Function Algebras Over Groups

FUNCTION ALGEBRAS OVER GROUPS V. MURUGANANDAM

Abstract. One of the primary aspects of harmonic analysis is

the study of functions on a homogenous space by means of suitable group actions. The other end of this thread is to introduce various

classes of functions on groups and study their properties vis-a-vis the underlying groups.

In this compact course, we focus on this

concept with particular reference to the most useful and widely studied notion in harmonic analysis namely Amenability.

We

aim to  look beyond the amenability at the end of the course.

1.

Preliminaries

Denition 1.1. A topological space G is said to be a topological group if it is a group and the map (x, y) → xy−1 is continuous from G × G into G. Let us give some important classes of locally compact Hausdor groups.

Examples 1.2. (i) Rn is a locally compact group. (ii) For every n ≥ 1, Tn is a compact group. (iii) For every xed n, let GL(n, R) be the group consisting of all n × n invertible matrices with real entries. Then GL(n, R) is a locally compact group as it is an open subset of Rn2 . (iv) SL(n, R) = {x ∈ GL(n, R) : det(x) = 1} and O(n) consisting of orthogonal matrices are also locally compact groups as they form closed subsets of GL R).  (n,    a b (v) S = x = 0 1 : a > 0, b ∈ R . (vi) If G is the Heisenberg group given by     1 x z     0 1 y g= : x, y, z ∈ R   0 0 1 0This notes is based on a series of lectures given in the Workshop and a centenary conference on Analysis and Applications, IISc Mathematics Initiative (IMI), I.I.Sc, Bangalore, from May, 14-23, 2009.

The author thanks R. Lakshmi Lavanya for

writing down the notes. 1

2

V. MURUGANANDAM

then G is noncompact and nonabelian. (vii) Let F2 denote the free group generated by a, b, with ab 6= ba. It is a nonabelian discrete group. Arbitrary element of this group is of the form am bn ak · · · where m, n, k belong to Z. Cc (G)

Let

and

Cb (G)

denote the space of all continuous functions

with compact support and continuous bounded functions on

G, respec-

tively. Let

Clu (G), Cru (G) and Cu (G) denote the space of all left uniformly

continuous functions, right uniformly continuous functions and uniformly continuous functions on

G.

Measure algebra. Denition 1.3. Let X be a locally compact space. A Borel measure µ is said to be regular if (1) µ(K) < ∞, for every compact set K. (2) For every Borel set E , µ(E) = inf{ µ(U ) : U is open and E ⊆

1.1.

(3)

U }.  If E is any Borel set and µ(E) < ∞, then µ(E) = sup µ(K) : K is compact, K ⊆ E .

Denition 1.4. A Banach space A over C is called a Banach algebra if A is an algebra satisfying kxyk ≤ kxkkyk, ∀ x, y ∈ A

and is called Banach? -algebra if it admits an involution x → x? on A such that kx? k = kxk for all x in A. Let

G

be a locally compact Hausdor group. If

space of complex Borel measures on

G,

M (G)

denotes the

then it forms a Banach Space.

In what follows, we briey show that the underlying group structure gives rise to two additional structures on

M (G)

µ∗ν

is called the

?

with respect to which

µ, ν are in M (G), then Z Z φ(x) d(µ ∗ ν)(x) = φ(xy) dµ(x) dν(y).

forms an Banach

Z

M (G),

algebra. If

convolution of the measures.

For every

dene

µ and ν , µ ∗ ν

M (G) and kµ ∗ νk ≤ kµk kνk . x belongs to G, then δx denotes the Dirac measure x. If x, y are in G, then Z Z Z Z φ d(δx ∗ δy ) = φ(uv) dδx (u) dδy (v) = φ(xy) = φ dδxy .

belongs to

Recall that if

at

3

Therefore,

M (G).

δx ∗ δy = δxy . See that δe is the multiplicative µ → µ? on M (G) is dened by Z Z ? φ dµ = φ(x−1 ) dµ(x).

It is easy to see that 1.2.

identity of

The involution

Group algebra.

δe

is the identity for

M (G).

One of the milestones in the history of abstract

Harmonic analysis was the existence of left invariant measure on a general locally compact Hausdor group that reads as follows. A Borel

m on G is said to be left invariant if m(xE) = m(E) for every set E, and for all x in G.

measure Borel

Theorem 1.5. (Haar, Von Neumann) Let G be a locally compact Hausdor group. There exists a non-zero, positive, left invariant, regular Borel measure on G. Moreover, it is unique up to a positive constant. Such a measure is called Haar measure and the corresponding integral is called Haar integral. Loomis[11] for a proof. Let

dx

We refer to the books by A. Weil[18], always denote the Haar integral.

Let us see some examples of Haar measures. (1) For any subset

E

of a discrete group, if

the number of elements in

counting measure.

E,

m(E)

is dened to be

then it is a Haar measure called

(2) The Lebesgue measure is the Haar measure for the group

Rn ,

as it is translation invaraint. R 2π 1 (3) For the Torus T, f (eıx )dx is the Haar integral. 2π 0 1 (4) If G = GL(n, R) then verify that dg11 dg12 · · · dgnn denes (det g)n a Haar measure. 1 (5) If G denotes the group given in 1.2(v) then 2 dadb is the Haar a measure. (6) The Lebesgue measure

dg = dxdydz is the Haar measure for the Heisenberg group given in 1.2(v). Let

Lp (G), 1 ≤ p < ∞

denote as usual, the Banach space consisting R f such that G |f (x)|p dx is nite. Similarly

of all measurable functions ∞ one denes L (G).

Theorem 1.6. (Lusin). For every p, 1 ≤ p < ∞, Cc (G) is dense in Lp (G).

Proposition 1.7. If f in Lp (G), 1 ≤ p < ∞ is xed then the map x →x f from G into Lp (G) is continuous.

4

V. MURUGANANDAM

Proof.

f ∈ Cc (G). Then, f is uniformly continuous. f. Let  > 0 be given. Since G is locally compact, there exists a neighborhood W of identity e such that W is compact. Choose a neighborhood U of e such that U ⊆ W and if x, y ∈ G are such that x ∈ U y then r  |f (x) − f (y)| < , µ(KW ) Let us assume that

Let K be the support of

µ

being the xed Haar measure. Then for every

x, y ∈ G

such that

x ∈ Uy

, we have

||x f −y f ||p < .

Use Lusin's theorem to show that the result holds for arbitrary elep ment of L (G).  A Haar measure need not be right invariant. In fact, for a xed

x

R

dmx by h f, dmx i = G f (yx)dy then one can see that it is left integral. Therefore, there exists a function x → ∆(x) called

if we dene

modular function satisfying the following: Z

Z

−1

f (y)dy

f (yx)dy = ∆(x ) G

G

The modular function of G can easily be seen to be a homomorphism + from G into R . By Proposition 1.7, we see that ∆ is continuous. A group

G

is called

Unimodular if

∆(x) = 1 ∀x ∈ G. It is trivial that any abelian group or discrete group is unimodular. As

∆

is a continuous homomorphism, any compact group is unimodular.

In fact all groups enumerated in 1.2 except the group are unimodular.

∆(x) = a−1 ,

 if

x=x=

a b 0 1

S

in example (v)

 .

Recall that by Radon-Nikodym theorem we can view

L1 (G)

as a

M (G) consisting of all measures which are absolutely 1 continuous with respect to Haar measure. We show that L (G) is a Banach ? subalgebra of M (G). 1 For every f, g ∈ L (G), since Z Z Z φ(x)d(f ∗ g)(x) dx = φ(xy)f (x)g(y) dx dy closed subspace of

G

G G Z Z

= G G

φ(x)f (y)g(y −1 x) dy dx,

5

we have

Z f ∗ g(x) =

f (y)g(y −1 x) dy.

(1.1)

G Similarly, the involution of

M (G),

restricted to

L1 (G),

is given by

f ? (x) dx = f (x−1 ) d(x−1 ). That is,

f ? (x) = ∆(x−1 )f (x−1 ).

(1.2)

Summarizing we observe that with the convolution product (1.1) and L1 (G) forms an Banach ?-algebra called

group

the involution (1.2),

algebra.

Moreover it is a two-sided ideal in

M (G)

and closed under

involution. In fact

Z µ ∗ g(x) =

g(y −1 x) dµ(y).

G

Z f ∗ µ(x) =

∆(y −1 )f (xy −1 ) dµ(y).

G 1 One can easily see that if G is commutative then L (G) is a com? mutative Banach -algebra. With a little more eort, one can prove the 1 converse also. That is, if L (G) is commutative with the convolution product then the underlying group is commutative. Let us also remark that if G is discrete then δe is the identity for 1 L (G). We recall that the converse also holds. That is L1 (G) has identity if and only if

G

is discrete.

But nevertheless, the group algebra of a general locally compact group, has bounded approximate identity.

We briey construct one

such as follows. Let

{Vα }α∈ I

be a neighbourhood system consisting of compact neigh-

bourhoods at e. Set

fα = where

µ

is the Haar measure and

contained in 1

Vα .

Then

{fα }α∈I

gα , µ(Vα ) gα is in Cc (G)

(1.3) with support of

gα

is

is a bounded approximate identity for

L (G).

Proposition 1.8. If f belongs to Lp (G), 1 ≤ p < ∞ then lim kfα ∗ f − f kp = 0 = lim kf ∗ fα − f kp α

Proof.

Cc (G) one can prove using Proposition 1.7 and the follows by Lusin's theorem. 

For any

general case

f

α

in

6

V. MURUGANANDAM

Let us end our discussion about the group algebra by recalling some important properties of group algebras of commutative groups.

G is commutative. A continuous group homomorT, is called a character. The set of all characters b. It can be seen that G b forms a locally comby G

We assume that phism from

G

into

of G is denoted

pact abelian group under pointwise multiplication and compact - open 1 topology and is called as dual group of G. Let ∆(L (G)) denote the set 1 of all non-zero complex homomorphisms of L (G).

Theorem 1.9. The map χ to τχ from Gb in to ∆(L1 (G)) given by Z τχ (f ) =

f (x)χ(x)dx ∀f ∈ L1 (G),

(1.4)

G

denes a homeomorphism from Gb into ∆(L1 (G)), with Gelfand topology.

Denition 1.10. For every f ∈ L1 (G) the Fourier transform of f is dened to be a function on Gb given by fˆ(γ) =

Z f (x)h x, γ i dx.

(1.5)

G

Proposition 1.11. The Fourier transform is a injective homomorb and its range is dense in C0 (G) b . phism from L1 (G) to C0 (G) Theorem 1.12 (Fourier Inversion formula). Let G be a locally compact abelian group. Then there exists a Haar measure dγ on Gb satisfying the following. b , then If f ∈ L1 (G) and fb ∈ L1 (G) Z fb(γ)γ(x) dγ

f (x) =

a. e

(1.6)

b G

Theorem 1.13. [Plancherel theorem] Let G be a locally compact abelian group. If f ∈ L1 (G) ∩ L2 (G) then kf kL2 (G) = kfˆkL2 (G) b .

The Fourier transform extends to a unitary operator from L2 (G) onto 2

b L (G). For the proofs and more details we refer to Loomis[11] and the recent book Folland [7]

7

1.3.

Representation theory.

Denition 1.14. A representation of G on a Hilbert space H is a homomorphism π from G into GL(H), the group consisting of all invertible operators in BL(H), such that the map x× → π(x)ξ

from G → H is continuous for every ξ in H.. H is called the representation space. We π is unitary if π(x) is unitary for every x

Here tation

say that the represen-

G.

in

That is,

h π(ξ)u, π(x)η i = h ξ, η i ∀x ∈ G ∀ ξ, η ∈ H. A representation

G,

(π, H)

is said to be an

irreducible representation of

G-invariant subspace. (π, H) is said to be cyclic with a cyclic vector ξ if the linear span of {π(x)ξ : x ∈ G} is a dense subspace of H. Any two (unitary) representations (π1 , H1 ) and (π2 , H2 ) are said to be (unitarily) equivalent if there exists an invertible bounded (unitary) operator T : H1 → H2 such that if

H

does not have any non-trivial closed

A representation

T ◦ π1 (g) = π2 (g) ◦ T, ∀ g ∈ G. On any Hilbert space one can have trivial representation. That is, for every

x

in

G,

dene

π(x) = IH .

But this apart, there is a built-in

representation for every group, given as follows. For any

x

in

G,

if

λ(x)

denotes the operator on

λ(x)f (y) = f (x−1 y) ∀y ∈ G then map

λ(x) is unitary as x → λ(x)f denes

and

L2 (G)

given by

∀f ∈ L2 (G),

Haar measure is left invariant. Moreover, the a unitary representation of

G

and is known as

left regular representation of G. Notation Let Gb be the set of equivalence classes of irreducible unitary e be the set of equivalence classes of unitary G and let G of G.

representation of representation

Remark 1.15. 1. If H is a Hilbert space let H denote the conjugate Hilbert space. If (π, H) is a representation dene π¯ , H by π¯ (x) = π(x). It is called conjugate representation. 2. Let (πi , Hi ), i = 1, 2 be two unitary representations of a group G. Let H = H1 ⊕ H2 be the direct sum of Hilbert spaces. If we dene π(x)(ξ1 , ξ2 ) = (π1 (x)ξ1 , π2 (x)ξ2 ), then it forms a unitary representation called the direct sum of (π1 , H1 ) and (π2 , H2 ).

8

V. MURUGANANDAM

ˆ 2 of H1 and H2 obtained by 3. Let H be the tensor product H1 ⊗H completing the algebraic tensor product H1 ⊗ H2 by dening the inner product h ξ1 ⊗ ξ2 , η1 ⊗ η2 i = h ξ1 , η1 ih ξ2 , η2 i. Recall that if T1 and T2 are bounded linear maps on H1 and H2 respectively then there exists a bounded linear map T1 ⊗ T2 such that T1 ⊗ T2 (ξ1 ⊗ ξ2 ) = T1 (x)ξ1 ⊗ T2 (x)ξ2 , and kT1 ⊗ T2 k = kT1 k kT2 k . If π1 and π2 are two unitary representations of G then π1 ⊗ π2 on ˆ 2 given by H1 ⊗H π1 ⊗ π2 (x) = π1 (x) ⊗ π2 (x) denes a unitary representation on G. It is called tensor product of π1 ˆ 2 ). and π2 and is denoted by (π1 ⊗ π2 , H1 ⊗H

Denition 1.16. Let n o ˜ . B(G) = πξ, η : ξ, η ∈ H, π ∈ G By the above remark, we observe that

B(G) is a subalgebra of Cb (G)

closed under complex conjugation having identity. This space is going to be an important object of study throughout this course.

Denition 1.17. Let A be a Banach ?- algebra. Any ? -homomorphism π of A into BL(H) for some Hilbert space H is called ? -representation of A. Remark 1.18. Let us recall that if A be a Banach ?- algebra and B is a C ? -algebra and if φ : A → B is a ? -homomorphism then kφ(x)k ≤ kxk

for every x ∈ A. (see Takesakai [17] for a proof.) In particular any ? representation of L1 (G) is norm decreasing. Any

?

representation

π

is said to be

of the subspace spanned by

[π(A)(H)]

non-degenerate

if the closure

H. Similarly, we can among the ?-representations of

is equal to

dene irreducibility and equivalence

A. A

?-representation (π, H)

of

L1 (G)

is non-degenerate if and only if

the following holds. For any bounded approximate identity L1 (G), π(fα ) → I in strong operator topology.

{fα }

in

Theorem 1.19. Suppose (π, H) is a unitary representation of G. For every f in L1 (G) if we dene π˜ (f ) on H by Z hπ ˜ (f )(ξ) η i =

f (x)h π(x)ξ, η idx, G

9

then f → π˜ (f ) denes a non-degenerate ?-representation of L1 (G). Moreover the correspondence π → π˜ is bijective from the equivalence classes of all unitary representations of G and the equivalence classes of all non-degenerate ? -representations of L1 (G). Proof.

Suppose that

π

is a unitary representation of

G.

If we dene

π ˜

by

Z hπ ˜ (f ) =

f (x)π(x)dx,

(1.7)

G that is, for every

ξ, η

in

H,

if

Z hπ ˜ (f )(ξ) η i =

f (x)h π(x)ξ, η idx, G

?-homomorphism.

then it can be easily seen to be a

{fα }

Let

ξ

in

H

be the bounded approximate identity given in (1.3). Fix

 > 0 Since x → π(x)ξ is continuous at e there x ∈ Vα0 ⇒ kπ(x)ξ − ξk < . For every α > α0 ,

Z

k˜ π (fα )ξ − ξk = fα (x)π(x)ξ dx − ξ

and

such that,

exists

Vα0

ZG ≤ |fα (x)| kπ(x)ξ − ξk dx Vα

Z |fα (x)| dx.

< . Vα

= . Hence

π ˜ (fα ) → I

in strong operator topology and so

π ˜ |L1 (G)

is non-

degenerate.

1 Conversely suppose that ρ : L (G) → BL(H) is a non-degenerate ? 1 -representation of L (G). 1 Set K = [ρ(L (G)(H)]. Then K is dense in H. Dene

π(x)(ρ(f )ξ) = ρ(δx ∗ f )(ξ). Since

δx ∗ fα ∗ f → δx ∗ f,

for all

f ∈ L1 (G),

we have

π(x)(ρ(f )ξ) = lim kρ(δx ∗ fα )kkρ(f )(ξ)k α

≤ kρ(f )(ξ)k. π(x) is bounded on K. Hence π(x) gets extended H. It is easy to see that π(xy) = π(x) ◦ π(y). Since

Therefore by 1.18, as an operator on

10

V. MURUGANANDAM

kξk = kπ(x−1 )π(x)(ξ)k ≤ kπ(x)ξk ≤ kξk, π f, g in L1 (G),

is unitary. Since for every

ρ(f ) ◦ ρ(g) = ρ(f ∗ g)(ξ) Z = f (y)ρ(δy ∗ g) G Z = f (y)π(y)(ρ(g))dy G

= π ˜ (f )(ρ(g)) we have

ρ(f ) = π ˜ (f ).



˜ corresponding to λ is given by left conRemark 1.20. For example λ, volution operators, since ˜ )(g)(x) = λ(f

Z

Z λ(y)(g)(x)f (y)dy =

G

g(y −1 x)f (y)dy = f ∗ g(x).

G

˜ ) = 0 then f = 0 This representation is faithful in the sense that if λ(f 1 in L (G). 2.

Positive definite functions

Denition 2.1. A function φ : G → C is said to be positive denite if for all c1 , c2 , ..., cn ∈ C and x1 , x2 , ..., xn ∈ G, n X

ci c¯j φ(x−1 j xi ) ≥ 0.

i,j=1

Example 2.2. If (π, H) is any unitary representation of G then for any ξ H function φ = πξ,ξ is positive denite since for any choice of xi and ci as in the denition n X

ci c¯j φ(x−1 j xi ) = h η, η i

i,j=1

where η =

P

i ci π(xi )(ξ).

Note that, taking n=1 and

c1 = 1 in Denition 2.1, we have φ(e) ≥ 0.

Proposition 2.3. Let φ be a positive denite function. Then (1) (2)

Proof.

φ(x−1 ) = φ(x) |φ(x)| ≤ φ(e), ∀ x ∈ G. (φ(x−1 i xj ))1≤i,j≤n is positive semi deφ(e) φ(x) . Since A = A? , and det(A) ≥ 0, φ(x−1 ) φ(e)

By hypothesis, the matrix

nite. Consider

 A=

the result follows.

11



Theorem 2.4. Let φ be a continuous function on G. Then the following are equivalent (1) φ is positive denite. ? (2) φ is bounded and h φ, f ∗ f i ≥ 0, ∀ f ∈ Cc (G). (3)

Proof.

h φ, µ? ∗ µ i ≥ 0, ∀ µ ∈ M (G).

If

µ

is a measure with nite support, that is if

µ =

n P

α i δx i ,

i=1

observe that

n X

?

h φ, µ ∗ µ i =

ci c¯j φ(x−1 j xi ).

i,j=1

h φ, µ? ∗ µ i ≥ 0. Let f belong to Cc (G). Then there exists {µα } of measures with nite support such that µα converges to f (y)dy in the weak?-topology.

Assume that

φ

is positive denite. Then

Therefore, (1) implies (2). Let us prove (2) implies (3).

Let us suppose that

µ

has compact

f in Cc (G) the function µ ∗ f belongs to Cc (G). {fα } is a bounded approximate identity such that each fα belongs to Cc (G), then µ ∗ fα belongs to Cc (G) and it converges to µ in the weak ?-topology. Therefore (3) is true in this case. If µ belongs to M (G), then there exists {µα } in M (G) with compact support such that {µα } converges to µ in the weak ?- topology.

support. Then for any If

Therefore (2) implies (3).



It is trivial that (3) implies (1).

Notation:

Let

functions on

G.

P (G) denote If φ belongs

the set of all continuous positive denite to

P (G)

then observe that

φ¯

belongs to

P (G).

Remark 2.5. In Example 2.2 we have seen that any matrix coecient belonging to a unitary representation is positive denite. Now we shall show that these are the only positive denite functions. In other words, we show that if φ belongs to P (G), then there exists a cyclic representation (π, H) with cyclic vector ξ such that φ(x) = πξ, ξ . 2.1.

GNS construction.

Theorem 2.6. Let φ be any continuous positive denite function on G. Then there exists a cyclic representation (π, H) with the cyclic vector ξ such that φ(x) = h πφ (x)u, u i locally almost everywhere.

12

V. MURUGANANDAM

Proof.

Dene

h ·, · iφ

L1 (G)

on

by

Z

?

¯ (y)dxdy. φ(x−1 y)g(x)f

h f, g iφ = h g ∗ f i = G

1 It is easy to see that it denes a sesquilinear form on L (G). If N = 1 {f ∈ L (G) : h f, f iφ = 0} , one can see by Cauchy Schwarz inequality 1 1 one can see that N = {f ∈ L (G) : h f, g iφ = 0∀g ∈ L (G)} . There1 fore it forms a closed subspace of L (G). Moreover, since

h x f,x g iφ = h f, g iφ we see that

(2.1)

N

is invariant under left translation. 1 Let H0 denote the quotient space L (G)/N . Complete it to get a

H. We shall dene L1 (G)/N take

Hilbert space

f˜ belongs

to

a representation

−1 f = π(x)(f˜) = xg

Then

π

π

on

H

as follows. If

˜

x−1 f

G.

extends to a unitary representation of

Let us show that (π, H) is cyclic. If {fα } is a bounded approximate 1 identity of L (G) then take a subnet if necessary to conclude that f˜α converges to a vector

ξ

weakly in

H.

Then

h f˜, ξ iφ = limh f˜, f˜α iφ = limh fα? ∗ f, φ i = α

Since

Z

Z

α

h g˜, f˜ iφ =

R R G

φ(x−1 y)g(y)dy =

G

Z

f (x)φ(x)dx.

(2.2)

G

¯ φ(x−1 y)f (x)g(y)dxdy

and

φ(y)g(xy)dy = h π(x−1 )˜ g , ξ iφ = h g˜, π(x)ξ i

G

G we see by (2.2) that

h g˜, f˜ iφ =

Z

¯ g˜, π(x)ξ i f (x)h

G

= h g˜, π(f )(ξ) iφ . Hence

[π(L1 (G))ξ]

is total in

H

(2.3)

and so the representation is cyclic.

Finally,

h ξ, f˜ i = limh f˜α f˜ i, = limh f˜α , π(f )ξ i = h ξ, π(f )(ξ) i α α R Therefore, by (2.2), f (x)φ(x)dx = h π(f )(ξ), ξ i G



Remark 2.7. Using the preceding theorem, we conclude that the vector space B(G) dened in 1.16 is in fact linear span of continuous positive denite functions.

13

3.

C?

algebras of groups

Denition 3.1. An Banach ?- algebra A is said to be C ? -algebra if the involution of A satises the additional condition kx? xk = kxk2 ,

Remark 1.

Let

X

x ∈ A.

be a compact Hausdor space. The space

continuous complex valued functions on

X

C(X)

of

is a unital Banach algebra

with the uniform norm. The map f → f¯ is an involution that makes C(X) into a C ? -algebra. Similarly if X is a locally compact noncompact

C0 (X) consists of continuous functions which ? vanish at innity forms a C -algebra without identity. ∞ ? 2. L (X, dµ) for any measure µ, is a C -algebra. 3. Let H be a Hilbert space. Then the unital Banach algebra BL(H) is ? a C algebra with the operator norm and the involution given by the ? map T → T . In general any norm closed ?-subalgebra of BL(H) is a ? C -algebra. Hausdor space then

Theorem 3.2. If k·k0 is dened on L1 (G) by kf k0 = sup {kπ(f )k : π is any non-degenerate ? -representation} (3.1) then it denes a norm on L1 (G). Moreover, the completion of L1 (G) with respect to this norm is a C ? -algebra. Proof. Clearly, kλf + gk0 ≤ |λ| kf k0 + kgk0

kf k0 = 0. Then λ(f ) = 0 where λ : L1 (G) → BL(L2 (G)) is the left regular representation. As λ is a faithful representation, we 0 have f = 0. Therefore k.k is indeed a norm.  Suppose that

Denition 3.3. The C ? -algebra obtained above is called full C ? -algebra of G and is denoted by C ? (G). The following theorem gives yet another way to realize

C ? (G).

Let

us rst recall

Theorem 3.4. A C ? -algebra has suciently many irreducible representations to separate points of A. That is, for every x ∈ A, x 6= 0 there exists an irreducible representation π of A such that π(x) 6= 0. Theorem 3.5. let A denote the C ? (G) n -algebra obtained o by completing 1 00 00 b L (G) with k · k where kf k = sup kπ(f )k : π ∈ G . Then C ? (G) and A are isometrically ? isomorphic

14

V. MURUGANANDAM

Proof.

Ψ : (L1 (G), k.k0 ) → (A, k.k00 ) . Then kΨ(f )k ≤ kf k0 . So Ψ ? ? extended into a -homomorphism from C (G) into A. We claim that Ψ ? b is injective. Suppose Ψ(f ) = 0, ∀ f ∈ C (G). Then π(f ) = 0, ∀ π ∈ G. By the preceding Gelfand-Raikov Theorem, f = 0. Therefore Ψ is ? injective. Since an injective ?-homomorphism between C -algebras is Dene

an isometry, we have

∀ f ∈ C ? (G).

kΨ(f )k = kf k, ? Therefore, Ψ(C (G)) is closed in C ? -completion of (L1 (G), k.k00 )

A.

Thus

Ψ(C ? (G)) = A.

Hence

A

is



Proposition 3.6. Suppose that the group G is abelian. Then the full ˆ C ? -algebra of G is identied with C0 (G). Proof.

If

(π, H)

is a unitary irreducible representation of

G

then by

Schur's lemma it is given by one-dimensional representation. That is, there exists a character

π

going to

χπ

identies

χπ such that π(x)ξ = χπ (x), for every x ∈ G. ˆ with the dual group given in Section 1. G

Now

Z π(f ) =

Z f (x)π(x) dx. =

G where

χˇπ = χπ (x−1 ).

f (x)χπ (x) dx. = fb(χˇπ ), G

Therefore,

kf k0 = sup kπ(f )k = sup kfb(χˇπ )k = kf k∞ , b π∈ G

since

b χˇ ∈ G.

b χ∈ G

b C ? (G) is the completion of (L1 (G), k.k0 ) and ∆(L1 (G)) = G, ? b : k.k∞ − norm}. Since we have C (G) is the completion of {fb ∈ C0 (G) b is dense in L1 (G), we have C ? (G) = C0 (G). b {f : fb ∈ Cc (G)}  Since

Remarks 3.7. 1. If ρ is any ? -representation of L1 (G), then ρ gets extension to a ? -representation of C ? (G). 2. If ρ1 and ρ2 are two non-degenerate representations of L1 (G) then they are equivalent if and only if their extensions to C ? (G) are equivalent. 3. Summarizing by we observe by Theorem 1.19 and the preceding theorem that there is a bijective correspondence between the unitary representations of G and non-degenerate ? -representations of C ? (G) such that irreducible ones go into irreducible ones. Moreover this identication respects equivalence relation among the representations.

15

From the proof of Theorem 3.2 we observe that instead of taking all unitary representations in the equation (3.1), if we take the left regular ? representation alone, we get another C -algebra.

Denition 3.8. The closure of ? -subalgebra λ(L1 (G)) in BL(L2 (G)) is a C ? -algebra and is called reduced C ? -algebra of G. We denote this C ? -algebra by Cλ? (G). We recall that if

G

is abelian then

ˆ f (γ) ≤ kλ(f )k

for every

f

in

0

ˆ So kf k = kλ(f )k for every f in L1 (G), so L1 (G) and for every γ in G. ? ? ∼ that C (G) = Cλ (G). Similarly if G is compact, by Peter-Weyl theory we observe that ˆ and so C ? (G) ∼ kπ(f )k ≤ kλ(f )k for every π in G = Cλ? (G). ? The question that for which groups, these two C -algebras are isometrically ?-isomorphic gives rise to a class of groups called amenable groups, which are to be discussed below.

Weak containment: Denition 3.9. Suppose that G is a locally compact group. Let Σ ⊆ e π ∈ G. e We say that π is said to be weakly contained in Σ if the G, corresponding ? -representation π of C ? (G) is weakly contained in corresponding Σ. We denote it by π  Σ.

3.1.

π is a representation of A and Σ is a collection of A then π is weakly contained in Σ denoted by π  Σ

We recall that if representations of

if any of the equivalent conditions in the following theorem is satised.

Theorem 3.10. (2) (3)

(4)

ker(π) ⊇

T

ker(ρ) n o e . kπ(x)k ≤ sup kρ(x)k : ρ ∈ G For every ξ ∈ Hπ , there exists a net {φα } consisting of the matrix coecients belonging the representations in Σ such that {φα } converges to πξ, ξ in the weak? topology. In fact Every positive form πξ,ξ associated to π is weak? limit of linear sum of positive linear form associated to Σ. Every state of A associated with π is a weakstar limit of states which are sums of positive forms associated with Σ. (1)

ρ∈Σ

See Theorem 3.4.4 of Diximier [4].

For more details regarding the ? weak containment among the representations of a C -algebra we refer to Section 3.4 of Diximier [4]. If we take

Σ = {λ}

then we observe by the preceding discussion ? ? and Theorem 3.5, that C (G) and Cλ (G) are isometrically isomorphic

16

V. MURUGANANDAM

if and only if every irreducible unitary representation of contained in

G

is weakly

λ.

The following theorem is useful in understanding the weak containment among the representations of

G.

Theorem 3.11. Let P1 (G) = {ψ ∈ P (G) : kψk = ψ(1) = 1} . On P1 (G) the weak? topology and the topology of uniformly convergence on compact sets are equivalent. See Theorem 13.5.2. of Diximer [4].

Theorem 3.12. Let G be a locally compact group and π belong to G˜ ˜ Then the following are equivalent: and Σ ⊆ G. (1)

π  Σ.

Every positive denite function ψ associated to π is limit of sum of positive denite functions associated to Σ with respect to the topology of uniformly convergence on compact sets. If π is further assumed to be irreducible, then the above if equivalent to the following. 3. If πξ,ξ for some ξ in H, is the limit of sum of positive denite functions associated to Σ with respect to the topology of uniformly convergence on compact sets. (2)

Proof.

The proof follows by Theorem 3.10, Theorem 3.11.

 3.2.

Fourier and Fourier Stieltjes algebra.

Theorem 3.13. Let G be a locally compact group. Let C ? (G) denote the C ? -algebra of G. The Banach space dual [C ? (G)]? of C ? (G) is given by {πξ,η : ξ, η ∈ Hπ , π is a unitary representation of G} . (3.2) We shall rst prove the following lemma.

Lemma 3.14. If φ is a positive linear form L1 (G), then φ gets extended to a positive linear form φ0 on C ? (G). The map φ → φ0 is bijective and kφk = kφ0 k.

Proof.

Let

itive form

τ : L1 (G) → C ? (G) denote on a Banach ?-algebra with

the imbedding. As every pos-

bounded approximate identity 1 is continuous (See [1, p396]). Therefore any positive form on L (G) 0 ? 0 ? gets extended to φ on C (G) such that, φ = φ ◦ τ. Let x ∈ C (G). 0 1 0 We need to see that kφk = kφ k. If f ∈ L (G) then |φ(f )| = |φ (f )| ≤

17

1/2

|φ0 (e)|1/2 kf ? ∗ f kC ? (G) ≤ kφ0 kkf k1 .

Therefore

kφk ≤ kφ0 k.

Let

x ∈ A.

Then,

|φ0 (x)| = |φ(x)| ≤

kφk1/2 (φ(x? x))1/2 .

≤ kφk1/2 kφ0 k1/2 kx? xkC ? (G) ≤ kφk1/2 kφ0 k1/2 kxk2 . kφ0 k ≤ kφk1/2 kφ0 k1/2 ⇒ kφ0 k1/2 ≤ kφk1/2 ⇒ kφ0 k ≤ kφk. kφk = kφ0 k as claimed. 

e Therefore, Hence

Let us prove the theorem.

Proof.

Let us recall Jordan Decomposition Theorem for

C ? -algebras.

(see Theeorem (3.2.5) of Pederson [13]). For each hermitean functional φ on C ? -algebra A, there exist positive elements φ+ and φ− such that

φ = φ+ −φ− and kφk = kφ+ k+kφ− k. By GNS construction theorem for C ? -algebras we see that if φ is any positive linear form, then there exists a unique (up to unitary equivalence) cyclic ?-representation (π, H, ξ) with the cyclic vector ξ, satisfying φ(x) = h π(x)(ξ), ξ i ∀ x ∈ A. Therefore, we observe that any element in the dual of a

C ? -algebra

is

a linear combination of positive linear forms. By the preceding lemma,



and Remark 3.7 the result follows.

Remark 3.15. The vector space B(G) is identied with the Banach space dual [C ? (G)]? of the C ? -algebra of G and is called Fourier-Stieltjes algebra of G. In particular, if

G

is abelian then

ˆ ? = M (G). ˆ B(G) w [C0 (G)]

In fact

the identication is given by the inverse Fourier-Stieltjes transform. That is

B(G) = {φ = µ ˆ:

for some

ˆ µ ∈ M (G)},

with

kφkB(G) = kµk ,

where the inverse Fourier-Stieltjes transform is given by

Z µ ˆ(γ) =

γ(x)dµ(x). ˆ G

But then this is precisely Fourier stieltjes algebra on

G. Refer the clas-

sical book by Rudin [15] for more details on harmonic analysis over abelian groups. The result that it forms a Banach algebra when

G

is

18

V. MURUGANANDAM

abelian was extended to all non-abelian groups by Eymard [6].

The

following theorem is due to Eymard.

Theorem 3.16. B(G) forms a Banach algebra with unity under pointwise product. Proof. We have already seen that B(G) forms an algebra with unity under pointwise product. By the preceding remark it forms a Banach space. In order to show that the norm satises Banach algebra condition we need the following fact due to Eymard. If

φ

belongs to

B(G)

then

kφk = inf{kξkkηk : φ = πξ,η }, where the inrmum is taken over all

πξ,η

(3.3)

such that

φ = πξ,η .

(In fact

the minimum is attained.) Using the above equation it is easy to show that for all

φ, ψ

in

B(G).

kφ · ψk ≤ kφk kψk , 

Denition 3.17. Let G be locally compact group. The closure of the ideal B(G)∩Cc (G) in B(G) is called the Fourier algebra and is denoted by A(G). When

G

is abelian, recall that Fourier algebra

b A(G) = {fˇ : f ∈ L1 (G)} where the

kf kL1 (G) ˆ .

fˇ denote

the inverse Fourier transform of

f

and



fˇ = A(G)

It is one of the classical results of abelian harmonic analysis

that whose proof can be found in Rudin [15]. Again, when ization of

G

A(G)

is abelian, one can arrive at the following character-

using Plancherel theorem.

 A(G) = f ∗ g˜, where

g˜(y) = g(y −1 ).

f, g ∈ L2 (G) ,

This result was extended to all locally compact

groups by Eymard.

Theorem 3.18

. Let G be a locally compact group. Then

(Eymand)

 A(G) = λf,g : f, g ∈ L2 (G) ,

where λf,g is the matrix coecient associated to the left regular representation λ of G. 1.

A(G)

A(G) = B(G) B(G). Equivalently, A(G) = B(G)

has identity if and only if

two sided ideal in

since

A(G)

is a

if and only if

G

is compact. 2.

A(G)

algebra.

is a commutative regular, semi simple Tauberian Banach

19

3. The canonical embedding of by

τx (φ) = φ(x)

G in ∆(A(G)) namely, x into τx given

is a bijective homeomorphism.

For the proofs of all these results we refer to Eymard [6]. 4.

Amenable groups

Denition 4.1. A linear map m : L∞ (G) → C is said to be a mean on L∞ (G), if m(f ) ≥ 0 for all f ≥ 0 in L∞ (G) and m(1) = 1. Moreover, a mean is said to be a left invariant mean if m(x f ) = m(f ) ∀f ∈ L∞ (G), ∀x ∈ G.

Denition 4.2. A locally compact group G is said to be L∞ (G) has a left invariant mean.

amenable

if

Amenable groups were rst introduced by John von Neumann in 1929 in his study of Banach-Tarski paradox. (See a fairly recent book by Runde[16] for a discussion on Banach-Tarski paradox.) But it was M.M. Day [2], who baptized the name. As we are going to see below, amenable groups form a vast collection of groups, which include for instance abelian groups, solvable groups and compact groups. Any compact group is amenable. In fact the normalized Haar measure is the required left invariant mean. That is, if then

m

is easily seen to be a left invariant mean.

h f, m i = 1

We shall use Markov- Kakutani xed point theorem

R G

f (x)dx

to show that

any abelian group is amenable.

∞ denotes the set of all means on L (G), then it forms a nonempty ? convex set. Moreover, it is a weak -compact set as it is a subset of the ∞ unit ball of the dual of L (G). ∞ ? ∞ ? For all x ∈ G, dene ρ(x) : L (G) → L (G) by If

K

h f, ρ(x)(F ) i = h

x−1 f, F

i ∀F ∈ L∞ (G)? .

ρ(x) is continuous, linear on L∞ (G)? and leaves K invariant. Since ρ(xy) = ρ(x) ◦ ρ(y), for all x, y in G and G is abelian, {ρ(x)}x∈G forms a commuting family. If m is a xed point, then m is left invariant. Then it is easy to see that

The following theorem is useful. We refer to Runde [16] for the proof.

Theorem 4.3. Let G be a locally compact group. Then the following are equivalent, (1) G is amenable. 1Markov-

Kakutani xed point theorem: If

a topological vector space and if in

K

is a compact convex subset of

is a commuting family of continuous linear

K into itself, then there exists a point p in K such that F. (See Dunford and Scwartz[5, Theorem V.10.6] for a proof.)

mappings which map

T (p) = p for all T

F

20

V. MURUGANANDAM

(2) (3)

Cu (G) has a left invariant mean. Cb (G) has a left invariant mean.

Corollary 4.4. Let Gd denote G with discrete topology. Then if Gd is amenable then G is amenable. Proof. Suppose Gd is amenable. Then l∞ (G) has left invariant mean, say

m.

Cb (G) ⊆ l∞ (G). Then m|Cb (G) Cb (G). Therefore G is amenable.

Observe that

mean on

is left invariant



The following properties of amenable groups are useful to generate more examples of amenable groups and nonamenable groups.

Lemma 4.5. Let H be a locally compact group, and let φ : G → H be a continuous, open homomorphism with φ(G) is dense in H. If G is amenable then H is amenable. Proof. Let m denote a left invariant mean on Cb (G). Dene a continuous homomorphism

φ? : Cb (H) → Cb (G)

by

?

φ (f )(g) = f (φ(g)), g ∈ G. If we dene

m ˜

on

Cb (H)

by

h f, m ˜ i = h φ? (f ), m i, then it is easy to see that

m ˜

is left invariant mean on

Cb (H). 

Corollary 4.6. Let G be amenable, and let N be a closed normal subgroup of G. Then G/N is amenable. Proposition 4.7. Let H be a closed subgroup of G. Then there is a Bruhat function for H. That is, there exists a function β : G → C associated to H, called Bruhat function satisfying the following: (1) β - is continuous and positive. (2) For all compact set K, support of (β|KH ) is compact. R (3) For all g ∈ G, β(gh) dh = 1. H

Theorem 4.8. Let H be a closed subgroup of G. If G is amenable, then H is also amenable. Proof.

If β denotes Cb (H) → l∞ (G) by

the Bruhat function associated to

Z T φ(g) = G

β(g −1 h)φ(h) dmH (h),

g ∈ G.

H,

dene

T :

21

Cb (H) into Cb (G) mapping 1H into 1G . Let φ belongs to Cb (G). Then we rst show that T (φ) is a continuous function on G. Fix g0 ∈ G. If V is a compact neighbourhood V of g0 , then by condition (2) of the preceding proposition, β|V H is uniformly continuous. There exists a neighbourhood W of e such that We show that

T

is a contraction from

the constant function

| β|V H (g) − β|V H (h)| < . for all

h

in gW. In particular if g ∈ g0 W, then g ∈ V H and −1  β(g h) − β(g0−1 h) < . mH (support of (β|V H ))kφk∞

| T φ(g) − T φ(g0 ) |≤ . If m is a left invariant mean on Cb (G), dene m ˜ on Cb (H) by h φ, m ˜ i = h T φ, m i. It is routine to check that m ˜ is left invariant mean.  Now it can be seen that

Theorem 4.9. Let N be a closed normal subgroup such that both N and G/N are amenable. Then G is amenable. Proof. Let mN be the left invariant mean on Cb (N ). Dene T : Cb (G) → Cb (G)

by

T φ(g) = h ( g−1 φ)|N , mN i. Then T (φ) is continuous, and kT φk∞ ≤ kφk∞ , mapping the constant function 1G into itself. Use the left invariance of mN to conclude that T φ(gh) = T φ(g), for all g in G, and h in H. fφ on G/N by Therefore T φ induces a function say T Tfφ(˜ g ) = T φ(g). Now if

m ˜

is a left invariant mean on

Cb (G/N )

then dene

m

on

Cb (G)

by

h φ, m i = h Tfφ, m ˜ i. One can verify that

m

denes a left invariant mean on

Cb (G). 

For a detailed proof of these theorems, we refer to Greenleaf [8] and Runde [16].

Remark 4.10. Any solvable group is solvable. Since any abelian group is amenable, we infer by preceding theorem, that Gd is amenable and so by Corollary 4.4, G is amenable. Now let us give some examples which are not amenable.

22

V. MURUGANANDAM

Theorem 4.11 Proof.

(Von-Neumann)

. F2 is not amenable.

Suppose on the contrary that there exists a left invariant mean

L∞ (G). Let W (x) = {w ∈ F2 : w starts with x} . Then F2 = {e} ∪ W (a) ∪ W (a−1 ) ∪ W (b) ∪ W (b−1 ) and the union is disjoint. −1 −1 If w belongs to F2 \ W (a), then a w belongs to W (a ). That is, w belongs to aW (a−1 ). Therefore, F2 = W (a) ∪ aW (a−1 ). Similarly F2 = W (b) ∪ bW (b−1 ). Now, if χE denotes the characteristic function of a set E, then m

on

1 = = ≥ = = So,

m(1) m({e}) + m(χW (a) ) + m(χW (a−1 ) ) + m(χW (b) ) + m(χW (b−1 ) ) m(χW (a) ) + m(χaW (a−1 ) ) + m(χW (b) ) + m(χbW (b−1 ) ) m(χW (a)∪aW (a−1 ) ) + m(χW (b)∪bW (b−1 ) ) m(χF2 ) + m(χF2 ) = 2.

m cannot be a left invariant mean.

That is,

F2



is not amenable.

Using Theorem 4.8 and the preceding theorem, we conclude that

Corollary 4.12. Any locally compact group that contains F2 as a closed subgroup, is not amenable. SL(2, R) is not since the the closed subgroup     amenable 1 0 1 2 is isomorphic and b = elements a = 2 1 0 1

For instance, generated by with

F2 .

Similarly we see that cause

SL(n, R) contains GL(n, R) is seen

reason

SL(n, R) is SL(2, R) as

not amenable for all

n > 2,

be-

closed subgroup. With the same

to be not amenable.

In fact, any connected

noncompact semisimple Lie group is not amenable. From the above we learn that a locally compact group containing

F2

is not amenable. It is worthwhile to know whether there exist dis-

crete groups which are non-amenable groups and do not contain

F2 . We

wish to point out that the existence of such groups remained a fundamental open problem (sometimes called von Neumann problem) until Olshanskii[12] established such groups in 1980. There exists another class of innite discrete groups (à la Gromov) with Kazdhan property

T, (much stronger than non-amenability), group.

yet not having any free sub-

23

5.

Some characterizations of Amenable groups

Theorem 5.1. Let G be a locally compact group. Then the following are equivalent: (1) (2) (3)

(4)

G is amenable.

There exists a net {gi } in S(G) such that {h ∗ gi − gi } → 0 in the weak? topology of (L∞ (G))? , ∀ h ∈ S(G). There exists a net {gi } in S(G) such that, lim kh ∗ gi − gi k1 = i

0, ∀ h ∈ S(G). For given  > 0, for every compact set K, there exists g ∈ S(G) such that kλ(x)g − gk1 < , ∀ x ∈ K.

Remark 5.2. Condition (iv) of the preceding theorem is called P1 property. More generally, Denition 5.3. We say that a locally compact group G is said to have the property Pp , 1 ≤ p < ∞ if the following holds. For given  > 0, for every compact set K, there exists g ∈ Lp (G) such that g is non-negative and kgkp = 1 satisfying kλ(x)g − gkp < , ∀ x ∈ K.

Theorem 5.4. Let G be a locally compact group. Then G has property P1 if and only if G has the property Pp , ∀ p such that 1 ≤ p < ∞. See Reiter [14] and Runde [16] for the proofs of the theorems cited above.

Theorem 5.5. G is amenable if and only if there is a net {fi } in the unit sphere of L2 (G) such that {fi ∗ fei } converges to 1 uniformly on compact subsets of G. See Pederson [13, Proposition 7.3.8]. In fact we can say something more.

Theorem 5.6. Let G be a locally compact group. Then G has property P2 if and only if there exists {fi } ⊆ Cc (G) such that kfi k2 = 1 and fi ∗ f˜i → 1 uniformly on compact subsets of G. Proof.

Let

f ∈ Cc (G)

K

G and let  > 0 be given. Choose kf k2 = 1 and |(f ∗ f˜)(x) − 1| < 2 , ∀ x ∈ K.

be a compact subset of

be such that

24

V. MURUGANANDAM

(f ∗ f˜)(e) = 1. Furthermore,

λ(x)f − f 2 = h λ(x)f − f, λ(x)f − f i 2

Then we have

  = h λ(x)f, λ(x)f i + h f, f i − 2Re h λ(x)f, f i = 2 − 2Re(f ∗ f˜(x))
= 2Re 1 − (f ∗ f˜)(x) ≤ 2 1 − (f ∗ f˜)(x) < . If we take

g = |f |

then,



λ(x)g − g 2 = λ(x)f − f 2 < . 2 2 Therefore

G

has the property

P2 .

Conversely, suppose G has the property P2 . Then for given  > 0, for 2 any compact set K, there exists f ∈ L (G) such that kλ(x)f − f k2 <  , f ≥ 0 and kf k2 = 1, ∀ x ∈ K. Since Cc (G) is dense in L2 (G), given 2  > 0 there exists g ∈ Cc (G) with kgk2 = 1 such that kg − f k < 4 . Therefore, for any x ∈ K, we have

|g ∗ g˜(x) − 1| ≤ |g ∗ g˜(x) − f ∗ f˜(x)| + |f ∗ f˜(x) − 1| = g ∗ (˜ g − f˜)(x) + (g − f ) ∗ f˜(x)| + |f ∗ f˜(x) − 1|. Therefore


|g ∗ g˜(x) − 1| ≤ kg − f k kgk + kf k + kλ(x)f − f k ≤ . We deduce that there exists

g ∗ g˜ → 1

{g}

in

Cc (G) with kgk2 = 1 such that G. Hence the theorem. 

uniformly on compact subsets of

G be a locally compact group. Then G is amenable if and only if there exists {fi } ⊆ Cc (G) such that kfi k2 = 1 and fi ∗ f˜i → 1 uniformly on compact subsets of G. Let us recall an all time important theorem Let

in the representation theory of groups, namely, Godement's theorem.

Theorem 5.7 (Godement). Let φ be any square-integrable continuous positive denite function on G. Then there exists a square-integrable e function ψ such that φ = ψ ∗ ψ. Refer Dixmier [4, Theorem 13.8.6] for a proof and more details.

Theorem 5.8 (Hulanicki). Let G be a locally compact group. Then the following are equivalent. (1) G is amenable. (2) Every irreducible unitary representation of G is weakly contained in λ. (3) The trivial representation G is weakly contained in λ.

25

Proof.

We shall rst show that (1) and (2) are equivalent. Suppose

is amenable. Then there exists

{fi }

in

Cc (G)

such that

kfi k = 1

G

and

fi ∗ f˜i → 1 in topology of uniform convergence on compact sets. That is, if φi = fi ∗ f˜i then φi → 1 in topology of uniform convergence on compact sets. Hence 1  λ, by (3) of Theorem 3.12. Conversely, suppose that if 1  λ. Then by Theorem 3.12 and by Theorem 5.5 we observe that G is amenable. Now we shall show that (2) and (3) are equivalent. Of course one has to show only that (3) implies (2). is weakly contained in

λ.

Let the trivial representation

By the above paragraph

G

is amenable.

Therefore by the preceding remark, there exists a net of positive definite functions

{ψi }

in

Cc (G),

{ψi } → 1 in topology of unib and φ = πξ,ξ . Then Let π ∈ G

such that

formly convergence on compact sets.

{φψi } → φ and φψi belongs to Cc (G). By Godement's theorem, there 2 exists gi ∈ L (G) such that φψi = gi ∗ g ˜. Thus gi ∗ g˜i → φ in topology of uniform convergence on compact sets. Hence by Theorem 3.12, again we have that

π

is weakly contained in

λ. 

The preceding theorem is due to Hulaniki [9]. Finally,

Corollary 5.9. A group G is amenable if and only if C ? (G) ∼ = Cλ? (G). Lemma 5.10. If {uα } is a bounded approximate identity in A(G). Then {uα } → 1 uniformly on compact sets. Proof. Recall that for every φ = πξ,ξ in B(G), |φ(x)| = |h π(x)ξ, η i| ≤ kξk kηk. Therefore

Let

K

be

such that

kφk∞ ≤ kφk. a compact set. Since A(G) is regular, φ|K = 1. For every x ∈ K,

there exists

φ ∈ A(G)

|uα (x) − 1| = |(uα .φ(x))(x) − φ(x)| ≤ kuα .φ − φkA(G) . Hence

{uα } → 1

uniformly on compact sets.



Theorem 5.11 (Leptin). Let G be a locally compact group. If G is amenable if and only if A(G) has a bounded approximate identity. Proof. Let G be amenable. Then there exists {fi } ⊆ Cc (G) such that

kfi k2 = 1 and fi ∗ f˜i → 1 uniformly on compact subsets of G. Denote ψi by fi ∗ f˜i . We claim that ψi is a bounded approximate identity in A(G). Since Cc (G) ∩ P (G) spans a dense subsets of A(G) and kψi kA(G) ≤

26

V. MURUGANANDAM

kfi k2 kf˜i k2 = 1, it is sucient to show that lim kψi φ−φkA(G) = 0, ∀ φ ∈ i

Cc (G) ∩ P (G). Let φ ∈ Cc (G) ∩ P (G). Take χi = ψi φ. Then χi ∈ P (G). 2 For f ∈ Cc (G), we let ρ(f ) denote the bounded operator on L (G) given by convolution

f ∈ L2 (G).

ρ(f )g = g ∗ f, By Godement's theorem,

L2 (G)

ρ(χi ), ρ(φ) are positive and there exists hi , h ∈

such that

˜ i, φ = h ∗ h ˜ χi = hi ∗ h and

ρ(χi )1/2 (f ) = f ∗ hi , ρ(φ)1/2 (f ) = f ∗ h,

∀ f ∈ Cc (G).

lim khi −hk2 = 0. Let S = supp(φ). Then supp(χi ) ⊆ i ψi → 1 uniformly on S, we have

We shall show that

S.

Since

lim kχi − φk∞ = 0.

(5.1)

i

f ∈ Cc (G), let fy (x) = f (xy −1 ), x ∈ G. Then clearly, kfy k2 = ∆(y) kf k2 . Therefore for any f ∈ Cc (G), we have



ρ(χi ) − ρ(φ) = f ∗ (χi − φ) 2

Z


χi (y) − φ(y) ∆(y)−1 fy dy = For any 1/2

2

G

!

Z

χi (y) − φ(y) ∆(y)−1/2 dy

≤

kf k2

G

!

Z

∆(y)−1/2 dy

≤



χi − φ kf k2 . ∞

S

lim ρ(χi ) − ρ(φ) = 0

Therefore, by (5.1), we have ber

c = sup kρ(χi )k

i

and the num-

is nite. By approximating the function

[0, c] we see

lim ρ(χi )1/2 − ρ(φ)1/2 = 0.

polynomials uniformly on the interval

√ t

with

that

i

f ∈ Cc (G),


lim f ∗ (hi − h) 2 = lim ρ(χi )1/2 − ρ(φ)1/2 (f ) 2 = 0.

Thus for every

i

Since

i

f ? (x) = ∆(x)−1 f (x−1 ), x ∈ G,

we have

limh hi − h, f ∗ g i = limh f ? ∗ (hi − h), g i = 0, ∀ f, g ∈ Cc (G). i

i

27

Moreover,

Z

2

hi = 2

Z

2

| hi (x)| dx = G

G

Z =

hi (x)hi (x)dx

˜ i (x−1 )dx = (hi ∗ h ˜ i )(e). hi (x)h

G Since

Cc (G) ∗ Cc (G)

is dense in

L2 (G)

and

sup khi k22 = sup χi (e) < ∞,

we have

f ∈ L2 (G).

limh hi , f i = h h, f i, i

Also

lim khi k2 = lim χi (e) = φ(e) = khk2 . i

i

Therefore,

lim khi − hk22 = 2khk22 − 2 lim Reh hi , h i = 0. i

i

Now,



ψi φ − φ = kχi − φkA(G) A(G)

˜i − h ∗ h ˜ = hi ∗ h A(G)

˜ i − h) ˜ + (hi − h) ∗ hk ˜ A(G) = hi ∗ (h ≤ khi k2 khi − hk2 + khi − hk2 khk2   = khi − hk2 khi k2 + khk2 .

Therefore, lim ψi φ − φ = 0. Similarly we can get lim φψi − A(G) i i

φ = 0. Hence ψi is a bounded approximate identity in A(G) as A(G) claimed.

{uα } is a bounded approximate identity. We ? is weak dense in B(G). Let φ ∈ B(G). Consider ? is a bounded net in B(G). Any bounded set in B(G) is weak

Conversely, suppose that claim that

{φ.uα }

A(G)

uα → 1 uniformly on compact sets. There? fore φ.uα → φ. By Theorem 3.11, φ.uα → φ in weak topology. SinceA(G) ? is dense in B(G), uα ∈ A(G). Therefore A(G) is weak dense in B(G).

compact. By Lemma 5.10,

Consider

A(G)+ = {φ ∈ A(G) : φ We show that

A(G)+

is weak

is a convex set, there exists

?

is positive denite}.

dense in ?

T ∈ C (G)

B(G)+ = P (G).

Since

A(G)+

such that

h T, u i = 0, ∀ u ∈ A(G)+ . A(G) is weak? dense in B(G), T = 0. ? Since (Cc (G) ∩ A(G))+ is dense in A(G)+ , (Cc (G) ∩ A(G))+ is weak dense in B(G)+ . As 1 ∈ B(G), there exists φi ∈ Cc (G) ∩ A(G) and

As

28

V. MURUGANANDAM

bounded such that

? in weak topology.

φi → 1

Therefore

φi → 1

in

topology of uniform convergence on compact sets. By Godement's 2 theorem, there exists {fi } in L (G) such that φi = fi ∗ f˜i . Hence G is amenable. This completes the proof of the Theorem 5.11.

 The preceding theorem is due to Leptin [10].

But the proof given

here is adopted from what is given in Pederson [13]. See also Appendix of De Canniere and Haagerup [3].

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55 (1949),

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7. G. Folland, A course in abstarct harmonic analysis, CRC Press, 1994. 8. F.P. Greenleaf, Invariant means on topological groups and their applications, Von Nostrand Mathematical Studies, no. 16, Von Nostrand, Princeton, New York, 1960. 9. A. Hulanicki, Means and Følner's conditions on locally compact groups, Studia. Math.

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13. Gert K. Pederson, C -algebras and their automorphism groups, London Math. Society Monographs, no. 14, Academic Press, London-New york, 1979. 14. H. Reiter and J. D. Stegeman, Classical harmonic analysis and locally compact groups, second ed., Oxford University Press, Oxford New York, 2000.

15. W. Rudin, Fourier analysis on groups, Interscience, New York - London, 1962. 16. V. Runde, Lectures on amenability, Lecture Notes in Mathematics, vol. 1774, Springer, Berlin, Heidelber, New York, 2002. 17. M. Takesakai, Theory of operator algebras I, Springer, New York, 1979. 18. A.

Weil,

L'intègration

dans

Gauthier-Villars, Paris, 1938.

les

groupes

topologiques

et

ses

applications,

29

Department of Mathematics, Pondicherry University, Pondicherry 605 014 E-mail address :

[email protected]