2.1 Manifold Structure of P(H)
23
coincides with the w
∗ -topology from L(H )
∗ , which is ‘weaker’ than the normtopology of L(H )
∗ . The last mentioned topology coincides on P(H) with the
trace-norm topology given by the metric d 2 (x, y) ∝ T r|P x − P y |, what finishes the
proof.
2.1.5 We shall introduce now a manifold structure on P(H) consistent with the
topology of P(H). Let for 0 = x ∈ H
N x := {y ∈ P(H) : T r(P x P y ) = 0}
(2.1.7)
be an open neighbourhood of x ∈ P(H), and let [x]
⊥ be the complex orthogonal
complement of x in H. We shall define the mapping x : N x → [x]
⊥ by the formula
x (y) :=
x
2
(x, y)
(I − P x )y,
(2.1.8)
where y ∈ y.
2.1.6 Proposition. The mapping x is a homeomorphism of N x onto [x]
⊥ (with the
norm-topology of H). The set
{(N x ; x ; [x]
⊥
) : 0 = x ∈ H}
(2.1.9)
is an atlas on P(H) defining a complex-analytic manifold structure consistent with
the topology of P(H) (defined in 2.1.1).
Proof. Let 0 = x ∈ H. For any y j ∈ N x and any y j ∈ y j ( j = 1, 2) it is y 1 = y 2 iff
(x, y 2 )y 1 = (x, y 1 )y 2 , hence x is injective. For any z ∈ [x]
⊥ and y := z + x we
have y ∈ N x (since x = 0) and x (y) = z, hence x is bijective. For x = 1 and
z j ∈ [x]
⊥
, y j := z j + x ( j = 1, 2, ) the identity
1 − T r(P y 1 P y 2 ) =
1
(z 1 2 + 1)(z 2 2 + 1)
z 1 − z 2 2 + +z 2 2 (1 − P z 2 )(z 1 − z 2 ) 2
(2.1.10)
implies the bicontinuity of x . For z ∈ x 1 (N x 1 ∩ N x 2 ) it is
x 2 ◦
−1
x 1
(z) = =x 2
2
x 1 + z
(x 2 , x 1 + z)
− x 2
and we see that the mapping
x 2 ◦
−1
x 1
: x 1 (N x 1 ∩ N x 2 ) → x 2 (N x 1 ∩ N x 2 )
(2.1.11)
is a complex analytic function, compare e.g. [51, 71].