1. Let H = ` = e for the standard orthonormal basis

1. Let H = `2 (N), u the shift to the right: uen = en+1 for the standard orthonormal basis
{en }n ⊂ H. The C∗ -algebra T generated by u is called the Toeplitz algebra.
Show that the algebra K(H) of compact operators on H is contained in T . Hint: consider
the operators un (1 − uu∗ )u∗ m .
2. Let X and Y be Banach spaces, Y is finite dimensional. Assume we are given a norm
on X ⊕ Y extending the norms on X and Y . Show that X ⊕ Y is a Banach space.
3. Let A be a unital algebra, A∼ it unitization. Show that
(i) A∼ ∼
= A ⊕ C as algebras, namely, the isomorphism is a + λ1 7→ (a + λ1A , λ);
(ii) SpA∼ (a) = SpA (a) ∪ {0} for any a ∈ A.
4. Let a be a normal element in a C∗ -algebra A. Show that if f ∈ C(SpA∼ (a)) and
f (0) = 0, then f (a) ∈ A. Conclude that the map f 7→ f (a) is an isometric ∗-isomorphism
of C0 (SpA∼ (a) \ {0}) onto the C∗ -subalgebra of A generated by a.