Vol. 23, No. 12 (2000) 827–831 S0161171200003148 © Hindawi Publishing Corp.
SPATIAL NUMERICAL RANGES OF ELEMENTS
OF SUBALGEBRAS OF
C
0(X)
SIN-EI TAKAHASI
(Received 9 July 1998 and in revised form 26 January 1999)
To Professor Junzo Wada on his retirement from Waseda University
Abstract.WhenAis a subalgebra of the commutative Banach algebraC0(X)of all con-tinuous complex-valued functions on a locally compact Hausdorff space X, the spatial numerical range of element ofAcan be described in terms of positive measures.
Keywords and phrases. Numerical range, spatial numerical range.
2000 Mathematics Subject Classification. Primary 46J10.
1. Introduction and results. LetXbe a locally compact Hausdorff space andC0(X) the commutative Banach algebra (with supremum norm∞) of all continuous com-plex-valued functions onXwhich vanish at infinity. LetAbe a subalgebra (not neces-sarily closed) ofC0(X),A∗the dual space ofAandf∈A. IfAis unital, then
V1(A,f )≡m(f ):m∈A∗andm =m(1)=1 (1.1) is called the (algebra) numerical range of f and it is a nonempty compact convex subset of the complex planeC(cf. [1, page 52]). However ifAis nonunital, then the above definition is not meaningful. In this case, Gaur and Husain [2] introduced the following set:
V(A,f )=m(f g):∃m∈A∗andg∈A
such thatm = g∞=m(g)=1 (1.2)
and studied the spatial numerical range in a nonunital algebra. The setV (A,f ) is equal toV1(A,f )wheneverAis unital.
In [2], Gaur and Husain proved the following result. Theorem1.1. Letf be an element ofC0(X). Then
coR(f )⊆V(C0(X),f )⊆coR(f ), (1.3) wherecoandcodenote the convex hull and the closed hull, respectively, andR(f )is the range of the functionf.
In this paper, we describe spatial numerical ranges of elements of subalgebras of
Theorem1.2. LetAbe a subalgebra ofC0(X)andf∈A. Then
(i) V(A,f )= {f d|µ|:∃µ ∈M(X)and∃g∈Asuch thatµ = g∞=gdµ=
1} ⊆coR(f ), where|µ|denotes the total variation ofµ.
(ii) IfAhas the following property (#), thencoR(f )⊆V (A,f ).
(#) For any finite set{x1,...,xn} in X, there exists g∈A such that g∞ =1 and
g(x1)= ··· =g(xn)=1.
Corollary1.3. LetAbe a∗-subalgebra ofC0(X)andf∈A. Then
(i)
V (A,f )=
f dµ:∃µ∈M(X)and∃g∈A
such thatµ =1, µ≥0,0≤g≤1and
g dµ=1
. (1.4)
(ii) IfAhas the following property (##), then
V (A,f )=
f dµ: 0≤µ∈M(X), µ =1andsupp(µ)is compact
. (1.5)
(##)For any compact setE⊆X, there existsg∈Asuch that0≤g≤1andg(x)=1 for allx∈E. Heresupp(µ)denotes the support ofµ.
Remark1.4. If A= C0(X), thenA satisfies the desired properties appeared in Theorem 1.2 and Corollary 1.3. Hence, we have
V(C0(X),f )=
f dµ: 0≤µ∈M(X),µ =1 and supp(µ)is compact
(1.6)
and
coR(f )⊆V (C0(X),f )⊆coR(f ). (1.7)
2. Proofs of results
Proof of Theorem1.2. (i) By the Hahn-Banach extension theorem, we have
V(A,f )=
f g dµ:∃µ∈M(X)and∃g∈A
such thatµ = g∞=
g dµ=1
.
(2.1)
Now supposeµ∈M(X),g∈Aand µ = g∞=gdµ=1. Letµ=h· |µ|be the polar decomposition ofµ(cf. [4, Corollary 19.38]), then|h| =1. Since
1=
g dµ=
ghd|µ| ≤
|g|2d|µ|
|h|2d|µ| ≤ µ =1, (2.2)
it follows that
ghd|µ|=
|g|2d|µ|
and hence there exists a scalarλ∈Csuch thatg(x)=λh(x)|µ|a.e. onX. Therefore we have
1=
ghd|µ| =¯λ
|h|2d|µ| =¯λµ =¯λ, (2.4)
and so
f g dµ=
f ghd|µ| =
f|h|2d|µ| =f d|µ|. (2.5)
Consequently, we obtain that
V(A,f )=
f d|µ|:∃µ∈M(X)and∃g∈A
such thatµ = g∞=
g dµ=1
.
(2.6)
Next consider the following set:
S=
v∈M(X):v≥0 andv =
|g|2dv=1. (2.7)
ThenS is a weak∗-closed set. Also note that |g|2d|µ| =1 by the above arguments and hence|µ| ∈S. Moreover, we can easily see that any extreme point ofS is also an extreme point of{v∈M(X):v ≥0, v ≤1}. But since the extreme points of
{v∈M(X):v≥0,v ≤1}consist of 0 and{δx:x∈X}, it follows that the extreme points of S are contained in {δx:x∈X}, whereδx denotes the Dirac measure at
x∈X. Then by the Krein-Milman theorem, we haveS⊆co{δx:x∈X}and so|µ| ∈ co{δx:x∈X}. Hencef d|µ| =limλf dvλfor some net{vλ}in co{δx:x∈X}, and sof d|µ| ∈coR(f ). Therefore, we have
f d|µ|:∃µ∈M(X)and∃g∈Asuch thatµ = g∞=
g dµ=1
⊆coR(f ).
(2.8)
(ii) Letx1,...,xn ∈X and λ1≥0,...,λn ≥0 with λ1+ ··· +λn =1, and setµ =
λ1δx1+ ··· +λnδxn. Thenµis a positive measure onXwith norm of one such that
f dµ=λ1f (x1)+ ··· +λnf (xn). Assume thatAhas the property (#). Then we can take an elementg∈Asuch thatg∞=1 andg(x1)= ··· =g(xn)=1. Therefore
gdµ=1 and hence we conclude that coR(f )⊆V(A,f ). Proof of Corollary1.3. AssumeAisa∗-subalgebra ofC
0(X). (i) Set
W=
f dµ:∃µ∈M(X)and∃g∈A
such thatµ =1, µ≥0,0≤g≤1 and
g dµ=1
.
(2.9)
of C0(X), we have |g|2 =gg¯∈ A. Moreover, we have |g|2d|µ| =1 and hence
|g|2d|µ| =1 as observed in the proof of Theorem 1.2. Hence we conclude that
V (A,f )⊆Wby Theorem 1.2 again and henceV (A,f )=W.
(ii) Letµ∈M(X), g∈Aandµ = g∞=gdµ=1. Then we have(1−|g|2)d|µ|
=0 as observed in the proof of (i). It follows that|g(x)| =1|µ|a.e. onXand hence supp(|µ|)is compact. Therefore, we have
V (A,f )⊆
f dµ: 0≤µ∈M(X), µ =1 and supp(µ)is compact
. (2.10)
Now, suppose that 0≤v∈M(X), v =1 and supp(v)is compact and thatAhas the property (##). Then we can take an elementg∈Asuch that 0≤g≤1 andg(x)=1 for allx∈supp(v). Thereforev = g =gdv=1 and hence, by Theorem 1.2, we have
f dµ: 0≤µ∈M(X), µ =1 and supp(µ)is compact⊆V (A,f ) (2.11)
3. Examples. LetX=(0,1], the half open interval and leth∈C0(X)be such that
h(x)≠0 for allx∈X. Define
A= {hg:g∈C0(X)}. (3.1)
ThenAis an ideal (and hence subalgebra) ofC0(X). In this case,Ais neither closed nor unital. AlsoAhas the desired property: for any compact setE⊆X, there exists
g∈Asuch thatg∞=1 andg(x)=1 for allx∈E. In fact, lettE=min{x:x∈E} and so 0< tE≤1. Put
g0(x)=
xϕ(x)
tEϕ(tE)h(x), if 0< x≤tE, 1
h(x), iftE< x≤1,
(3.2)
whereϕ(x)=min(|h(tE)|x/tE,|h(x)|) (x∈X). Since|g0(x)| ≤x/tEϕ(tE)for 0<
x≤tE, a functiong0must be inC0(X). Setg=hg0and henceg∈Aandg(x)=1 for allx∈E. Also since|g(x)| =(x/tE)·(ϕ(x)/ϕ(tE))≤1·1=1 for 0< x≤tE, it follows thatg∞=1. ThereforeAhas the desired property and so by Theorem 1.2, we have
V(A,f )=f d|µ|:∃µ∈M(X)andg∈A
such thatµ = g∞=
g dµ=1 (3.3)
and
coR(f )⊆V(A,f )⊆coR(f ) (3.4) for everyf∈A. In particular, iff∈Ais real-valued, then we have
V(A,f )=
[α,β], if{x∈X:f (x)=0}≠∅,
(0,β]or[α,0), if{x∈X:f (x)=0} = ∅, (3.5)
Of course, this holds even ifA=C0(X), so we have the spatial numerical range of the functionf (x)=x(x∈X)with respect toC0(X)is equal toX=(0,1]. This fact has been observed in [2, Example 4.2].
Also, Ais not generally a ∗-subalgebra of C0(X). But ifh is real-valued, thenA becomes a∗-subalgebra ofC0(X)and soAhas the property(##).
Acknowledgements. The author thanks the referee for helpful comments and improving the paper. This research is partially supported by the Grant-in-Aid for Sci-entific Research, The Ministry of Education, Science and Culture, Japan.
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