2. SOME BASIC FACTS ABOUT Aut(X, µ) 9
can find large enough N, such that if n N, there are nth roots of 1, say
λ1,...,λm, with |λi λi| σ, and nth roots of −1, say λ1,...,λm, with
|λi λi| σ. Then if V (ei) = λiei,W (ei) = λi ei,V,W clearly work.
Addendum. Rosendal [Ro] has recently found the following simple
proof that all unitary equivalence classes in Aut(X, µ) are meager in the
weak topology.
For each infinite set I N, let A(I) = {T Aut(X, µ) : ∃i I(T
i
= 1)}.
By 2.2 and 2.3, A(I) is dense in (Aut(X, µ),w). Let V0 V1 . . . be a
basis of open nbhds of 1 and consider the set B(I, k) = {T Aut(X, µ) :
∃i I(i k & T
i
Vk}. It contains A(I \ {0,...,k}), so it is open dense.
Thus
C(I) =
k
B(I, k) = {T : ∃{in} I(T
in
1)}
is comeager and clearly invariant under unitary equivalence. If a unitary
equivalence class C is non-meager, it is contained in all C(I),I N infinite.
It follows that if T C, then T
n
1, so T = 1, a contradiction.
Notice that this proof shows that in any Polish group G in which the
sets A(I) = {g G : ∃i
I(gi
= 1)} are dense for all infinite I N, every
conjugacy class is meager. Another group that has this property is U(H)
(see the preceding remark) and this gives a simple proof that conjugacy
classes in U(H) are meager.
Denote by ERG the set of ergodic T Aut(X, µ).
Theorem 2.6 (Halmos [Ha]). ERG is dense in (Aut(X, µ),w).
Proof. Take X =
2N.
For T Aut(X, µ), define fT : X
XZ
by
fT (x)n = T
−n(x).
Then if S is the (left) shift action on
XZ,
S((xn)) = (xn−1),
clearly fT (T
n(x))
=
Sn(fT
(x)). So if νT = (fT )∗µ, νT is shift-invariant. We
will verify that T νT is continuous from (Aut(X, µ),w) to the space of
shift-invariant probability measures on
XZ,
which is a convex compact set
in the usual
weak∗-topology
of probability measures on the compact space
XZ.
Then
T ERG νT is an ergodic, shift-invariant measure.
But the ergodic shift-invariant measures are exactly the extreme points of
the convex, compact set of shift-invariant measures, so they form a set
and thus ERG is Gδ.
To check the continuity of T νT we need to verify that for some
uniformly dense D
C(XZ),
and any f D,T fdνT is continuous. By
Stone-Weierstrass we can take D to be the complex algebra generated by
the functions (xi)i∈Z χA(xn), for n Z,A X clopen. Thus it is enough
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