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Don John Prep
Don John Prep
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In my research in functional analysis, I came across this rather simple result:

For a normed algebra A over $ \mathbb{C} $ with unit, in which multiplication (right, right and left is commutative)are both continuous w.r.t the norm defined, such that A with this norm is a Banach space, there exists an equivalent norm with respect to which A is a Banach Algebra.

I was wondering, is the existence of a unit in A truly necessary, or can we guarantee the existence of an equivalent norm such that A is a Banach algebra?

In my research in functional analysis, I came across this rather simple result:

For a normed algebra A over $ \mathbb{C} $ with unit, in which multiplication (right and left is commutative) such that A with this norm is Banach space, there exists an equivalent norm with respect to which A is a Banach Algebra.

I was wondering, is the existence of a unit in A truly necessary, or can we guarantee the existence of an equivalent norm such that A is a Banach algebra?

In my research in functional analysis, I came across this rather simple result:

For a normed algebra A over $ \mathbb{C} $ with unit, in which multiplication , right and left are both continuous w.r.t the norm defined, such that A with this norm is a Banach space, there exists an equivalent norm with respect to which A is a Banach Algebra.

I was wondering, is the existence of a unit in A truly necessary, or can we guarantee the existence of an equivalent norm such that A is a Banach algebra?

Source Link
Don John Prep
Don John Prep
  • 221
  • 1
  • 10

A unital algebra with norm and continuous multiplication is a Banach algebra

In my research in functional analysis, I came across this rather simple result:

For a normed algebra A over $ \mathbb{C} $ with unit, in which multiplication (right and left is commutative) such that A with this norm is Banach space, there exists an equivalent norm with respect to which A is a Banach Algebra.

I was wondering, is the existence of a unit in A truly necessary, or can we guarantee the existence of an equivalent norm such that A is a Banach algebra?