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)| )},
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