
Chapter 5: Banach Algebra
... (2) From part (1) it is clear that x̂(n) = xn for all x ∈ ℓ1 . ere exists x ∈ ℓ1 such that xi ̸= xj for all pair i ̸= j. Since x̂ is continuous under Gelfand topology, it must be discrete. (3) For any non-empty subset I ⊆ Z+ , consider JI = {x ∈ ℓ1 : xn = 0 for all n ∈ I}. It is clear that JI is a cl ...
... (2) From part (1) it is clear that x̂(n) = xn for all x ∈ ℓ1 . ere exists x ∈ ℓ1 such that xi ̸= xj for all pair i ̸= j. Since x̂ is continuous under Gelfand topology, it must be discrete. (3) For any non-empty subset I ⊆ Z+ , consider JI = {x ∈ ℓ1 : xn = 0 for all n ∈ I}. It is clear that JI is a cl ...