What does it mean to be strictly-strictly continuous?

In summary, strictly-strictly continuous refers to a function that is both strictly continuous and strictly increasing or decreasing. It differs from strict continuity in that it does not allow for any constant regions. A function cannot be strictly-strictly continuous at a single point, and this concept is important in various areas of mathematics. To determine if a function is strictly-strictly continuous, one must check for both strict continuity and strict monotonicity.
  • #1
CornMuffin
55
5
What does it mean to be "strictly-strictly" continuous?

I am unsure what it means to be "strictly-strictly" continuous. Is that the same thing as saying just "strictly" continuous?


Here is the context:
[itex]\alpha [/itex] is a unital [itex]*[/itex]-homomorphism from [itex]M(A)[/itex] to [itex]\mathcal{L}(A)[/itex] such that [itex]\alpha [/itex] is strictly-strictly continuous on the unit ball of [itex]M(A)[/itex]

([itex]M(A)[/itex] is the multiplier algebra of [itex]A[/itex], and [itex]\mathcal{L}(A)[/itex] is the set of adjointable operators on [itex]A[/itex])
 
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  • #2
In this context, it likely means that \alpha is continuous when taking the norm on the unit ball of M(A).
 

Related to What does it mean to be strictly-strictly continuous?

1. What is the definition of being strictly-strictly continuous?

Strictly-strictly continuous refers to a function that is both strictly continuous and strictly increasing or decreasing. This means that the function has no sudden jumps or breaks and is continuously increasing or decreasing without any plateaus or constant regions.

2. How is strictly-strictly continuity different from strict continuity?

Strict continuity only requires the function to have no sudden jumps or breaks, while strictly-strictly continuity also requires the function to be strictly increasing or decreasing. In other words, strictly-strictly continuous functions cannot have any constant regions.

3. Can a function be strictly-strictly continuous at a single point?

No, a function cannot be strictly-strictly continuous at a single point. This is because strictly-strictly continuity requires the function to be strictly increasing or decreasing on an interval, not just at a single point.

4. What is the importance of strictly-strictly continuity in mathematics?

Strictly-strictly continuity is important in mathematics because it allows us to define and study functions that are continuously increasing or decreasing without any constant regions. This concept is useful in many areas of mathematics, including calculus and analysis.

5. How can we determine if a function is strictly-strictly continuous?

To determine if a function is strictly-strictly continuous, we need to check if it is both strictly continuous and strictly increasing or decreasing. This can be done by analyzing the function's graph or by using mathematical definitions and proofs.

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