0=1, h∞=1. (10)

Thenf:=h(gε)g∈Ᏸ(δ)has all required properties in Lemma 9.

Lemma10. Forx0∈K(δ)andε (0< ε <1), there exists a sequence{fn} ⊂Ᏸ(δ) such that

_{fn≤1, fn}

∞≤_n1, fn

x0=0, lim

n→∞δ

fn(x)=0 x≠x0,1>δfnx0>1−ε,

(11)

anddx0:=δ(fn)(x0)is independent ofn.

Proof. SinceKsatisfies the first countable axiom, there is a family{Un}of open neighborhood ofx0such thatUi+1⊂Uiand ∞1Un= {x0}. Then there exists a family {Vn}of open neighborhood ofx0such thatVn⊂Un, and there isgn∈Ᏸ(δ)such that gn(x0)=1, 0≤gn≤1, gn=0 onK\Vn. (12) Thengn=δ(gn)=0 onK\Un. Sincex0is inK(δ), there is a functiongε(=gε∗)∈Ᏸ(δ) such that

_{gε<1, 1−ε=ηx0◦δ−ε <δ}

gεx0. (13)

For each cn:=min{(1− δ(gε)∞)/(1+ δ(gn)∞),1/n}, there is a functionhn∈ C1_([−gε

∞,gε∞])such that

hn∞≤cn, hngεx0=0, hn

gεx0=1, hn∞=1. (14)

Then everyfn:=hn(gε)gn∈Ᏸ(δ)has the properties required in Lemma 10.

LetWbe the compact Hausdorff spaceW=K×K×Twith the product topology. For f∈Ᏸ(δ), we define ˜f∈C(W )by

˜

fx,x ,z:=zf (x)+δ(f )x, (15)

for(x,x,z)∈W. Then we havef˜∞= f.

Proof of Theorem7. Let Wi :=Ki×Ki×T and Si = {f˜∈ C(Wi);f ∈Ᏸ(δi)} (i=1,2).

Define a linear isomorphism ˜T ofS1ontoS2by ˜

Tf˜:=T (f ) f˜∈S1. (16) Then ˜T is well defined sincef→f˜is a linear isomorphism.

We may assume thatT−1 _{=1 and 1<T<2. Then we haveT˜}−1 _{= T}−1 ₌₁ andT˜ = T<2. For(y0, y0, z0)∈W2, letΦ be a norm-preserving extension of

˜ T∗_{L(y0, y}

L(y0,y₀,z0)(f )˜ =f (y˜ 0,y0,z0) (f˜∈S2). Then, from the Riesz representation theorem, there exists a regular Borel measureµy0, y0, z0 _{onW1 such thatµ}y0, y0, z0 _{= Φ =} T˜∗_L(y0,y

0,z0) ≤ T<2 and

Φ(h)=

W1hdµ

y0, y0, z0 h∈CW₁. (17)

Hence we have

z0(T f )y0+δ2(T f )y0

= W1

˜

fx,x,zdµy0, y0, z0

= W1

zf (x)+δ1(f )(x)dµy0, y0, z0

(18)

forf∈Ᏸ(δ1).

In the following, we identifyΦandµy0, y0, z0.

µx0,x0,z0, where(x₀,x₀,z₀)∈W1, is also defined in a similar way. Then we have µx0,x₀,z0 _≤1.

The following lemma shows that for x0 ∈ K1(δ1), µy,y,z(K1× {x0} ×T), where (y,y,z)∈W2depends ony only, that is,µy,y,z(K1× {x0} ×T)is independent of y, z, and any choice of norm-preserving extension of ˜T∗_L(y,y,z).

Lemma11. (1)Forx0∈K1(δ1)andε (0< ε <1), let{fn} ⊂Ᏸ(δ1)be a sequence in Lemma 10. Then for(y,y,z)∈W2,

µy,y,z_K

1×{x0}×T=

₁

dx0

lim n→∞T˜

_˜

fny,y,z

=

₁

dx0

lim n→∞δ2

Tfny.

(19)

(2)Fory0∈K2(δ2)andε (0< ε <1), let{gn} ⊂Ᏸ(δ2)be a sequence in Lemma 10. Then for(x,x,z)∈W1,

µx,x,z_K

2×{y0}×T=

1 dy0

lim n→∞T˜

−1_{gn˜ x,x ,z}

=

1 dy0

lim n→∞δ1

T−1_{gnx .}

(20)

Proof. (1) Letµy,y ,z_{be a norm-preserving extension of ˜T}∗_L(y,y ,z).

lim

n→∞T (˜ fn)(y,y˜ ,z)=nlim→∞ _W1fn˜ dµ y,y,z₌

W1nlim→∞ ˜

fndµy,y,z

=

K1×{x0}×Tdx0dµ

y,y,z_=dx0µy,y,z_K

1×{x0}×T.

(21)

From the uniform boundedness ofT,

lim

n→∞T (˜ fn)(y,y˜ ,z)=nlim→∞

Thus, we have

dx0µy,y,z_K

1×{x0}×T=_nlim_→∞δ2T fny (23)

which implies that forx0∈K1(δ1), µy,y,z(K1×{x0}×T)depends ony ∈K2only. The statement (2) is also shown by the same argument as above.

Now, letM1be any real number with(1<)T<2M1<2. Let ˜K2:= {y∈K2:∃x∈ K1such that|µy,y,z(K1× {x} ×T)|> M1for everyz∈Tand every norm-preserving extensionµy,y,z _{of ˜T}∗_{L(y,y,z)}. Since µ}y,y,z _{= T˜}∗_{L(y,y,z) ≤ T<2M1, for y∈}

˜

K2, there can be at most onex∈K1with the property in the definition of ˜K2. Thus the mapρ1of ˜K2toK1is well defined byρ1(y):=xifxis related toyas above.

Next, we set M2:=1/(2M1). Let ˜K1:= {x∈ K1: ∃y ∈K2 such that|µx,x,z(K2× {y} ×T)|> M2 for everyz∈T and for every norm-preserving extension µx,x,z of (T˜−1₎∗_{L(x,x,z)}. Sinceµ}x,x,z _{= (T˜}−1₎∗_{L(x,x,z) ≤ T}−1 _{≤1, forx∈K˜1, there can} be at most oney∈K2with the property in the definition of ˜K1. Thus, the mapρ2of

˜

K1toK2is well defined byρ2(x):=yifyis related toxas above.

The following lemma shows that ˜Kicontains sufficiently many elements (hence, is nonempty).

Lemma12. (1)Forx0∈K1(δ1), there existsy0∈K˜2∩K2(δ2)such thatρ1(y0)=x0. (2)Fory0∈K2(δ2), there existsx0∈K˜1∩K1(δ1)such thatρ2(x0)=y0.

Proof. (1) Forx0∈K1(δ1)and 0< ε <1−M1, there exists a family{fn} ⊂Ᏸ(δ1) in Lemma 10 such that

_{fn≤1, fn}

∞≤_n1, fn

x0=0, lim

n→∞δ1

fn(x)=0 ∀x≠x0, 1−ε <dx0<1,

(24)

wheredx0 =δ1(fn)(x0). If limn→∞|T (˜ fn)(y,y˜ ,z)| ≤M1 for every(y,y ,z)∈W2, then

1−ε <dx0=lim n→∞fn

x0+δ1fnx0=_nlim_→∞fn˜x0,x0,1 =_nlim_→∞T˜−1∗_L(x0,x0,

1)T˜fn˜ =lim

n→∞

_W2T˜fn˜y,y,zdµx0,x0,1

≤

W2nlim→∞

T˜fn˜y,y,zdµx0,x0,1

≤M1µx0,x0,1≤M1.

(25)

This contradicts with 1−ε > M1.

Hence there exists(y0,y0,z0)∈W2such that

lim n→∞T˜

_˜

Then, from Lemma 11 we have for arbitraryz∈Tand any norm-preserving exten-sionµy0,y0,zof ˜T∗L_(y0,y

0,z),

M1<_nlim_→∞T˜fn˜y0,y0,z0=_nlim_→∞δ2T fny0

=_nlim_→∞T˜fn˜y0,y0,z0=dx0µy0,y0,zK1×{x0}×T <µy₀,y₀,z_K

1×{x0}×T.

(27)

Thus,y0∈K˜2∩K2(δ2)andρ1(y0)=x0.

(2) For y0 ∈ K2(δ2) and 0< ε < 1−M2T, we take a family {gn} ⊂ Ᏸ(δ2) in Lemma 10. The remainder of the proof is completed by the same way as above.

Now, we state another important lemma which holds without the first countability axiom.

Lemma13. Ifx0∈K˜1andρ2(x0)∈K2(δ2), thenx0∈K1(δ1). Proof. Letµx0,x0,1_{be a norm-preserving extension of(T˜}−1₎∗_L(x0,x0,

1). Sinceµx0,x0,1 is regular, Since for allεsuch that 0< ε < M2/(M2+3+ T−1∞) there is an open

neighborhoodUεofρ2(x0)such that

_µx0,x0,1_K2×Uε\ρ

2x0×T< ε. (28) Forε,Uεandρ2(x0), we take a functionf∈Ᏸ(δ2)in Lemma 9, then

f ≤1, f∞≤ε, fρ2x0=0,

f=δ2(f )=0 onK2\Uε, 1>δ2(f )ρ2x0>1−ε. (29) Since

_{K2×{ρ2(x0)}×T}zf (y)dµx0,x0,1

≤ f∞µx0,x0,1 ≤ε,

_{K2×{ρ2(x0)}×T}δ2(f )ρ2x0dµx0,x0,1

=δ2(f )ρ2x0µx0,x0,1K2×{ρ2(x0)}×T > (1−ε)M2,

(30)

we have

_{K2×{ρ2(x0)}×T}f dµ˜ x0,x0,1 ≥

_{K2×{ρ2(x0)}×T}δ2(f )ρ2x0dµx0,x0,1

−

K2×{ρ2(x0)}×Tzf (y)dµ

x0,x0,1

> (1−ε)M2−ε >0.

(31)

From this and

_{K2×(Uε\{ρ2(x0)})×T}f dµ˜ x0,x0,1

≤f˜∞µx0,x0,1K2×

we have

K2×Uε×T

˜

f dµx0,x0,1

≥

K2×{ρ2(x0)}×T

˜

f dµx0,x0,1

−

K2×(Uε\{ρ2(x0)})×T

˜

f dµx0,x0,1

> (1−ε)M2−2ε >0.

(33)

Since

K2×(K2\Uε)×T ˜

f dµx0,x0,1

=

K2×(K2\Uε)×T

zf (y)dµx0,x0,1

≤ f∞µx0,x0,1≤ε,

(34)

we get

T˜−1_f˜x

0,x0,1=T˜−1∗L(x0,x0,1)f˜=

_W2f dµ˜ x0,x0,1

≥

K2×Uε×T

˜

f dµx0,x0,1

−

K2×(K2\Uε)×T

˜

f dµx0,x0,1

≥(1−ε)M2−3ε >0.

(35)

Thus

_δ1

T−1_{(f )x}

0=T˜−1f˜x0,x0,1−T−1(f )x0 ≥T˜−1_f˜x

0,x0,1−T−1(f )x0 ≥(1−ε)M2−3ε−εT−1

∞>0,

(36)

that is,x0∈K1(δ1). This completes the proof.

Lemma14. Ify0∈K˜2∩K2(δ2), thenρ1(y0)∈K˜1∩K1(δ1)andρ2(ρ1(y0))=y0. Proof. Let ρ1(y0)=x0 (y0∈ K˜2∩K2(δ2)). If x0 ∈K˜1 and ρ2(x0)= y0, then x0∈K1(δ1)from Lemma 13. Hence, suppose that eitherx0is not in ˜K1orx0∈K˜1 andρ2(x0)≠y0. Then there existsz0∈Tsuch that|µx0,x0,z0(K2×{y0}×T)| ≤M2.

Let P :=sup{|µx,x,z_(K

2× {y0} ×T)|;(x,x,z)∈W1}(≤1). Since y0∈K2(δ2), we have P =sup{|µx,x,z_{(K2× {y0} ×T)|;(x,x,z) ∈W1}by Lemma 11. Since P > M2} by Lemma 12 and 0<T−M1< M1, there exists(x1,x1,z1)∈W1such that

_µx1,x1,z1_{K2×{y0}×T>maxM2,T−M1P/M1. (37)}

Then, for arbitraryz∈Tand any norm-preserving extensionµx1,x1,z_,

_µx1,x1,z_{K2×{y0}×T> M}

2, (38)

by Lemma 12. Fory0∈K2(δ2)andε (0< ε <1), there exists a family{gn} ⊂Ᏸ(δ2) in Lemma 10. Then, sincey1≠y0,

0= lim n→∞

z1gny1+δ2gny1=_nlim_→∞gn˜ y1,y1,z1 =_nlim_→∞T˜∗_{L(y1,y1,z1)T˜}−1_{gn˜ = lim}

n→∞ _W1T˜

−1_{gn˜ dµ}y1,y1,z1

=_nlim_→∞

K1×{x1}×TT˜

−1_{gn˜ dµ}y1,y1,z1_+lim

n→∞ _{K1×(K1\{x1})×T}T˜

−1_{gn˜ dµ}y1,y1,z1_. (39)

Now, by Lemma 11,

nlim→∞ _K1×{x1}×TT˜

−1_{gn˜ dµ}y1,y1,z1

=

K1×{x1}×Tnlim→∞T˜

−1_{gn˜ dµ}y1,y1,z1

=

K1×{x1}×T

dy0µx,x1,z_{K2×{y0}×Tdµ}y1,y1,z1

=

K1×{x1}×Tdy0µ

x1,x1,z1_K

2×{y0}×Tdµy1,y1,z1

=dy0µx1,x1,z1_K

2×{y0}×Tµy1,y1,z1K1×{x1}×T >dy0·

T−M1P M1 ·M1=

_dy

0P

T−M1.

(40)

On the other hand,

nlim→∞ _{K1×K1\{x1}×T}T˜

−1_{gn˜ dµ}y1,y1,z1

=

K1×K1\{x1}×Tnlim→∞T˜

−1_{gn˜ dµ}y1,y1,z1

=

K1×(K1\{x1})×Tdy0µ

x,x,z_K

2×{y0}×Tdµy1,y1,z1

≤dy0Pµy1,y1,z1_K

1×K1\{x1}×T =dy0Pµy1,y1,z1_K1×K

1×T−µy1,y1,z1K1×{x1}×T ≤dy0PT−µy1,y1,z1_{K1×{x1}×T<dy0PT−M1.}

(41)

This contradicts to

0= lim

n→∞ _K1×{x1}×TT˜

−1_{gn˜ dµ}y1,y1,z1_+lim

n→∞ _{K1×(K1\{x1})×T}T˜

−1_{gn˜ dµ}y1,y1,z1_{. (42)}

Thusx0∈K˜1andy0=ρ2(x0)=ρ2(ρ1(y0)).

Lemma15. ρiis continuous onKi(δi) (i=1,2).

Proof. We show thatρ1is continuous. Suppose thatρ1is discontinuous aty0∈ K2(δ2). Then there exists a sequence{yn} ⊂K2(δ2)such that yn →y0∈K2(δ2), butxn:=ρ1(yn)is not converge toρ1(y0)=x0. There exists an open neighborhood V1(⊂K1(δ1))ofx0such that for everyn0there isn(≥n0)withxnoutsideV1. Since µy0,y0,1_{is regular, forε (0< ε < (2M1−T)/(T+2M1+10))there exists an open} neighborhoodU1(⊂V1)ofx0such that

_µy0,y0,1_K

1×U1\{x0}×T< ε, U1⊂V1. (43) Forx0,U1, andε, by Lemma 9, there exists a functionf∈Ᏸ(δ1)such that

f ≤1, f∞≤ε, fx0=0,

1>δ1(f )x0>1−ε, f=δ1(f )=0 onK1\U1. (44)

Since

K1×{x0}×T

zf (x)dµy0,y0,1

≤ f∞µy0,y0,1≤2ε,

K1×{x0}×Tδ1(f )(x0)dµ

y0,y0,1

= |δ1(f )(x0)µy0,y0,1K1×{x0}×T > (1−ε)M1,

(45)

we have

K1×{x0}×T ˜

f dµy0,y0,1

> (1−ε)M1−2ε > ε. (46)

From (46) and

K1×(U1\{x0})×T

˜

f dµy0,y0,1

≤f˜∞µy0,y0,1K1×(U1\{x0})×T≤ε, (47)

we have

K1×U1×T ˜

f dµy0,y0,1

≥

K1×{x0}×T ˜

f dµy0,y0,1

−

K1×(U1\{x0})×T

˜

f dµy0,y0,1

≥(1−ε)M1−3ε >2ε

_{K1×(K1\U1)×T}f dµ˜ y0,y0,1 =

_{K1×(K1\U1)×T}zf (x)dµy0,y0,1

≤ f∞µy0,y0,1≤2ε.

Thus

_{T (}˜ _{f )(y}˜ _0,y0,1)=T˜∗_L(y 0,y0,1)

_˜

f= W1

˜

f dµy0,y0,1

≥

K1×U1×T

˜

f dµy0,y0,1

−

_{K1×(K1\U1)×T}f dµ˜ y0,y0,1

> (1−ε)M1−5ε >0.

(49)

Now, sinceyn→y0inK2, then(yn,yn,1)→(y0,y0,1)inW2. There existsn0such that∀n(> n0)implies|T (˜ f )(yn,yn,˜ 1)|> (1−ε)M1−5ε. Fixn1(≥n0)such thatxn1= ρ1(yn1)lies outsideV1. Sinceµyn1,yn1,1is regular, there exists an open neighborhood U2(⊂K1)ofxn1such that

_µyn₁,yn₁,1_{K1×U2\{xn1}×T< ε, U}

1∩U2=φ. (50) Forxn1,U2, andε, we takeg(∈Ᏸ(δ1))in Lemma 9 such that

g ≤1, g∞≤ε, g(xn1)=0,

1>δ1(g)(xn1)>1−ε, g=δ1(g)=0 onK1\U2. (51) By the same way as above, we have

K1×U2×T ˜

gdµyn₁,yn₁,1

> (1−ε)M1−3ε >0,

K1×(K1\U2)×Tgdµ˜

yn₁,yn₁,1

=

K1×(K1\U2)×Tzg(x)dµ

yn₁,yn₁,1

≤ g∞µyn1,yn1,1 ≤2ε.

(52)

Then

T˜g˜yn1,yn1,1=T˜∗_L(yn 1,yn1,1)

˜ g=

W1g dµ˜

yn₁,yn₁,1

≥

K1×U2×Tg dµ˜

yn₁,yn₁,1

−

_{K1×(K1\U2)×T}g dµ˜ yn1,yn1,1

> (1−ε)M1−5ε >0.

(53)

Thus, if we choose a complex number λ0∈T such that ˜T (f )(yn1˜ ,yn1,1) and λ0 (T (˜ g))(yn1˜ ,yn1,1)have equal arguments, then

_{f+λ0g=maxf∞,g∞}

+maxδ1(f )∞,δ1(g)∞

≤1+ε, (54)

This is a contradiction. Therefore, ρ1 is continuous onK2(δ2). A similar argument shows thatρ2is continuous onK1(δ2).

Remark16. There is not a nonzero closed∗-derivation inC(D)(Dis the Cantor set). However, we can obtain similar results forC(1)_{(X)(X: a compact subset ofR) by} the same way as above.

References

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Soc.32(1995), no. 1, 85–91. MR 96c:46024. Zbl 910.46018.

[6] T. Matsumoto and S. Watanabe, Extreme points and linear isometries of the domain of a closed∗_{-derivation in C(K), J. Math. Soc. Japan48 (1996), no. 2, 229–254.} MR 96m:46129. Zbl 860.46013.

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Toshiko Matsumoto:2-19-1143Ikego Zushi,249-0003, Japan