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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). 1