The algebras of bounded and essentially bounded
Lebesgue measurable functions
Raymond Mortini, Rudolf Rupp
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THE ALGEBRAS OF BOUNDED AND ESSENTIALLY
BOUNDED LEBESGUE MEASURABLE FUNCTIONS
RAYMOND MORTINI AND RUDOLF RUPP
Abstract. Let X be a set in Rn with positive Lebesgue measure. It
is well known that the spectrum of the algebra L∞ (X) of (equivalence
classes) of essentially bounded, measurable functions on X is an extremely disconnected compact Hausdorff space. We show, by elementary methods, that the spectrum M of the algebra Lb (X) of all bounded
measurable functions on X is not extremely disconnected, though totally
disconnected. Let ∆ = {δx : x ∈ X} be the set of point evaluations and
let g be the Gelfand topology on M . Then (∆, g) is homeomorphic to
(X, Tdis ), where Tdis is the discrete topology. Moreover, ∆ is a dense
subset of the spectrum M of Lb (X). Finally, the hull h(I), (which is
homeomorphic to M (L∞ (X))), of the ideal of all functions in Lb (X)
vanishing almost everywhere on X is a nowhere dense and extremely
disconnected subset of the Corona M \ ∆ of Lb (X).
22.2.2016
Introduction
Given a set X ⊆ Rn of positive Lebesgue measure σ(X), let
n
o
L ∞ := f : Df ⊆ X → C σ(X \ Df ) = 0, f -Lebesgue meas., ||f ||e.s < ∞ ,
where
||f ||e.s := inf η > 0 : σ({x ∈ X : |f (x)| ≥ η}) = 0
is the essential supremum of f on X. L ∞ is called the set of essentially
bounded functions on X. It is obvious that if f is a bounded continuous function on X, then ||f ||∞ = ||f ||e.s . Define on L ∞ the following equivalence
relation:
f ∼ g ⇐⇒ f − g ∈ N := {f ∈ L ∞ : f ≡ 0 a.e.}.
Let [f ] := f + N denote the equivalence class associated with f , and put
|| [f ] ||∞ := inf{||g||e.s : g ∼ f }.
Then
(0.1)
||[f ]||∞ = ||f ||e.s
1991 Mathematics Subject Classification. Primary 46J10, Secondary 54G05; 54C20.
Key words and phrases. bounded Lebesgue measurable functions; essentially bounded
functions; spectra and maximal ideal spaces; extremely disconnected space; totally disconnected space.
1
2
RAYMOND MORTINI AND RUDOLF RUPP
for any representant f ∈ L ∞ . Moreover, if L∞ = L∞ (X) = {[f ] : f ∈
L ∞ }, then
(L∞ , +, ×, •, ||[ · ]||∞ )
S
is a commutative unital Banach algebra such that ||[f ]||2∞ = ||[f ]2 ||∞ for
every f ∈ L∞ . In particular, L∞ is a uniform algebra. L∞ is the classical
algebra of (equivalence classes) of essentially bounded functions. It is well
known that its spectrum (or maximal ideal space) M (L∞ ) is an extremely
(also called extremally) disconnected compact Hausdorff space (see e.g.
[2])
and that (L∞ , || · ||e.s ) is isomorphic isometric to C(M (L∞ )), || · ||∞ .
In this note we investigate its cousin algebra, Lb (X, C), of all bounded
Lebesgue measurable functions on X endowed with the supremum norm
||·||∞ . Let us emphasize that Lb (X, C) is strictly contained in L ∞ , and that
L ∞ contains unbounded functions. As far as we are aware of, a description
of the maximal ideal space M of Lb (X, C) has never been given in literature.
To our surprise, it will turn out that M is not extremely disconnected, but
only totally disconnected. We shall also study the hull h(I) of the ideal I of
all functions in Lb (X) vanishing almost everywhere on X. How big is this
set? It will be shown that this set is a nowhere dense subset of the Corona
M \ ∆ of Lb (X, C), where ∆ = {δx : x ∈ X} is the set of point functionals
on Lb (X). Another result will tell us that the restriction of the Gelfand
topology g to ∆ is the discrete topology and that ∆ is dense in M .
Our approach is entirely self-contained and thus accessible to graduate
students. The results could also be derived by using the advanced theory of
Banach lattices, Boolean algebras and Dedekind complete spaces, necessitating though the knowledge of a large amount of classical results. Usually,
these advanced methods are given in monographs on C ∗ -algebras and operator algebras, as for example [1, 10, 7].
It will finally be observed that the topological stable rank (and hence the
Bass stable rank) of Lb (X) is one.
1. Some topological tools
Here we list, for the reader’s convenience, several standard results needed
for our proofs. As usual, let dim X denote the covering dimension of a compact space X and C(X) := C(X, C) the space of all continuous, complexvalued functions on X. A proof of the following well-known result is given
in [8, p.113].
Theorem (A). Let X be a compact Hausdorff space. The following assertions are equivalent:
(1) dim X = 0;
(2) X is totally disconnected;
(3) The closed-open sets form a basis for the topology on X.
Let us note that a topological space X is called extremely disconnected
if the closure of any open set is open. Or equivalently, if two disjoint open
BOUNDED MEASURABLE FUNCTIONS
3
sets have disjoint closures. In view of Theorem A it is clear that extreme
disconnectedness implies total disconnectedness. A function of the form
f :=
n
X
αj χEj ,
j=1
αj ∈ C, Ej ∩Ek = ∅ for j 6= k, is called a step-function (or a simple function).
The following standard results can easily be shown:
Lemma (B). Simple functions have the following properties:
P
i) Let s = nk=1 λk χEk be a simple function on a set Y (where Ej ∩
Ek = ∅ for j 6= k), ε > 0, and let F be the union of those Ek where
|λk | ≥ ε. Then
U := {y ∈ Y : |s(y)| < ε} = {y ∈ Y : χF c (y) = 1},
where F c denotes the complement of F in X.
ii) If in the case Y ⊆ Rm the sets Ej are measurable with positive measure, then E := F c is measurable and has positive measure whenever
U 6= ∅.
iii) If, in the case of a topological space Y , the sets Ej are closed-open,
then E is closed-open, too.
Lemma (C). Let Y be a topological space, X ⊆ Y and U an open subset of
Y . Then
(1) X ∩ U ⊆ X ∩ U .
(2) If, additionally, X is dense in Y , then U = X ∩ U .
(3) If X is dense in Y , and if x0 is an isolated point in X, then x0 is
an isolated point in Y , whenever Y is Hausdorff.
A commutative unital Banach algebra A is said to have the topological
stable rank n if n is the smallest integer such that the set
n
o
n
X
xj fj = 1A
Un (A) := (f1 , . . . , fn ) ∈ An : ∃(x1 , . . . , xn ) ∈ An :
j=1
of invertible n-tuples in A is dense in An . We refer to [6] for detailed proofs
concerning the relation of this concept, introduced by Rieffel, to other notions of stable ranks.
2. The spectrum of the algebra of bounded Lebesgue
measurable functions
Let ℓ∞ (X, C) be the set of all bounded, complex-valued functions on
X and let Lb (X, C) be the subset of all bounded Lebesgue measurable
functions. Endowed with the usual pointwise
algebraic operations and the
supremum norm, ℓ∞ (X, C), +, ×, •, ||·||∞ becomes a uniform algebra with
S
4
RAYMOND MORTINI AND RUDOLF RUPP
Lb (X, C) as a closed subalgebra 1. Complex conjugation being an involution, we actually see that these algebras are commutative C ∗ -algebras. Many
properties of our two algebras above therefore are known consequences of
the general abstract theory and can be found for instance in [10, 7]. Our
main object will be Lb (X, C). Its companion algebra, the algebra of all
bounded Borel measurable functions on completely regular spaces is briefly
mentioned in [10, III.1.25]. We shall not consider that algebra here. Let
us recall the following fact, which immediately follows from the classical
theory of Banach algebras. Observe that M (A) denotes the spectrum of a
commutative unital complex Banach algebra A.
Proposition 2.1. Let I be the ideal of all functions in Lb (X, C) which
vanish almost everywhere on X and let h(I) ⊆ M (Lb (X, C)) be its hull.
Then
(i) (L∞ , || • ||e.s ) and (Lb (X, C)/I, || • ||q ) are isomorphic isometric.
(ii) M (Lb (X, C)/I) is homeomorphic with h(I) and M (L∞ ).
We now present the new results of our note. Note that those parts dealing
with the algebra ℓ∞ (X, C) of all bounded functions on X are well-known (see
for example [5, 11]); we only include them to facilitate a comparison of both
algebras which appear here and in order to get a better view in which aspects
‘our’ algebra Lb (X, C) differs from the standard one, ℓ∞ (X, C).
Theorem 2.2. Let A denote either ℓ∞ (X, C) or Lb (X, C). Then the following assertions hold:
P
(1) Un (A) = (f1 , . . . , fn ) ∈ An : inf x∈X nj=1 |fj (x)| ≥ δ > 0 .
(2) A is isometric isomorphic to C(M (A), C).
(3) Let ∆ := {δx : x ∈ X} be the set of evaluation functionals on A.
Then the topological space (∆, g) is a discrete space.
(4) Under the discrete topology, X̃ := (X, Tdis ) can be continuously embedded into (M (A), g).
(5) The set ∆ is dense in (M (A), g).
(6) Any point in ∆ is an isolated point in M (A) and M (A)\∆ is closed.
(7) M (ℓ∞ (X, C)) is homeomorphic to the Stone-Čech compactification
of the discrete space (X, Tdis ) and M (ℓ∞ (X, C)) is extremely disconnected, hence totally disconnected.
(8) M (Lb (X, C)) is totally disconnected, but not extremely disconnected.
(9) The hull of the closed ideal I of all functions in Lb (X, C) which vanish almost everywhere on X is a nowhere dense subset of the corona
C := M (Lb (X, C)) \ ∆ of Lb (X, C) and is extremely disconnected.
(10) dim M (A) = 0.
(11) bsr A = tsr A = 1.
1 The closedness follows from the fact that a pointwise (and a fortiori a uniform)
convergent sequence of measurable functions is measurable.
BOUNDED MEASURABLE FUNCTIONS
Proof. (1) If for some gj ∈ A the identity
1≤
Pn
n
X
Pn
j=1 gj fj
||gj ||∞ |fj | ≤ M
n
X
5
= 1 holds, then
|fj |,
j=1
j=1
and so j=1 |fj | is bounded away from zero. For the converse, just consider
the functions
fj
gj := Pn
|fk |2
Pn k=1
which do belong to A and satisfy j=1 gj fj = 1.
b || · ||M (A) ) (see
(2) As a uniform algebra A is isometric isomorphic to (A,
e.g. [3, p. 185]). Since the spectrum σ(f ) of a real-valued function in A is a
subset of the reals (note that f − λ is invertible whenever λ 6∈ R), we deduce
from the C-linearity of the Gelfand transform that for every f = u + iv ∈ A
and x ∈ M (A)
fb(x) = u
b(x) −i vb(x) = fb(x).
|{z} |{z}
∈R
∈R
b || · ||M (A) ) is a self-adjoint, uniformly closed, point separating
Hence (A,
b = C(M (A)).
subalgebra of C(M (A)). By the Stone-Weierstrass theorem, A
(3) Given x0 ∈ X, let f be the characteristic function of {x0 }. Then f ∈
A. The assertion now follows from the following identities. Let 0 < ε < 1/2.
Then
Uε,f (δx0 ) ∩ ∆ = {m ∈ M (A) : |m(f ) − δx0 (f )| < ε} ∩ ∆
= {δx , x ∈ X : |f (x) − f (x0 )| < ε}
= {δx0 }.
Thus {x0 } is an isolated point in ∆.
(4) The map
(
(X, Tdis ) → (M (A), g)
x
7→ δx
is clearly an embedding by (3).
(5) That ∆ is dense in M (A) follows from (1) using the usual arguments
given e.g. in [3, p. 191].
(6) Because by (5), ∆ is dense in M (A), we deduce from Lemma (C3),
that {x0 } is open in M (A), too. Since ∆ is a union of open singletons, it is
an open subset of M (A). Hence M (A) \ ∆ is closed.
(7) Since X̃ carries the discrete topology, we see that ℓ∞ (X, C) = Cb (X̃, C).
Because X̃ is a completely regular space, we get from the Stone-Čech Theorem that M (ℓ∞ (X, C)) = β X̃. Moreover, β X̃ is extremely disconnected
(see [4] and [1, p.437]).
(8) We first show that M (Lb (X, C)) is totally disconnected. Let m0 ∈
M (Lb (X, C)) and let U be an open neighborhood of m0 . Choose a basic
6
RAYMOND MORTINI AND RUDOLF RUPP
open set (in the Gelfand topology)
V := Vε,f1 ,...,fn (m0 ) =
n
\
{m ∈ M (Lb (X, C)) : |fbj (m) − fbj (m0 )| < ε}
j=1
such that m0∈ V ⊆ U . By [9,p. 15], there are measurable step-functions
sj such that sj − fj + fbj (m0 )∞ < ε/2. Hence
m0 ∈
n
\
j=1
m ∈ M (Lb (X, C)) : |b
sj (m)| < ε/2 ⊆ V ⊆ U.
Now, for fixed j, sj =
PN (j)
k=1
λk,j χEk,j for pairwise disjoint subsets Ek,j of
X, (k = 1, . . . , N (j)). Given a measurable set E 2, we claim that
χ
cE = χM
(2.1)
for some uniquely determined open-closed subset M = ME of M (Lb (X, C)).
In fact, χ2E = χE is an idempotent in Lb (X, C). Hence σ(χE ) ⊆ {0, 1}. The
obvious cases are σ(χE ) = {0}, respectively σ(χE ) = {1}, leading readily
to ME = ∅, respectively ME = X. So assume that σ(χE ) = {0, 1}. Now
σ(f ) = fb(M (Lb (X, C))) for any f ∈ Lb (X, C). If we let
M := {m ∈ M (Lb (X, C)) : χ
cE (m) = 1},
we see that χ
cE = χM . Since χ
cE , and hence χM , is continuous, M is openclosed. This yields (2.1). We conclude that
N (j)
sbj =
X
λk,j χMk,j
k=1
for pairwise disjoint open-closed subsets Mk,j of M (Lb (X, C)). By Lemma
(B),
{m ∈ M (Lb (X, C)) : |b
sj (m)| < ε/2} = {m ∈ M (Lb (X, C)) : χFj (m) = 1}
T
for some closed-open sets Fj ⊆ M (Lb (X, C)). Accordingly, if F := nj=1 Fj ,
then
m0 ∈ F = {m ∈ M (Lb (X, C)) : χF (m) = 1} ⊆ U.
Since F is closed-open, χF is continuous on M (Lb (X, C)) and it can be
written as F = {c
χS = 1} for some measurable subset S ⊆ X. In fact, by
(2), there is f ∈ Lb (X, C) such that fb = χF . Now χ2F = χF ; hence fb2 = fb
2 − f = 0. Since the uniform algebra L (X, C) is semi-simple,
and so f\
b
2
f = f . Hence f is a characteristic function of some measurable subset S of
X. We conclude that the closed-open sets of the form
(2.2)
{m ∈ M (Lb (X, C)) : χ
cS (m) = 1}, S ⊆ X measurable,
2 This set may have measure zero
BOUNDED MEASURABLE FUNCTIONS
7
form a basis for the Gelfand topology. Since M (Lb (X, C)) is a compact
Hausdorff space, the total disconnectedness is just an equivalent property
(see Theorem A).
Next we show that M (Lb (X, C)) is not extremely disconnected. Let E
be a non-measurable subset of X and let F := X \ E. Since X̃ carries
the discrete topology, E and F are open sets in X̃. Hence, by (4), ∆E :=
{δx : x ∈ E} ⊆ ∆ and ∆F := {δx : x ∈ F } ⊆ ∆ are disjoint open sets in
M (Lb (X, C), g). In view of achieving a contradiction, suppose that their
closures in M (Lb (X, C), g) are disjoint. Since by (2), Lb (X, C) is a normal
algebra on its spectrum, there is f ∈ Lb (X, C) such that fb = 1 on ∆E and
fb = 0 on ∆F . Since ∆E ∪ ∆F = ∆, we deduce that
fb−1 ({1}) ∩ ∆ = {δx : x ∈ X, fb(δx ) = 1} = ∆E .
Hence f −1 ({1}) = E and so E is a measurable set, because f is a measurable
function. A contradiction. We conclude that M (Lb (X, C)) is not extremely
disconnected.
(9) Because for any x0 ∈ X, the characteristic function of {x0 } is measurable, we see that x0 6∈ h(I), the hull of I. Hence
∅=
6 h(I) ⊆ M (Lb (X, C)) \ ∆ = C.
Next we show that h(I) is nowhere dense in C. Assuming the contrary,
there is a (relatively) open set V in C such that V ⊆ h(I). Let m0 ∈
χS = 1} in
V . Then, by (2.2), there is a basic closed-open set U := {b
M (Lb (X, C)), where S ⊆ X is measurable, such that
m0 ∈ {m ∈ C : χ
bS (m) = 1} ⊆ V ⊆ h(I).
Now the denseness of ∆ in M (Lb (X, C)) implies that
U ∩ ∆ = U = U.
Hence S is infinite (because otherwise U ∩ ∆ = ∆S := {δx : x ∈ S} is a
closed set contained in ∆, but m0 ∈ C ∩ U ). Let (xn ) be a sequence of
distinct points in S and consider the ideal
J := {f ∈ Lb (X, C) : f (xn ) = 0 for almost every n}.
Then J is a proper ideal. Any maximal ideal containing J is necessarily
contained in the corona C. Hence h(J) ⊆ C. Also
R := {δxn : n ∈ N} \ ∆ ⊆ U \ ∆ = U \ ∆ ⊆ V ⊆ h(I).
Now the function u, given by u(x) = 0 for x ∈ X \ {xn : n ∈ N} and
u(xn ) = 1 for n ∈ N, is measurable. Thus u belongs to Lb (X, C) and u is
equal to 0 almost everywhere. Consequently, u ∈ I and so u
b ≡ 0 on h(I).
But u
b(δxn ) = 1 implies that u
b(m) = 1 for every m in the weak-∗-closure of
{δxn : n ∈ N}. In particular, u
b = 1 on R; a contradiction. We conclude that
h(I) is nowhere dense in C.
8
RAYMOND MORTINI AND RUDOLF RUPP
That h(I) is extremely disconnected, is a consequence of Proposition 2.1
and the fact that h(I) is homeomorphic with the extremely disconnected set
M (L∞ ) ([2, p. 18]).
(10) That dim M (A) = 0 follows from Theorem A and the fact that M (A)
is totally disconnected.
(11) Let f ∈ A. Given ε > 0, the functions
(
f if |f | ≥ ε
Fε :=
ε if |f | < ε
belong to A, are invertible, and uniformly approximate f . Hence tsr A = 1.
But 1 ≤ bsr A ≤ tsr A. Thus bsr A = 1, too.
Proposition 2.1 and (9) show in particular that M (L∞ ) is embedded as a
nowhere dense subset into the Corona C := M (Lb (X, C)) \ ∆ of Lb (X, C).
Acknowledgements
We thank the referee for drawing attention to reference [10].
BOUNDED MEASURABLE FUNCTIONS
9
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4
Université de Lorraine, Département de Mathématiques et Institut Élie
Cartan de Lorraine, UMR 7502, Ile du Saulcy, F-57045 Metz, France
E-mail address: [email protected]
Fakultät für Angewandte Mathematik, Physik und Allgemeinwissenschaften,
TH-Nürnberg, Kesslerplatz 12, D-90489 Nürnberg, Germany
E-mail address: [email protected]