Contents

Chapter 1. Integration on topological groups 3 1. The Haar measure 3 1.1. The Riesz-Markov Theorem 6 2. The Modular function 7 3. The involutive Banach algebra M_{b}(G) 10 4. The Banach algebra L^{1}(G) 12

2 CONTENTS

CHAPTER 1 Integration on topological groups

1. The Haar measure

From now on we will always assume that G is a locally compact, Hausdorff topological group. Let X be a topological space. Denote by C_{c}(X) the space of compactly supported continuous functions on X. For K ⊂ X compact, let

C_{K}(X) = {f ∈ C_{c}(X) | Supp(f) ⊂ K} .

Then C_{c}(X) = ∪_{K}C_{K}(X). Make C_{K}(X) into a Banach space with the norm

||f||_{K} = ||f||_{∞} := sup_{x} |f(x)| .

We have C_{c}(X) = ∪_{K}C_{K}(X). We give C_{c}(X) the inductive limit topology. Then C_{c}(X) is a locally convex, Hausdorff, topological vector space. Let C_{∞}(X) be the space of continuous function that vanishes at infinity, .i.e., f ∈ C_{∞}(X) if for each ϵ > 0 there exists a compact set K such that |f(x)| < ϵ for all x ∉ K. With the norm ||f||_{∞} : sup |f(x)| the space C_{∞}(X) is a Banach space. If K ⊂ X is compact and f ∈ C_{K}(X) then ||f||_{K} = ||f||_{∞}. Therefore the inclusion C_{c}(X) ↪ C_{∞}(X) is continuous. Let M(X) := C_{c}(X)^{∗} and M_{b}(X) := C_{∞}(X)^{∗} where ^{∗} stands for dual space. Then M_{b}(X) ⊂ M(X) and M_{b}(X) is a Banach space with norm

||μ|| = sup_{f∈C_{∞}(X), ||f||_{∞}=1} |μ(f)| .

Definition 1. The space M(X) is the space of complex Radon measures on X. The space M_{b}(X) is the subspace of bounded Radon measures on X. A measure μ is called real if μ(f) ∈ ℝ for all real valued f ∈ C_{c}(X) and positive if μ(f) ≥ 0 for all f ≥ 0. Let M_{+}(X) be the space of positive Radon measure on X.

Example 1.1. Let X = ℝ^{n}. Define λ : C_{c}(ℝ^{n}) by

λ(f) := ∫_{ℝ^{n}} f(x) dx_{1} … dx_{n} .

If K ⊂ ℝ^{n} is compact, then the volume, Vol(K), is finite. If Supp(f) ⊂ K then

|λ(f)| ≤ Vol(K)||f||_{∞} .

Hence λ is a Radon measure, the Lebesgue measure on ℝ^{n}. If φ integrable with respect to λ. Define

λ_{φ}(f) := λ(φf) = ∫ f(x)φ(x) dλ(x) , f ∈ C_{∞}(X) .

Then

|λ_{φ}(f)| ≤ ||f||_{∞}||φ||_{1} .

It follows that λ_{φ} is a bounded measure.

Example 1.2. For x ∈ X let ε_{x}(f) = f(x). As |f(x)| ≤ ||f||_{∞} it follows that ε_{x} is a bounded measure, the Dirac measure at x. Let {x_{j}}_{j∈J}, J ⊂ ℕ, be a finite or countable sequence of points in X. Let r_{j} ∈ ℂ, j ∈ C, be a sequence such that ∑ |r_{j}| = C < ∞. Define μ = ∑_{j∈J} r_{j}ε_{x_{j}}. Thus

μ(f) = ∑_{j∈J} r_{j}f(x_{j}) , f ∈ C_{∞}(X) .

Then |μ(f)| ≤ ∑_{j∈J} |r_{j}| ||f||_{∞} = C||f||_{∞}. Hence μ is a bounded measure.

Example 1.3. Let {x_{j}} be a sequence in X such that if K is a compact set, then K only contains at most finitely many x_{j}s. Let {r_{j}} be any sequence of complex numbers and define

μ(f) := ∑ r_{j}f(x_{j}), f ∈ C_{c}(X) .

If K ⊂ X is compact let C_{K} = ∑_{x_{j}∈K} |r_{j}| < ∞ as this is only a finite sum. Then |μ(f)| ≤ C_{K}||f||_{K}. Thus μ is a measure. Notice that μ is only a finite measure if ∑ |r_{j}| < ∞.

Assume that the topological group G acts separately continuously on X. Let f : X → ℂ and a ∈ G. Define L_{a}f : x ↦ f(a^{−1}x). Then L_{ab}f(x) = f(b^{−1}a^{−1}x) = L_{b}f(a^{−1}x) = L_{a}(L_{b}f)(x). As L_{e}f = f it follows that G acts on the space of functions on X. If f is continuous then L_{a}f is also continuous. Furthermore

Supp(L_{a}f) = aSupp(f)

Hence L_{a}f ∈ C_{c}(X) if and only if f ∈ C_{c}(X) have compact support if and only if f has compact support. Also L_{a}C_{∞}(X) = C_{∞}(X). By duality we can now define an action of G on M(X) and M_{b}(X) by

a · μ(f) = L_{a}μ(f) := μ(L_{a^{−1}}f)

or

∫_{X} f(x)d(a · μ) = ∫_{X} f(ax) dμ(x) , f ∈ C_{c}(X) .

Definition 2. A measure μ ∈ M(X) is quasi-invariant if for each a ∈ G there is a number c(a) ∈ ℝ such that a · μ = c(a)μ. The measure μ is invariant if a · μ = μ for all a ∈ G, i.e. μ ∈ M(X)^{G}.

We apply this now to the group G. As G acts on itself both from the right and the left we have a left and right action of G on C_{c}(G),

L_{a}f(x) = f(a^{−1}x) and R_{a}f(xa) .

We say that μ is left-invariant if L_{a}μ = μ for all a ∈ G and right-invariant if R_{a}μ = μ for all a ∈ G. Notice that if G is abelian then a measure is left invariant if and only if it is right invariant. Let f be a function on G. Define f^{∨}(x) := f(x^{−1}). Then f^{∨} ∈ C_{c}(G) if f ∈ C_{c}(G) and Supp(f^{∨}) = Supp(f)^{−1}. By duality we define ^{∨} : M(G) → M(G) by

μ^{∨}(f) = μ(f^{∨}) .

Lemma 1.1. Let μ ∈ M(G) then μ is left invariant if and only if μ^{∨} is right invariant.

Proof. Exercise.

Definition 3. A measure μ ∈ M_{+}(G), μ ≠ 0 is called a left Haar measure on G if L_{a}μ = μ for all a ∈ G. It is called a right Haar measure on G if R_{a}μ = μ for all a ∈ G.

Theorem 1.2 (Haar). Every locally compact topological group has a left Haar measure and a right Haar measure.

Theorem 1.3 (von Neumann). Let μ and ν be left Haar measure on the locally compact topological group G. Then there exists a positive number c such that μ = cν.

Example 1.4. If G = ℝ^{n} then the Lebesgue measure is left and right invariant. Let G = ℝ^{+} and let

μ(f) = ∫_{0}^{∞} \frac{f(x)}{x} dx .

Then

μ(L_{y}f) = ∫_{0}^{∞} \frac{f(x/y)}{x} dx
= ∫_{0}^{∞} \frac{f(u)}{yu}ydu
= μ(f)

by letting u = x/y and hence x = yu and du = y^{−1}dx. It follows that μ is left and right invariant.

Example 1.5 (GL(n, ℝ)). Recall that GL(n, ℝ) can be viewed as an open dense subset in M(n, ℝ)≃^{Ψ}ℝ^{n^{2}}. Define a measure μ on GL(n, ℝ) by

∫_{GL(n,ℝ)} f(X) dμ(X) : = ∫_{Ψ(GL(n,ℝ))} f(Ψ^{−1}(x))| det( Ψ^{−1}(x))|^{−n}dλ(x)
= ∫_{Ψ(GL(n,ℝ))} f ∘ Ψ^{−1}(x) | det( Ψ^{−1}(x))|^{−n}dλ(x)

where dλ is the Lebesgue measure on ℝ^{n^{2}}. Let C, X ∈ GL(n, ℝ) and denote as before by x_{1}, …, x_{n} the column vectors of X. Then the matrix CX is given by.

CX = (Cx_{1}, …, Cx_{n})

Hence multiplication by C on GL(n, ℝ) corresponds via Ψ to the n^{2} × n^{2} matrixL_{C} = ( \begin{matrix}C & & & & 0 \\ & . & . & . \\ 0 & & & & C \end{matrix} )

which has determinant det(C)^{n}. It follows that

∫ f(CX) dμ(X) = ∫ f(CΨ^{−1}(x)) |det Ψ^{−1}(x)|^{−n} dλ(x)
= ∫ f(Ψ^{−1}(L_{C}(x))) |det Ψ^{−1}(x)|^{−n} dλ(x)
= | det C|^{n} ∫ f(Ψ^{−1}(L_{C}(x))) |det Ψ^{−1}(L_{C}x)|^{−n} dλ(x)
= ∫ f(Ψ^{−1}(x)) |det Ψ^{−1}(x)|^{−n} dλ(x)

Hence μ if left invariant.

From now on λ will be a fixed left Haar measure on G. We will sometimes simply write dx for dλ(x), i.e.

∫_{G} f(x) dλ(x) = ∫_{G} f(x) dx .

For 1 ≤ p ≤ ∞ let L^{p}(G) = L^{p}(G, λ) be the space of measurable functions f : G → ℂ such that

||f||_{p} := (∫_{G} |f(x)|^{p} dx)^{1/p} < ∞ .

Let N (G) be the space of function with ||f||_{p} = 0. This is exactly the space of measurable functions that are zero almost every where. Hence N (G) does not depend on p. Finally we let

L^{p}(G) := L^{p}(G)/N (G) .

Then L^{p}(G) is a Banach space and L^{p}(G)^{∗} = L^{q}(G) where \frac{1}{p} + \frac{1}{q} = 1 and 1 < p < ∞. We let L_{loc}(G) be the set of functions that are integrable over every compact subset of G and

L_{loc}(G) = L_{loc}(G)/N (G) .

We will use standard theorems from measure theory without further comments. The following will be used quite often.

Lemma 1.4. Let φ ∈ L_{loc}(G). Define λ_{φ} : C_{c}(G) → ℂ by

< λ_{φ}, f >= ∫_{G} f(x)φ(x) dλ .

Then λ_{φ} is a Radon measure. λ_{φ} is finite if and only if φ ∈ L^{p}(G). Furthermore the following holds:

1. λ_{φ} = 0 if and only if φ ∈ N (G). Hence φ ↦ λ_{φ} is well defined on L_{loc}(G);
2. |λ_{φ}| = λ_{|φ|};
3. ||λ_{φ}|| = ||φ||_{1} if φ ∈ L^{1}(G) .

Proof. Let K ⊂ G be compact and c_{K} := ∫_{K} |φ(x)| dx. If Supp(f) ⊂ K it follows that |f(x)φ(x)| ≤ ||f||_{∞}|φ(x)| on K and hence

| ∫_{K} f(x)φ(x) dx| ≤ c_{K}||f||_{∞} .

If φ ∈ L^{1}(G) then the same argument shows that λ_{ϖ} is finite. We leave it as an exercise to show that finite implies φ ∈ L^{1}.

1) If φ = 0 a.e. then fφ = 0 a.e. for f ∈ C_{c}(G) and hence λ_{φ} = 0. Assume that λ_{φ} = 0. 2)

3) follows from (2).

1.1. The Riesz-Markov Theorem. Many authors define measure in a different way as a σ-additive maps on a σ-algebra. Integrations is then defined first in terms of step functions and then extended to L^{1} using limiting processes. But by the following theorem of Riesz and Markov both methods are equivalent.

Theorem 1.5 (Riesz-Markov). Let X be a locally compact Hausdorff topological space and ν a positive linear functional on C_{c}(X). Then there is a Borel measure μ on X such that

I(f) = ∫ f(x) dμ(x)

for all f ∈ C_{c}(X). The measure μ may be taking to be quasi regular or inner regular. In each of those cases μ is unique.

Notice that any positive functional is continuous: Let K ⊂ G be compact. Choose φ ∈ C_{c}(X) such that φ|K = 1. Let f ∈ C_{c}(X) with Supp(f) ⊂ K. First we notice that if f is real, then |f| ± f ≥ 0 and hence

|I(f)| ≤ I(|f|) .

If f is complex then we can still show that |I(f)| ≤ I(|f|). In general we have ||f||_{∞}φ − |f| ≥ 0. Hence with C = I(φ) it follows that

|I(f)| ≤ I(|f|) ≤ C||f|| .

Thus I is continuous. This observation shows that the study of quasi regular or inner regular Borel measures on X is equivalent to the study of positive (continuous) linear forms on C_{c}(X).

Exercise

In the following X and Y stands for locally compact T_{2} spaces and G stands for a locally compact Hausdorff topological group. If μ is a measure on X and h ∈ L_{loc}(X) then μ_{h} is the measure f ↦ μ(hf).

1. Let λ ∈ M_{f}(G). Define

||λ|| := |μ|(G).

Show that || · || is a norm on M_{f}(G) and that (M_{f}(G), || · ||) is a Banach space.

2. Let G be the ax + b-group. Show that the measure

∫_{G} f dλ := ∫_{0}^{∞} ∫_{ℝ} f(a, b) \frac{dbda}{a^{2}} .

is left invariant.

3. Proof Lemma 1.1

4. Let G be a discrete group. Show that the counting measure μ(f) = ∑_{g∈G} f(g) is left and right invariant.

5. Show that C_{c}(X) is dense in C_{∞}(X).

6. Let X_{∞} = X ∪ {∞} be the one-point compactification of X and let C(X_{∞}) be the space of continuous functions on X_{∞}. Show that

C_{∞}(X) = {f ∈ C(X_{∞}) | f(∞) = 0}

and that C(X_{∞}) corresponds to the space of bounded continuous functions on X such that lim_{x→∞} f(x) exists. Finally show that every measure in M_{b}(X) extends to a continuous linear form on C(X_{∞}).

7. Let Y ⊂ X be closed. Show the following:

(a) Let K ⊂ X be compact. Then K ∩ Y is compact in Y.

(b) f ∈ C_{c}(X) then f|Y ∈ C_{c}(Y).

(c) The map C_{K}(X) ∋ f ↦ f|Y ∈ C_{K∩Y} (X) is continuous for all K ⊂ X compact.

(d) If μ is a measure on Y, then μ^{Y} (f) := μ(f|Y) is a measure on X.

The measure μ^{Y} is called the extension of μ to X.

8. Let U ⊂ X be open. Show the following:

(a) If K ⊂ U is compact, then K is compact in X

(b) If K ⊂ U is compact, then the canonical injection C_{K}(U) ↪ C_{K}(X) is continuous.

(c) If μ is a measure on X then μ_{U} := μ|C_{c}(U) is a measure on U.

The measure μ_{U} is called the restriction of μ to U.

9. Let λ be the Lebesgue measure on ℝ. Show the following: (a) C_{c}(ℝ^{+}) ∋ f ↦ μ(f) := ∫_{0}^{∞} \frac{f(x)}{x} dx is a measure on ℝ^{+}.

(b) The measure μ can not be extended to a measure on ℝ.

10. Let X and Y be locally compact topological spaces and φ : X → Y a homeomorphism. Let μ be

a measure on X. Show the following:

(a) If f ∈ C_{c}(Y) then f ∘ φ ∈ C_{c}(X).

(b) The map C_{c}(Y) ∋ f ↦ φ^{∗}(μ)(f) := μ(f ∘ φ) ∈ ℂ is a measure on Y.

11. Let μ ∈ M(X). Show the following:

(a) Then there exists a function h ∈ L_{loc}(X) such that μ = |μ|_{h}. Furthermore |h(x)| = 1 μ-a.e.

(b) If g ∈ L_{loc}(X, |μ|) then |μ_{g}| = |μ|_{|g|}.

12. Let G be a locally compact Hausdorff topological group and let λ be a left invariant Haar measure

on G. Show the following:

(a) G is compact if and only if λ(G) < ∞.

(b) G is discrete if and only if λ({e}) > 0.

13. Let μ be a Radon measure on X. For 1 ≤ p < ∞ show that C_{c}(X) is dense in L^{p}(X).

14. Let K ⊂ G be a compact subgroup, X = G/K, and let κ : G → X be the canonical map a ↦ aH. Define a linear form μ_{K} on C_{c}(X) by

μ_{K}(f) = ∫ f ∘ κ(a) dλ(a) .

Show that μ_{K} is a positive, G-invariant Radon measure on X.

15. Let H ⊂ G be a closed subgroup and X = G/H. Assume that μ ≠ 0 is a positive Radon measure on X and let λ_{H} be a left Haar measure on H. For f ∈ C_{c}(G) define

f^{♭}(xH) := ∫_{H} f(xh) dλ_{H}(h) .

Show the following:

(a) f^{♭} is well defined and f^{♭} ∈ C_{c}(X).

(b) Define a linear form on C_{c}(G) by

ν(f) : = ∫_{X}∫_{H} f(xh) dhdμ(xH)
= ∫ f^{♭}(x) dμ(x)

Then ν is a positive, non-zero, G-left invariant Radon measure on G.

16. Let H_{+} = {z ∈ ℂ | Im(z) > 0} = SL(2, ℝ)/SO(2). Define a lnear map μ on H_{+} by

μ(f) = ∫_{−∞}^{∞}∫_{0}^{∞} f(x + iy) \frac{dydx}{y^{2}} .

Show that μ is a positive, non-zero, SL(2, ℝ)-invariant Radon measure on H_{+}.

2. The Modular function

From now on G denotes a locally compact topological group and λ = λ_{L} a left invariant Haar measure on G. Let a ∈ G, then R_{a}λ is also left invariant. Hence there exists a positive number Δ(a) such that R_{a}λ = Δ(a)λ. Thus

∫ f(xa) dλ(x) = Δ(a)^{−1} ∫ f(x) dλ(x)

for all f ∈ C_{c}(G).

Definition 4. The function Δ = Δ_{G} : G → ℝ^{+} with

∫ f(xa) dμ(x) = Δ(a)^{−1} ∫ f(x) dμ(x) (2.1)

is called the modular function for G.

Lemma 1.6. Let λ be a left Haar measure on G. Then there exists a positive function f ∈ C_{c}(G) such that λ(f) > 0.

Proof. As λ ≠ 0 there is a function f ∈ C_{c}(G) such that λ(f) ≠ 0. Write f = f_{R} + if_{I} with f_{R}, f_{I} real. Then λ(f_{R}) ≠ 0 or λ(f_{I}) ≠ 0, so we can assume that f is real. Then decompose f = f_{+} − f_{−} where f_{+} = max{f, 0} and f_{−} = max{−f, 0}. Obviously f_{+}, f_{−} ∈ C_{c}(G). Furthermore either λ(f_{+}) ≠ 0 or λ(f_{−}) ≠ 0. We can therefore assume that f is non-negative. But then λ(f) > 0 as λ is positive.

Lemma 1.7. Let λ be a left Haar measure and f ∈ C_{c}(G), f ≠ 0. Then λ(f) > 0.

Proof. We can assume that f(e) ≠ 0. We can assume that f(e) = 1. Assume that λ(f) = 0. Let U = Supp(f). Then U is compact and U^{o} ≠ ∅. Let V = {x ∈ G | f(x) > 1/2}. Then V is open, e ∈ V, and V̄ is compact. Let g ∈ C_{c}(G), g > 0. Then there exists a_{1}, …, a_{n} ∈ G such that Supp(g) ⊂ ∪_{j=1}^{n}a_{j}V. Suppose that g(x) ≠ 0. Then we can write x = a_{j}y with y ∈ V. Hence L_{a_{j}}f(x) = f(y) > 1/2, or 2||g||_{∞}L_{a_{j}}f(x) > ||g||_{∞}. Thus

g ≤ 2||g||_{∞} ∑ L_{a_{j}}f .

It follows that

∫ g(x) dλ(x) ≤ 2||g||_{∞} ∑ ∫ L_{a_{j}}f(x) dλ(x) = 2n||g||_{∞} ∫ f(x) dλ(x) = 0 .

Hence λ(g) = 0 for all positive functions g ∈ C_{c}(G). But this obviously contradicts Lemma 1.6.

Lemma 1.8. Let K, L ⊂ G be compact. Then KL ⊂ G is compact.

Proof. The set K × L ⊂ G × G is compact. As the multiplication map μ : G × G → G, μ(a, b) = ab is continuous it follows that the image μ(K × L) = KL is compact.

Lemma 1.9. The modular function is a continuous homomorphism from G into ℝ^{+}.

Proof. It is clear that Δ : G → ℝ^{+} is a homomorphism, so it suffice to show that Δ is continuous at e. Let f ∈ C_{c}(G) be such that λ(f) = 1. Then

|Δ(a) − 1| = | ∫ f(xa^{−1}) − f(x) dλ(x)| ≤ ∫ |f(xa^{−1}) − f(x)| dμ(x) .

Let U ∈ N (e) be symmetric and compact. Then Supp(f)U = V is compact. Let φ ∈ C_{c}(G) be such that φ|V = 1. Then

|f(xa^{−1}) − f(x)| ≤ 2||f||_{∞}φ(x)

which is integrable. It therefore follows from the Lebesgue Dominated Convergence Theorem that

lim_{U∋a→e} |Δ(a) − 1| ≤ ∫ lim_{U∋a→e} |f(xa^{−1}) − f(x)| dλ(x) = 0

Hence Δ is continuous.

Definition 5. The group G is called unimodular if Δ = 1, i.e., every left invariant measure is also right invariant.

Example 1.6 (Unimodular groups). The following type of groups are unimodular:

1. Abelian groups;

2. A group G is almost compact if G/Z(G) is compact. Here Z(G) is the center of G. Every almost compact group is unimodular. In particular every compact group is unimodular.

Proof. As xz = zx for all x ∈ G and z ∈ Z(G) it follows that Δ|Z(G) = 1. Hence Δ factor to a homomorphism Δ : G/Z(G) → ℝ^{+}. But G/Z(G) is a compact group. Hence Δ(G/Z(G)) is a compact subgroup of ℝ^{+}. But the only compact subgroup of ℝ^{+} is the trivial one {1}. Hence Δ(G/Z(G)) = {1}.

Assume that G is compact, then the constant function 1 : x ↦ 1, is in C_{c}(G) and thus integrable with respect to any Haar measure. We can therefore normalize the Haar measure on G such that ∫ dλ = λ(G) = 1.

3. The Heisenberg group.

4. The general linear group GL(nℝ).

Example 1.7 (Non unimodular groups). The following groups are not unimodular:

1. The ax + b-group (see exercise 1).

2. The group of upper triangular matrices \{( \begin{matrix}a_{1} & & x_{ij} \\ 0 & ... \\ 0 & 0 & a_{n} \end{matrix} ) | a_{j} ∈ ℝ^{+}, x_{ij} ∈ ℝ\}, see exercise 5.

Lemma 1.10. Let G be a locally compact topological group with left Haar measure λ and right Haar measure λ_{R}. Denote by Δ the modular function on G. Then the following holds.

1. L_{a}λ_{R} = Δ(a)λ_{R}.

2. The measure f ↦ ∫ \frac{f(x)}{Δ(x)} dλ(x) is right invariant.

3. ∫ f(x^{−1}) dλ(x) = ∫ \frac{f(x)}{Δ(x)} dλ(x) for all f ∈ C_{c}(G). Thus λ^{∨} = λ_{1/Δ}.

4. Let f^{∨} := f(x^{−1})/Δ(x). Then ||f|| = ||f^{∨}|| .

Proof. Recall that the measure λ^{∨} : f ↦ ∫ f(x^{−1}) dλ(x) is right invariant. Hence there is a constant c > 0 such that λ_{R} = cλ^{∨}. We can therefore assume that λ_{R} = λ^{∨}. Let f ∈ C_{c}(G), f > 0, such that f(e) > 0. Let φ(x) := f(x)f(x^{−1}). Then φ ∈ C_{c}(G), φ > 0, φ(x) = φ(x^{−1}), and φ(e) > 0. Then λ(φ) = λ_{R}(φ)

1) As L_{a}λ_{R} is right invariant it follows that L_{a}λ_{R} = c(a)λ_{R}. Thus

c(a) ∫ φ(x) dλ_{R}(x) = ∫ φ(ax) dλ_{R}(x)
= ∫ φ(ax^{−1}) dλ(x)
= ∫ φ(xa^{−1}) dλ(x) (because φ(x) = φ(x^{−1}))
= Δ(a) ∫ φ(x) dλ(x)
= Δ(a) ∫ φ(x) dλ_{R}(x)

By Lemma 1.7 we have λ(φ) ≠ 0, hence Δ(a) = c(a).

2) We have

∫ R_{a}f(x) dλ_{1/Δ}(x) = ∫ f(xa)\frac{1}{Δ(x)} dλ(x)
= Δ(a) ∫ \frac{f(xa)}{Δ(xa)} dλ(x)
= ∫ \frac{f(x)}{Δ(x)} dλ(x)
= λ_{1/Δ}(f)

by (2.1). It follows that λ_{1/Δ} is right-invariant.

3) As λ^{∨} = λ_{R} is right invariant it follows by (2) that λ^{∨} = cλ_{1/Δ} for some c > 0. Take f > 0 such that f^{∨} = f. Then λ^{∨}(f) = λ(f). As λ^{∨∨} = λ = c(λ_{1/Δ})^{∨} it follows that

λ(f) = c ∫ f(x^{−1})/Δ(x) dλ(x)
= c ∫ f(x)/Δ(x) dλ(x)
= cλ^{∨}(f)

Hence c = 1.

(4) follows from (3).

Exercise

1. Show that the modular function for the ax + b group is given by Δ(a, b) =

2. Prove Lemma 1.1

3. Let G be a discrete group. Show that the counting measure μ(f) = ∑_{g∈G} f(g) is left and right invariant.

4. Show that GL(n, ℝ) is unimodular.

5. For a = (a_{1}, …, a_{n}) ∈ ℝ^{n}, a_{1} ≠ 0, and x = (x_{ij}) ∈ M(n, ℝ) with x_{ij} = 0 if i ≥ j let

A(a, x) = ( \begin{matrix}a_{1} & & x_{ij} \\ 0 & ... \\ 0 & 0 & a_{n} \end{matrix} ) ∈ GL(n, ℝ)

and let S be the subgroup of all GL(n, ℝ) consisting of all A(a, x), i.e., S is the group of all uuper triangular matrices. Find a left invariant measure of the form

μ(f) = ∫_{0}^{∞}⋯∫_{0}^{∞}∫_{−∞}^{∞}…∫_{−∞}^{∞} f(A(x, x))h(a, x)dx_{n−1,n} … x_{12}da_{1} … da_{n}
= ∫_{−∞}^{∞}⋯∫_{−∞}^{∞}∫_{0}^{∞}…∫_{0}^{∞} f(A(a, x))k(a, x) da_{1} … da_{n}dx_{n−1,n} … x_{12}

with continuous functions h and k. Then find the modular function for G.

6. Let G be a topological group. Let H = G × ℝ with the multiplication

(a, t)(b, s) = (ab, Δ(a)s + t) .

Show that the measure

f ↦ ∫ ∫ f(a, s)Δ(a)^{−1} dsda

is left and right invariant. In particular any locally compact Hausdorff topological group is a closed subgroup of a unimodular group.

3. The involutive Banach algebra M_{b}(G)

In this section G is a locally compact Hausdorff topological group with left Haar measure λ. Let M_{b} = M_{b}(G) be the Banach space of bounded measures on G. Let μ_{1}, …, μ_{n} ∈ M_{b} and f ∈ C_{c}(G). Then the function

G × … × G ∋ (x_{1}, …, x_{n}) ↦ f(x_{1} · . . . · x_{n}) ∈ ℂ

is integrable with respect to μ_{1} ⊗ … ⊗ μ_{n}. Define μ_{1} ⋆ … ⋆ μ_{n} : C_{c}(G) → ℂ by

μ_{1} ⋆ … ⋆ μ_{n}(f) := ∫_{G} … ∫_{G} f(x_{1} · . . . · x_{1}) dμ_{1}(x_{1}) … dμ_{n}(x_{n}) .

Lemma 1.11. Let μ, ν, σ, μ_{1}, …, μ_{n} ∈ M_{b}(G). Then μ_{1} ⋆ … ⋆ μ_{n} ∈ M_{b}(G) and ||μ_{1} ⋆ … ⋆ μ_{n}|| ≤ ||μ_{1}|| · . . . · ||μ_{n}||. Furhtermore the following holds:

1. The map

M_{b}(G) × M_{b}(G) ∋ (μ, ν) ↦ μ ∗ ν ∈ M_{b}(G)

is bilinear and bounded.

2. μ ⋆ ν(f) = ∫_{G} ∫_{G} f(xy) dν(y)dμ(x)

3. (μ ⋆ ν) ⋆ σ = μ ⋆ (ν ⋆ σ)

4. Let a ∈ G and ε_{a} be the point mass at a. Then ε_{a} ⋆ ν = L_{a}ν and ν ⋆ ε_{a} = R_{a^{−1}}ν. In particular ε_{e} ⋆ ν = ν ⋆ ε_{e} = ν.

5. If μ and ν are real, then μ ⋆ ν is real.

6. If both μ and ν are positive, then μ ⋆ ν is positive.

3. THE INVOLUTIVE BANACH ALGEBRA M_{b}(G) 11

Proof. Let f ∈ C_{c}(G). Then (x_{1}, …, x_{n}) ↦ f(x_{1} · . . . · x_{n}) is continuous and vanishes at infinite. Furthermore

|μ_{1} ⋆ … ⋆ μ_{n}(f)| ≤ ∫ ∫ |f(x_{1} · . . . · x_{n})| d|μ_{1}|(x) … d|μ_{n}|(y) ≤ ||f||_{∞}||μ_{1}|| … ||μ_{n}|| .

Thus μ_{1} ⋆ … ⋆ μ_{n} ∈ M_{b}(G).

1) That μ, ν ↦ μ ∗ ν is bounded follows directly from the first part. Let f ∈ C_{c}(G) and r ∈ ℝ. Then

μ ∗ (rν + σ)(f) = ∫ ∫ f(xy)dμ(x)d(rν + σ)(y)
= r ∫ ∫ f(xy)dμ(x)dν(y) + ∫ ∫ f(xy)dμ(x)dσ(y)
= rμ ∗ ν(f) + μ ∗ σ (f) .

Thus μ ∗ (rν + σ) = rμ ∗ ν + μ ∗ σ. One shows in the same way that (rν + σ) ∗ μ = r(ν ∗ μ) + σ ∗ μ.

2) This follows by Fubini’s theorem because the function

G × G ∋ (x, y) ↦ f(xy) ∈ ℂ

is integrable with respect to μ ⊗ ν.

3) Let f ∈ C_{c}(G). Then

(μ ⋆ ν) ⋆ σ(f) = ∫ ∫ f(xy) d(μ ⋆ ν)(x)dσ(y)
= ∫ ∫ (∫ f(uty) dμ(u)dν(t)) dσ(y)
= ∫ (∫ ∫ f(uty) dν(t)dσ(y)) dμ(u)  (by Fubini’s Theorem)
= ∫ ∫ f(us) d(ν ⋆ σ)(s)dμ(u)
= μ ⋆ (ν ⋆ σ)(f).

4) Let f ∈ C_{c}(G). Then

ε_{a} ⋆ ν(f) = ∫ ∫ f(xy) dε_{a}(x)dν(y)
= ∫ f(ay) dν(y)
= < L_{a}ν, f > .

Thus ϵ_{a} ∗ ν = L_{a}ν. Similarly

ν ⋆ ε_{a}(f) = ∫ ∫ f(xy) dε_{a}(y)dν(x)
= ∫ f(xa) dν(x)
= < ν, R_{a}f >
= < R_{a^{−1}}ν, f > .

The other parts of the lemma are clear.

Definition 6. Let μ_{1}, …, μ_{n} ∈ M_{b}(G), then the measure μ_{1} ⋆ … ⋆ μ_{n} is called the convolution of μ_{1}, …, μ_{n}.

Let μ ∈ M_{b}(G). Define μ^{∗} : C_{c}(G) → ℂ by

μ^{∗}(f) := \overline{∫ f̄(x^{−1}) dμ(x)} .

Lemma 1.12. Let μ, ν ∈ M_{b}(G). Then μ^{∗} ∈ M_{b}(G) and the following holds:

1. μ^{∗∗} = μ

2. ||μ^{∗}|| = ||μ||,

3. μ ↦ μ^{∗} is conjugate linear.

4. (μ ⋆ ν)^{∗} = ν^{∗} ⋆ μ^{∗};

Proof. Obviously μ^{∗∗} = μ and μ ↦ μ^{∗} is complex linear. Furthermore μ^{∗} is linear and |μ^{∗}(f)| ≤ ||f||_{∞}||μ||. Hence μ^{∗} ∈ M_{b}(G) and ||μ^{∗}|| ≤ ||μ||. As μ^{∗∗} = μ it follows that ||μ^{∗}|| = ||μ||. Let f ∈ C_{c}(G). Then

ν^{∗} ⋆ μ^{∗}(f) = ∫ ∫ f(yx) dν^{∗}(y)dμ^{∗}(x)

= \overline{∫ \overline{(∫ f(yx^{−1}) dν^{∗}(y))} dμ(x)}

= \overline{∫ (∫ \overline{f(y^{−1}x^{−1})} dν(y))} dμ(x)

On the other hand we have

(μ ⋆ ν)^{∗}(f) = \overline{∫ \overline{f(s^{−1})} d(μ ⋆ ν)(s)}
= \overline{∫ ∫ \overline{f(y^{−1}x^{−1})} dμ(x)dν(y)}

It follows now by Fubini’s Theorem that (μ ⋆ ν)^{∗} = ν^{∗} ⋆ μ^{∗}.

Definition 7. Let A be a n algebra over ℂ. An involution on A is a map ^{∗} : A → A such that the following holds:

1. a^{∗∗} = a for all a ∈ A

2. a ↦ a^{∗} is conjugate linear

3. (ab)^{∗} = b^{∗}a^{∗} An algebra A with involution ^{∗} is called an involutive algebra.

Definition 8. Let A be an algebra and || · || a norm on A. Then A is called a normed algebra if ||ab|| ≤ ||a|| ||b|| for all a, b ∈ A. If ^{∗} is an involution on A then (A, || · ||,^{∗}) is called a normed involutive algebra if ||a^{∗}|| = ||a|| for all a ∈ A. If (A, || · ||) is a Banach space then (A, || · ||,^{∗}) is called an involutive Banach algebra. A is an involutive Banach algebra with identity if there is an element e ∈ A such that ea = ae = a for all a ∈ A.

We can now collect our results on (M_{b}(G), || · ||,^{∗}) together in the following theorem:

Theorem 1.13. Let G be a locally compact group. Then (M_{b}(G), || · ||,^{∗}) is a involutive Banach algebra with identity element ε_{e}. Furthermore M_{b}(G) is abelian if and only if G is abelian.

Proof. The only statement that we still have to prove is that M_{b}(G) is abelian if and only if G is abelian. Assume first that G is abelian. Then f(xy) = f(yx) for all f ∈ C_{c}(G) and x, y ∈ G. This implies that μ ∗ ν = ν ∗ μ for all μ, ν ∈ M_{b}(G). Assume now that M_{b}(G) is abelian. Then ε_{xy} = ε_{x} ⋆ ε_{y} = ε_{y} ⋆ ε_{x} = ε_{yx}. If G is not commutative we can find x, y ∈ G such that xy ≠ yx. But {xy} is compact Hence there is a f ∈ C_{c}(G) such that f(xy) ≠ f(yx) which implies that ε_{xy} ≠ ε_{yx}. Thus G is abelian.

Exercise

In the following λ will always stand for a left-invariant Haar measure on the topological group G. We let M_{b}(G) denote the space of finite Radon measures on G.

4. The Banach algebra L^{1}(G)

In the section G is a locally compact Hausdorff topological group with a fixed left Haar measure λ. We then have an isometric embedding

L^{1}(G) ∋ f ↦ λ_{f} ∈ M_{b}(G) .

In this section we will discuss the relation between this embedding and the algebraic operations from last section.

4. THE BANACH ALGEBRA L^{1}(G) 13

Lemma 1.14. Let f, g ∈ L^{1}(G). Then

x ↦ f(x)g(x^{−1}y)

is integrable for almost all y ∈ G,

f ∗ g(y) := ∫ f(x)g(x^{−1}y) dμ(x)

is in L^{1}(G), and

||f ∗ g||_{1} ≤ ||f||_{1} ||g||_{1} .

Furthermore we have

λ_{f∗g} = λ_{f} ∗ λ_{g} .

Proof. We have by Funbini’s Theorem

∫ ∫ |f(x)||g(x^{−1}y)| dλ(x) dλ(y) = ∫ ∫ |f(x)||g(x^{−1}y)| dλ(y)dλ(x)
= ∫ |f(x)| (∫ |g(x^{−1}y) | dλ(y)) dλ(x)
= ||g||_{1} ∫ |f(x)| dλ(x)
= ||g||_{1}||f||_{1}
< ∞

as λ is left invariant. It follows that

G × G ∋ (x, y) ↦ f(x)g(x^{−1}y) ∈ ℂ

is integrable with respect to λ ⊗ λ. In particular

∫_{G} f(x)g(x^{−1}y) dμ(x)

is finite for almost all y ∈ G and y ↦ ∫ f(x)g(x^{−1}y) dμ(x) is integrable. By the above calculation we also get:

∫ |f ⋆ g(y)| dλ(y) ≤ ∫ ∫ |f(x)||g(x^{−1}y)| dλ(x) dλ(y) = ||f||_{1}||g||_{1} .

Let h ∈ C_{c}(G). Then

< λ_{f} ⋆ λ_{g}, h >= ∫ ∫ h(xy) dλ_{f}(x)dλ_{g}(y)
= ∫ ∫ h(xy)f(x)g(y) dλ(x)dλ(y)

As |h(xy)f(x)g(y)| ≤ ||h||_{∞}|f(x)||g(y)| it follows by Fubini’s Theorem that

∫ ∫ h(xy)f(x)g(y) dλ(x)dλ(y) = ∫ ∫ h(xy)g(y) dλ(y)f(x) dλ(x)
= ∫ ∫ h(y)g(x^{−1}y) dλ(y)f(x) dλ(x)
= ∫ h(y) ∫ f(x) g(x^{−1}y) dλ(x) dλ(y)
= ∫ h(y)f ⋆ g(y) dλ(y)
= < h, λ_{f⋆g} >

where we have used Fubini’s theorem one more time. Theorem 1.15. Let f : G → ℂ. Define f^{∗}(x) := \overline{f(x^{−1})}/Δ(x). Then the following holds:

1. f^{∗} ∈ L^{1}(G) if and only if L^{1}(G) and ||f|| = ||f^{∗}||..

2. (L^{1}(G), || · ||_{1},^{∗}) is a involutive Banach algebra.

3. λ_{f^{∗}} = λ_{f}^{∗}. Thus L^{1}(G) ∋ f ↦ λ_{f} ∈ M_{b}(G) is a norm-preserving ∗-homorphismus.

Proof. (1) is an exercise. (2) follows from (1). Let g ∈ C_{c}(G). Using that λ is real and Lemma 1.10 we get:

λ_{f}^{∗}(g) = \overline{∫ \overline{g(x^{−1})} dλ_{f}(x)}
= \overline{∫ \overline{g(x^{−1})} f(x) dλ(x)}
= ∫ g(x^{−1}) \overline{f(x)} dλ(x)
= ∫ g(x)\overline{f(x^{−1})}Δ(x)^{−1} dλ(x)
= ∫ g(x)f^{∗}(x) dλ(x)
= λ_{f^{∗}}(g) .

The claim follows now from the fact that ||λ_{f}|| = ||f||_{1} for all f ∈ L^{1}(G).

Exercise

In the following λ will always stand for a left-invariant Haar measure on the topological group G. We let M_{f}(G) denote the space of finite Radon measures on G.

1. Let f ∈ L^{1}(G) and μ ∈ M_{b}(G). Define f ⋆ μ so that f ⋆ μ is a function and λ_{f⋆μ} = λ_{f} ⋆ μ. Find a formula for f ⋆ μ. Similarly find μ ⋆ f such that λ_{μ∗f} = μ ⋆ λ_{f}.

2. Let 1 ≤ p < ∞. If f ∈ L^{p}(G) and μ ∈ M_{b}(G) then f ∗ μ, μ ∗ f ∈ L^{p}(G) and

||f ∗ μ||_{p}, ||μ ∗ f||_{p} ≤ ||f||_{p}||μ|| .

3. Let H be a closed subgroup of G and let X = G/H. Assume that μ is a positive, G-invariant measure on X. Show that if f ∈ L^{1}(G) and g ∈ L^{1}(X, μ) then

G ∋ a ↦ f(a)g(a^{−1}x) ∈ ℂ

is integrable with respect to λ for μ-almost all x ∈ X. Furthermore (a) X ∋ y ↦ f ∗ g(x) := ∫ f(a)g(a^{−1}x) dλ is integrable with respect to μ

(b) ||f ∗ g||_{1} ≤ ||f||_{1}||g||_{1}.

4. Let 1 ≤ p < ∞. Show that the map L : G → B(L^{p}(G)), L_{a}f(b) := f(a^{−1}b) is strongly continuous, i.e.,

lim_{a→b} ||L_{a}f − L_{b}f||_{p} = 0

for all f ∈ L^{p}(G).

5. Let f ∈ L^{1}(G) and g ∈ C_{c}(G). Then f ∗ g is continuous.

6. Let f, g ∈ L^{2}(G). Then f ∗ g is defined and f ∗ g ∈ C_{∞}(G).

7. Assume that there exists a sequence of open set U_{n} ∈ N (e), n ∈ ℕ, such that ∩U_{n} = {e}. We can

then assume that U_{n+1} ⊂ U_{n}. Show the following:

(a) There exists a sequence of compactly supported functions {f_{n}}_{n∈ℕ} such that Supp(f) ⊂ U_{n} and ∫ f_{n} = 1.

(b) Let f ∈ L^{1}(G) then

lim_{n→∞} f_{n} ∗ f = f .