1. Let T ∈ B(H) be a positive operator. Define a function f by
(
1, t > 0
f (t) =
0, t ≤ 0.
Show that f (T ) is the projection onto Im T .
2. Recall that a partial isometry is an operator u on H such that there exists a direct sum
decomposition H = H1 ⊕ H2 such that u|H1 is isometric and u|H2 = 0. Show that u∗ u is the
projection onto H1 .
Conversely, show that if u is an operator such that u∗ u is the projection onto a subspace H1
of H, then u is a partial isometry with initial space H1 . In particular, the above definition
of a partial isometry is equivalent to that in the book.
3. Let X be a compact space, I an ideal in C(X), K = {x ∈ X | f (x) = 0 ∀f ∈ I}. Show
that C(X)/I is isomorphic to C(K).
4. Let A be a C∗ -subalgebra of B(H), T ∈ A, T = u|T | the polar decomposition of T
in B(H). Show that uf (|T |) ∈ A for any continuous function f such that f (0) = 0.
5. Assume I is an ideal in a C∗ -algebra A. Recall that the norm on A/I is defined by
kx + Ik = inf kx − yk.
y∈I
Show that for any x ∈ A the infimum is attained at some y ∈ I. Hint: consider first the case
when x ≥ 0 and A = C ∗ (x); in the general case use the previous exercise.
6. Prove the following variant of the Stone-Weierstrass theorem. Let X be a compact
space, A ⊂ C(X) a C∗ -subalgebra separating points of X (so for any s, t ∈ X, s 6= t, there
exists f ∈ A such that f (s) 6= f (t)). Then either A = C(X) or there exists t ∈ X such that
A = {f | f (t) = 0}.
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