Show that L^1(R) a Banach algebra Commutative

Marla Payton

Marla Payton

Answered question

2022-01-07

Show that L1(R) a Banach algebra Commutative

Answer & Explanation

Juan Spiller

Juan Spiller

Beginner2022-01-08Added 38 answers

We will define the multiplicative operation of Banach algebra in case of L1(R) and prove its commutativity.
Let: AL1(R) be a Banch algebra
Let's define the multiplication: A×AA as:
f,gA:f×g=f(τ)g(tτ)dτ
{This is also called the convolution operation}
Noe observe that:
g×f=g(τ)f(tτ)dτ
Now substituting: tτx we get:
g×f=t+tg(tx)f(x)d(tx)
g×f=g(tx)f(x)(dx)
g×f=f(x)g(tx)dx
g×f=f(x)g(tx)dx
Again substituting: xτ we obtain:
g×f=f(τ)g(tτ)dτ
g×f=f×g
Multiplication in A is Commutative
L1(R) is Commutative
censoratojk

censoratojk

Beginner2022-01-09Added 46 answers

Good explanation, thanks a lot!

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