By A. C. M. van Rooij

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**Example text**

Show that σ (ab) and σ (ba) differ by at most the point 0. ) Calculate σ (L), σ (R), σ (L R), and σ (R L) when R and L are, respectively, the right and left shifts on the usual Banach space 1 (N). 6. Let M be the operator (αn ) → (αn /(n + 1)) on 2 , and let R be the right shift operator. Set T = M R. Prove that T is compact and that σ (T ) = {0}. Does T have any eigenvalues? Calculate T n for n ∈ N, and check that limn→∞ T n 1/n = 0. ) 2 7. Take A = 1 (ω) for the weight ω = (ωn ), where ωn = e−n (n ∈ Z+ ).

Let A be a unital C ∗ algebra such that A has no proper closed ideals of ﬁnite codimension. Then every homomorphism from A into a Banach algebra is continuous. 4. The above result does not cover the commutative C ∗ -algebras of the form C( ) for a compact, inﬁnite space . It was a question of Kaplansky (1949) whether or not every homomorphism from C( ) is automatically continuous; this is equivalent to the question whether every algebra norm · on C( ) is equivalent to the uniform norm | · | . ) A major advance was due to Bade and Curtis (1960).

3. 4. 5. the kernel of F. Show that I ⊂ k(h(I )). Prove that the map F → k(h(F)) is a closure operation on the family of Gelfand-closed subsets of A . The topology it deﬁnes is the hull-kernel topology. By considering the disc algebra, show that this topology need not be Hausdorff. Prove that the hull-kernel topology coincides with the Gelfand topology if and only if A is regular, in the sense that, for each closed subset F of A and each ϕ ∈ A \ F, there exists f ∈ A with f (F) = {0} and f (ϕ) = 1.