MATH 656 - Spring 2016
Homework Assignment 2
Due: Tuesday March 22, 2016
1. Let X be a Banach space, Y ⊆ X a closed linear subspace and π : X → X/Y the
canonical quotient map. Show that π ∗ : (X/Y )∗ → X ∗ is an isometry and
π ∗ ((X/Y )∗ ) = Y ⊥ .
2. Let X, Y be Banach spaces. Show that the following statements are equivalent:
(a) T ∈ K(X, Y ).
(b) T (B) is compact for every bounded set B ⊂ X.
(c) For any bounded sequence (xn )n ⊂ X, T xn has a convergent subsequence in Y .
(d) T (b1 (X)) is totally bounded. (I.e., For all > 0, there exists x1 , . . . , xn ∈ b1 (X)
such that T (b1 (X)) ⊆ ∪i B (xi ). Here, B (xi ) is the open ball with radius centered at xi .)
3. Let Y be a finite dimensional subspace of a Banach space X. Show that there exists
a closed subspace Z ⊂ X such that Y = X ⊕ Z.
4. Let ϕ ∈ ΩA , where A is a non-unital Banach algebra. Show that if A has a contractive
(left) approximate identity (i.e., a net (eα ) ⊂ A such that eα a → a for all a ∈ A and
keα k ≤ 1), then kϕk = 1.
5. Consider the vector space C2 equipped with the norm k(λ1 , λ2 )k∞ = maxi |λi |, and the
product (λ1 , λ2 ) · (µ1 , µ2 ) = (λ1 µ1 , 0). Prove that C2 is a non-unital abelian Banach
algebra whose character space is a singleton, hence compact.
6. Let A = `2 (N)⊕C with k(an )+αk = k(an )k2 +|α|. Define (an )+α (bn )+β = (an bn ).
(a) Find the Jacobson radical rad(A).
(b) Show that A2 = `1 (N).
(c) Pick any non-zero (necessarily discontinuous) linear functional ψ on `2 (N) which
is zero on `1 (N). Define a new norm |||(an ) + α||| = k(an )k2 + |ψ((an )) − α|. Show
that (A, ||| · |||) is complete, making ||| · ||| a Banach algebra norm on A which is not
equivalent to the original.
7. Let A be a unital abelian Banach algebra.
(a) Show that σ(a + b) ⊂ σ(a) + σ(b) and σ(ab) ⊆ σ(a)σ(b) for all a, b ∈ A.
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(b) Show that if A contains a nontrivial idempotent (e = e2 ∈ A\{0, 1}), then ΩA is
disconnected.
(c) Let a1 , . . . , an generate A as a Banach algebra, and put
σ(a1 , . . . , an ) = {(ϕ(a1 ), . . . , ϕ(an )) : ϕ ∈ ΩA } ⊂ Cn .
Show that the natural map ΩA → σ(a1 , . . . , an ) is a homeomorphism.
8. Recall that for a LCA group G, the topology of compact convergence on G given by
the inclusion Ĝ ⊂ Cb (G) coincides with the topology of relative weak ∗-convergence
in L∞ (G). Show (by means of an example) why these two topologies do not always
coincide on b1 (Cb (G)).
9. Let A be a Banach algebra, let (eα )α ⊂ A (respectively (fβ )β ⊂ A) be a left (respectively right) bounded approximate identity for A. Show that A admits a two-sided
bounded approximate identity. (Hint: Consider elements of the form −fβ eα + eα + fβ ).
A remark on continuity of left translations on Cc (G):
Let G be a locally compact group and f ∈ Cc (G). In class, we claimed that the map
G → Cc (G);
y 7→ Ly f,
where (Ly f )(x) = f (y −1 x), is continuous for the k · k∞ -norm. Here we give a proof of this
result.
Proposition 0.1. Let f ∈ Cc (G). Then limy→e kLy f − f k∞ = 0.
Proof. Fix f ∈ Cc (G) and > 0 and let K = supp(f ). For each x ∈ K, there is a symmetric
neighborhood Ux of e such that
|f (y −1 x) − f (x)| < /2
(y ∈ Ux ).
Take a symmetric neighborhood Vx of e such that Vx2 ⊂ Ux . By compactness, there are
x1 , . . . , xn ∈ K such that
K ⊆ ∪i Vxi xi .
Put V = ∩i Vxi , and we claim
kLy f − f k∞ < (y ∈ V ).
−1
−1
−1
Indeed, if x ∈ K, then there is a j such that xx−1
j ∈ Vxj , so that y x = y (xxj )xj ∈ Uxj xj .
Thus
|f (y −1 x) − f (x)| ≤ |f (y −1 x) − f (xj )| + |f (xj ) − f (x)| < /2 + /2.
Similarly, if y −1 x ∈ K, then by symmetry (replacing y with y −1 and x with y −1 x in the
above argument)
|f (y −1 x) − f (x)| < .
Finally, if x, y −1 x ∈
/ K, then |f (y −1 x) − f (x)| = 0.
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