The Banach{Tarski Paradox and Amenability Lecture 12: Compact Groups 6 September 2011 Invariant means and amenability Definition Let G be a locally compact group. An invariant mean is a linear 1 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 2 G and f 2 L1(G) Definition A locally compact group G is amenable if it admits an invariant mean. Finite groups are amenable Let G be a finite group. Then L1(G) is the space of all functions f : G ! R. 1 We construct an invariant mean m : L (G) ! R by averaging. Given f : G ! R define 1 X m(f ) := f (g) jGj g2G 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 2 G and f 2 L1(G) Theorem Let G be a finite group. Then G is amenable. Compact groups We usually just write compact group for compact topological group. Every compact topological group G is locally compact, because every point g 2 G has an open neighbourhood (all of G) which is contained in a compact set (all of G). So compact groups have a Haar measure. A group G with the discrete topology 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 circle group T, with its usual topology, is compact. n 2. The additive groups R and R are not compact. × 3. The multiplicative group R is not compact. 4. GL(n; R) is not compact. 5. The subgroup of GL(2; R) given by x y G = : x 6= 0; y 2 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) < 1 This lemma follows from: Proposition Let G be a locally compact group with Haar measure µ. If K is a compact subset of G then 0 ≤ µ(K) < 1 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. 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 L1(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 2 G and f 2 L1(G) That is, m is an invariant mean on G. Theorem Let G be a compact group. Then G is amenable. The torus is compact n The n{dimensional torus T is defined to be the direct product of iθ n copies of the circle group T = fe : θ 2 Rg 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 G2 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 orthogonal group n O(n; R) := fA 2 Mn(R) j hAx; Ayi = hx; yi for all x; y 2 R g 2. The special orthogonal group SO(n; R) := fA 2 O(n; R) j det(A) = 1g 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-discrete group). 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 2 O(n; R). Then if the n vector v 2 R is a column of A, we have 2 2 1 = hv; vi = v1 + ··· + vn Thus each entry aij of A satisfies jaij j ≤ 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 unitary group n U(n; C) := fA 2 Mn(C) j hAz; Awi = hz; wi for all z; w 2 C g 2. The special unitary group SU(n; C) := fA 2 U(n; C) j det(A) = 1g 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 2 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 symplectic group n Sp(n) := fA 2 Mn(H) j hAz; Awi = hz; wi for all z; w 2 H g The group Sp(n) is compact hence amenable..