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The Banach–Tarski Paradox and Amenability Lecture 12: Compact Groups

6 September 2011 Invariant means and amenability

Definition Let G be a locally compact . An invariant mean is a linear ∞ functional m : L (G) → R such that: 1. m(f ) ≥ 0 if f ≥ 0

2. m(χG ) = 1 3. m(g · f ) = m(f ) for all g ∈ G and f ∈ L∞(G)

Definition A locally G is amenable if it admits an invariant mean. Finite groups are amenable

Let G be a finite group. Then L∞(G) is the space of all functions f : G → R.

∞ We construct an invariant mean m : L (G) → R by averaging. Given f : G → R define 1 X m(f ) := f (g) |G| g∈G

Then it is not hard to verify that m is linear and that 1. m(f ) ≥ 0 if f ≥ 0

2. m(χG ) = 1 3. m(g · f ) = m(f ) for all g ∈ G and f ∈ L∞(G) Theorem Let G be a finite group. Then G is amenable. Compact groups

We usually just write compact group for compact . Every compact topological group G is locally compact, because every point g ∈ G has an open neighbourhood (all of G) which is contained in a compact set (all of G). So compact groups have a . A group G with the discrete is compact if and only if G is finite. That is, discrete groups are compact if and only if they are finite. An infinite group G will always be non-compact in the discrete topology, but G infinite might (or might not) be compact in some other topology. Examples and non-examples of compact groups

n Recall that a subset A ⊂ R (with its usual topology) is compact if and only if A is closed and bounded. This fact explains the following examples. Examples

1. The T, with its usual topology, is compact. n 2. The additive groups R and R are not compact. × 3. The R is not compact. 4. GL(n, R) is not compact. 5. The of GL(2, R) given by x y   G = : x 6= 0, y ∈ 0 x−1 R

is not compact. Haar measure on compact groups Let G be a compact group with left-invariant Haar measure µ. Lemma The Haar measure of the compact group G is positive and finite. That is, for G compact with Haar measure µ,

0 < µ(G) < ∞

This lemma follows from: Proposition Let G be a with Haar measure µ. If K is a compact subset of G then

0 ≤ µ(K) < ∞

Compact sets having finite measure is often included in the definition of Haar measure. It can be proved from µ being a regular Borel measure using the Riesz Representation Theorem. Amenability of compact groups

Let G be a compact group. Let µ be a left-invariant Haar measure on G. We may as well normalise so that µ(G) = 1. ∞ We construct an invariant mean m : L (G) → R by averaging. Given a measurable, essentially bounded function f : G → R define 1 Z Z m(f ) := f dµ = f dµ µ(G) G G This integral exists and is well-defined on equivalence classes in L∞(G). Compact groups are amenable

From properties of the Haar integral, it is not hard to verify that m is linear and that 1. m(f ) ≥ 0 if f ≥ 0

2. m(χG ) = 1 3. m(g · f ) = m(f ) for all g ∈ G and f ∈ L∞(G) That is, m is an invariant mean on G. Theorem Let G be a compact group. Then G is amenable. The is compact

n The n–dimensional torus T is defined to be the direct product of iθ n copies of the circle group T = {e : θ ∈ R}

n ∗ ∗ T = T × · · · × T ≤ C × · · · × C | {z } | {z } n times n times

Since T is compact, Tychonoff’s Theorem (for finite products of n compact spaces) tells us that the torus T is compact. n Hence T is amenable. Direct products and Haar measure

Let G1 and be locally compact groups. Let G = G1 × G2 with the product topology. A basis for this topology is products U1 × U2 where for i = 1, 2, the set Ui is open in Gi . Then G is a locally compact group.

Now for i = 1, 2 let µi be a left-invariant Haar measure on Gi .A left-invariant Haar measure µ on G is then given by Z µ(A) = 1 dµ1 dµ2 A where A is a Borel subset of G. Orthogonal groups and special orthogonal groups

n Let h·, ·i be the usual inner product on R . Let Mn(R) be the set of all n × n matrices with entries in R. Definitions 1. The

n O(n, R) := {A ∈ Mn(R) | hAx, Ayi = hx, yi for all x, y ∈ R }

2. The special orthogonal group

SO(n, R) := {A ∈ O(n, R) | det(A) = 1}

We will see later that with the discrete topology, O(n, R) and SO(n, R) are amenable if and only if n ≤ 2. We first consider O(n, R) and SO(n, R) as non-discrete groups. Orthogonal groups are compact

We will show that O(n, R) is compact (as a non-). It suffices to show that O(n, R) is closed and bounded as a subset of n2 R .

To see that O(n, R) is closed, note that a matrix A is in O(n, R) if and only if it satisfies the matrix equation

AAt = I

That is, the entries of A satisfy n2 polynomial equations, one for each entry of I . So the orthogonal group can be viewed as the n2 subset of R which is the intersection of the set of solutions of 2 these n polynomial equations. Hence O(n, R) is a closed subset of n2 R . Orthogonal groups are compact

To see that O(n, R) is bounded, let A ∈ O(n, R). Then if the n vector v ∈ R is a column of A, we have

2 2 1 = hv, vi = v1 + ··· + vn

Thus each entry aij of A satisfies |aij | ≤ 1. So O(n, R) forms a n2 bounded subset of R .

Therefore O(n, R) is compact.

Thus O(n, R), as a non-discrete group, is amenable. Special orthogonal groups are compact

SO(n, R) is a closed subset of the compact set O(n, R).

Hence SO(n, R) is compact.

Thus SO(n, R), as a non-discrete group, is amenable. Connected components of compact groups

A compact group has finitely many connected components, since each connected component is open.

In O(n, R), the special orthogonal group is the connected component containing the identity, and there is one other component, which consists of orthogonal matrices with determinant −1. Unitary and special unitary groups are compact n Let h·, ·i be the usual Hermitian inner product on C

hz, wi = z1w1 + z2w2 + ··· + znwn

Let Mn(C) be the set of all n × n matrices with entries in C. Definitions 1. The

n U(n, C) := {A ∈ Mn(C) | hAz, Awi = hz, wi for all z, w ∈ C }

2. The

SU(n, C) := {A ∈ U(n, C) | det(A) = 1}

A similar argument to that for orthogonal and special orthogonal groups shows that unitary and special unitary groups are compact. Hence they are amenable. Sp(n) is compact Denote by H the quaternions z = a + bi + cj + dk where a, b, c, d ∈ R and i, j, k satisfy the equations i 2 = j2 = k2 = −1 ij = k jk = i ki = j We define z = a − bi − cj − dk n Let h·, ·i be the usual Hermitian inner product on H

hz, wi = z1w1 + z2w2 + ··· + znwn

Let Mn(H) be the set of all n × n matrices with entries in H. Definition The

n Sp(n) := {A ∈ Mn(H) | hAz, Awi = hz, wi for all z, w ∈ H }

The group Sp(n) is compact hence amenable.