奈良教育大学学術リポジトリNEAR
Some Conditions for Dirichlet Algebras
著者 JIMBO Toshiya
journal or
publication title
奈良教育大学紀要. 自然科学
volume 26
number 2
page range 15‑18
year 1977‑11‑15
URL http://hdl.handle.net/10105/2523
Some Conditions for Dirichlet Algebras
Toshiya Jimbo
{Department of Mathematics, Nara University of Education, Nara, Japan) (Received April 28, 1977)
1. Introduction and notations
Let X be a compact Hausdorff space and C(X) the algebra of all continuous complex- valued functions on X. For/ in C(X) and a compact subset Y of X, define
||/||r=max{|/fr)l :x<=Y).
A subalgebra A of C(X) is called a uniform algebra on X if it is a Banach algebra with the norm || \\x, contains the constants and separates the points of X. Let M(A) be the space of maximal ideals of A and F{A) the Shilov boundary of A. Let Re A= {Re/: /eA). The uni- form algebra A is called a Dirichlet algebra on X if ReA is uniformly dense in ReC(X).
Then we have X=r(A).
The purpose of this note is to seek the conditions in order that a uniform algebra be- comes a Dirichlet algebra.
For m in M(A), a representing measure fx on X for m is a positive regular Borel measure such that
m(f) =jxfdM for all / in A.
If A is a Dirichlet algebra on X, and if meM(A), then there is a unique representing measure on X for m.
Forfand g in C{Y), let [f,g ; Y] be the uniform algebra generated byfand g, i.e. it is the uniform closure on Y of all finite sums
E.-./60 Cijf'g*
with complex coefficients c,y. Let D be a closed unit disc in the complex plane C, and let /gC(D). For a point w inf(D) (the image of /) we denote by L(w;f) the level set
{z(=D : f(z)=w} of f.
The following results are known as the conditions in order that [z,f; D\=C(D).
Theorem (Melgelyan [4]). Iff in C(D) is real-valued, and if for every w in f(D) the interior of L(iv;f) is the empty set and C-L{w;f) is connected, then [z, f ; D]=C(D).
Theorem (Wermer [6]). Iff{z)=z+R(z), where
\R(z)-R{a)\<\z-a\
for all a, z in D -with a^z, then [z,f;D]=C(D).
We shall consider the analogues of these theorems.
15
16 Toshiya Jimbo
2. A Dirichlet algebra [z, f;D\
Let K be a compact set in C and P(K) the uniform closure on K of all polynomials in z. We denote by K and 8K the polynomially convex hull and the topological boundary of K in C, respectively.
Proposition 1. If f is a function in C(D) such that for each w in f(D), d{L(w,f)K)
=L(w,f), C-f(D) is connected and f(D) has no interior, then [z,f; D] is a Dirichlet al- gebra on D.
Proof. Let u be a real-valued continuous function on D. By Walsh's theorem it follows that P(L(w;f)) is a Dirichlet algebra on L(w;f) for each w in f(D). For each e^>0 we can take a polynomial pw in z with
\\u-Re^lUcw ;/3<Ce.
Then there exists an open neighborhood U(w) of w such that for each ze.f~l{U{w))
\u(z) -Repw(z) \<e.
Since/(D) is a compact set, it is covered by finite number of such sets U{w), denoted by
C/(wi), •E•E•E, U(wn), i. e.
/(D)C U^W.
Let K=f(D) and Vj=U(wj)r\K. For this covering {Vfi of K there are continuous func-
n
tions <pj on Vj such that O«Jp^l, jO;=O on X-Vy and S¥>/=! on X.
Jsal
Since P(X) = C(K) by Mergelyan's approximation theorem, there exist polynomials q,- such that
\ \<pj-Re qj\\K <± (ma.x \\pj\\D)-\ ||Im gy||jc <-^- (max||/.j||/>)-1.
n j n j
Let us put pj=pu>j. Then for z in D, we have
\ u{z) -Re -h pj(z)qj(f{z)) \
^I S yy(/(z)) {«(«) -Re/»/(*)} I
y=i
+ IRe E <pj(f(z))pj(z) -Re S pi(z)qi(f{*)) I
1=1 /=!
^e E »>/(/(*))+hIVj(/(z))-Re Qitfi*))1 1 RePi(z)I
J-1 }-1
+£ I Im/»y(z)| |Im?y(/"(*))l <«+«+e=3e.
;=1
Since E Pi<liof n belongs to [z,f; D\, this completes the proof.
Corollary 1. If f is a continuous real-valued function on D such that for each zv in f(D), 0(L(w;/)A)=L(w;/), then [z,f;D] is a Dirichlet algebra on D.
Corollary 2. Iff(z)= \z\ +R{z), where
\R(z)-R(a)\<\ \z\-\w\ \
for all z, w in D with wi=z, then [z,f;D] is a Dirichlet algebra on D.
Proof. If z,w&L(w;f), then \R(z)-R(w)\=\\z\-\w\\.
By the assumption this implies'|z| = \iv\.
Conversely, let )z| = |w|. Then there exist two sequences {zj}, {wj\ such that |zy+1|^|zy|
^1, |wy|^|wy+i|^l, zj l= Wj, j=l,2, -, and
limz/=z, limwj=w.
Hence we have
]f(z)-f(w) \ = \R(z)-R(w) \
=lim|R(zj) -R(u>j) |^lim| \zj\ -\Wj\ | =0.
Thus, z, weL(w;f). It follows that for each w in f(D), L{w;f) is either the circle centered at 0 or {0}. Since C-f(D) is connected, we have the corollary by Proposition 1.
Example 1. The uniform algebras [z, \z\";D\, n=l,2, •E•Eå ,are Dirichlet algebras on D.
Corollary 3 (Minsker [5]). LetfeC(X) and «>0. Iff separates the points of X,
then [f, \f\" ;X] is a Dirichlet algebra on X.
Proof. Without loss of generality, we may assume that ||/|U=1. Let K=f{X) and let A=[z, \z\";D~\. Since K is a subset of D and A is a Dirichlet algebra on D, it follows that the uniform closure of A\K is a Dirichlet algebra on K. If meReC(X), the function mo/"1 is uniformly approximable on K by the real parts ofpolynomials in z and \z\". Thus u belongs to the uniform closure of Re [/, |/|" ; X], and the corollary is proved.
Example 2 ([7]). Let/(*)=exp2jr*|*l and A=[z,f; D\.
Then A is not a Dirichlet algebra on D. In fact, let /x be the normalized Lebesgue meas- ure on the unit circle T. For each Borel set E of D we define the measure £t on D by
Ju(£)=(u(£nT). Let m be the homomorphism defined by
m(g)=g(0) for all g in A.
Then fj. and the Dirac measure are the representing measures for m. Thus A is not a
Dirichlet algebra.
3. The partition of unity for A
For a compact set K we put A\Y={f\ Y:feA).
We denote by A(Y) the uniform closure of A\Y.
Definition 1. Let A be the uniform algebra and {Vy}"=1 an open covering of X. We say that there exists a partition of unity for A subordinate to an open covering {Vj} if there are functions <pj in A such that the closure of the set {jeX: <pj(x)^O] is contained in Vj, O'=l,2, -,«)
0<;y>/^l and 2 fi=l on X.
Then we denote by {Vj, <pj} the partition of unity for A.
Under the observation of the proof of Proposition 1 and Example 1, we shall obtain the
following.
18 Toshiya Jimbo
Proposition 2. Let A be a uniform algebra on X. Let {Ka}« be a closed disjoint cover- ing of X and suppose that every A(K«) is a Dirichlet algebra on K«. Suppose that for an arbitrary open covering {Ua}a of X with UaZ)Ka there exists a partition of unity {Vj, <pj]for A such that for each j, KacVjCVj<zUa for some a. Then A is a Dirichlet algebra on X.
Proof. Let u be a real-valued continuous function on X. Since A(Ka) is a Dirichlet algebra, for each e>0 we can take a function fa in A such that
\u-Re/tt|<e on Ka.
Put C7.= {x<=X;\u(x)- (Refa)(x) \<e}.
Since {Ua}« is an open covering of X with Uaz>Ka, by the assumption of the theorem there exists a partition of unity {Vj, <p/}"=! •E If KadVjCzVj(zUa, we write /} iorfa.
Let F=f]fjv>j, then FeA and
k-ReF| =|ib (M-Re/M|
^ tl\tt-'RefJ\<p<e 'E <pi=e on X.
j=\ j=i
