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 A∼ ∼
= A ⊕ C as algebras,
namely, the isomorphism is a + λ1 7→ (a + λ1A , λ).