2 I. MEASURE PRESERVING AUTOMORPHISMS

equipped with the pointwise convergence topology) consisting of all T ∈

Iso(MALGµ,dµ) with T(∅) = ∅. (We only need to verify that an isometry

T with T(∅) = ∅ is an automorphism of the Boolean algebra MALGµ. First

notice that µ(A) = dµ(∅,A) = dµ(∅,T(A)) = µ(T(A)). Next we see that if

A, B ∈ MALGµ and A ∩ B = ∅, then T(A) ∩ T(B) = ∅. Indeed dµ(A, B) =

µ(A) + µ(B), thus dµ(T(A),T(B)) = dµ(A, B) = µ(A) + µ(B) = µ(T(A)) +

µ(T(B)), so T(A) ∩ T(B) = ∅. From this we obtain that T(∼ A) = ∼ T(A)

(where ∼ A = X \ A), since dµ(A, ∼ A) = 1, so dµ(T(A),T(∼ A)) = 1 and

T(A) ∩ T(∼ A)) = ∅, so T(∼ A) =∼ T(A). Finally, A ⊆ B ⇔ A ∩ (∼ B) =

∅ ⇔ T(A) ∩ (∼ T(B)) = ∅ ⇔ T(A) ⊆ T(B), so T is an automorphism of the

Boolean algebra.)

Identifying T ∈ Aut(X, µ) with the unitary operator on

L2(X,

µ) defined

by

UT (f) = f ◦ T

−1,

we can identify Aut(X, µ) (as a topological group) with a closed subgroup

of

U(L2(X,

µ)), the unitary group of the Hilbert space

L2(X,

µ), equipped

with the weak topology, which coincides on the unitary group with the

strong topology. With this identification, Aut(X, µ) becomes the group of

U ∈

U(L2(X,

µ)) that satisfy:

U(fg) = U(f)U(g) (pointwise multiplication)

whenever f, g, fg ∈

L2(X,

µ), i.e., the so-called multiplicative operators. It

also coincides with the group of U ∈

U(L2(X,

µ)) that satisfy

f ≥ 0 ⇒ U(f) ≥ 0,

U(1) = 1,

i.e., the so-called positivity preserving operators fixing 1. Note also that any

such U preserves real functions.

Let L0(X,

2

µ) =

C⊥

= {f ∈

L2(X,

µ) : fdµ = 0} be the orthogonal

of the space C = C1 of the constant functions. Clearly UT is the identity

on C and L0(X,

2

µ) is invariant under UT . So we can also identify T with

UT

0

= UT |L0(X,

2

µ) and view Aut(X, µ) as a closed subgroup of U(L0(X,

2

µ)).

The map T → UT of Aut(X, µ) into

U(L2(X,

µ)) is called the Koopman

representation of Aut(X, µ). It has no non-trivial closed invariant subspaces

in L0(X,

2

µ), i.e., the Koopman representation on L0(X,

2

µ) is irreducible (see

Glasner, [Gl2], 5.14).

Finally, for further reference, let us describe explicitly some open bases

for (Aut(X, µ),w). The sets of the form

VS,A1...An, = {T : ∀i ≤ n(µ(T(Ai)∆S(Ai)) )},

where A1,...,An ∈ MALGµ, 0,S ∈ Aut(X, µ), form an open basis for

w. This is immediate from the definition of this topology.

Next we claim that the sets of the form

WS,A1...An, = {T : ∀i, j ≤ n(|µ(S(Ai) ∩ Aj) − µ(T(Ai) ∩ Aj)| )},