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Concerning these three sentences.

A: We finally realized the importance of hard training.

I think the phrase, "the importance of hard training" has inside it the structure that "Hard training is important.
Am I right?

B: In the era of declining forests, an importance of green properties such as biodegradability and favorable CO2 balance grows.

I think the phrase, "an importance of green properties" doesn't have a structure that
"green properties are important".
I think "an importance of green properties" can be paraphrased to "an importance that green properties has".
Am I right?

C: 2005 witnessed the rise in importance of other European markets like those of the United Kingdom, Portugal, Italy and even that of France.

I think "importance of other European markets" can be paraphrased to "an importance that other European markets have".
Am I right?
Comments  
It's hard to understand your question because the meanings of be important and have importance are the same. The structure be important is more common.

Training is important. = Training has importance.
Green properties are important. = Green properties have importance.
European markets are important. = European markets have importance.

Maybe you are focusing on the difference between importance as need (first sentence) and importance as significance (the other two)? Looking at it that way, the first sentence is a little different from the other two, and the paraphrase with have importance is a little less elegant, that's true.

CJ
--What I mean is ------ I want to know if the sentence A and sentence B (I found after searching Google) are really correct in grammatical terms. When I ask native speakers if these two are correct, they ignore the sentences I quoted, and say "the" is definitely necessary before "importance.

So I want to ask again if these two are correct or wrong.

And I want to know the structural difference among these 3 sentences (A, B, and C)

I think in A, "the importance of hard training" is transformed by the structure that "Hard training is important" to make it more concise, while the other 2 sentences don't have such structures.

So another thing I want to know is if "an importance of green properties" in B means "an important thing that green properties has" and if "importance of other European markets in C" means "an important thing that other European markets have".
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A - yes, your interpretation is correct.

B is not grammatically correct, due to the use of 'an'. It should read - In the era of declining forests, the importance of green properties such as biodegradability and favorable CO2 balance grows.

This means that green properties are growing more important, now that we are in the era of declining forests.

I think the phrase, "an importance of green properties" doesn't have a structure that
"green properties are important". - this is the meaning.
I think "an importance of green properties" can be paraphrased to "an importance that green properties has". no.

C: 2005 witnessed the rise in importance of other European markets like those of the United Kingdom, Portugal, Italy and even that of France.

I think "importance of other European markets" can be paraphrased to "an importance that other European markets have".
Am I right?
No. European markets are becoming more important.
Hi Anon

First let me say that I agree with the other native speakers you've apparently asked, and I'd reword sentence B this way:
In an era of declining forests, the importance of green properties such as biodegradability and favorable CO2 balance grows.

As to your actual question, I'd say both B and C indicate that something became "more important". You can paraphrase B this way:

Green properties such as biodegradability and favorable CO2 balance are/grow more important (in an era of declining forests).

C: 2005 witnessed the rise in importance of other European markets like those of the United Kingdom, Portugal, Italy and even that of France.

Reworded:
European markets such as those of the United Kingdom, Portugal, Italy and even that of France became more important in 2005.

Edit:
Oops! I didn't realize that Nona had already answered.
YankeeHi Anon

First let me say that I agree with the other native speakers you've apparently asked, and I'd reword sentence B this way:
In an era of declining forests, the importance of green properties such as biodegradability and favorable CO2 balance grows.

As to your actual question, I'd say both B and C indicate that something became "more important". You can paraphrase B this way:

Green properties such as biodegradability and favorable CO2 balance are/grow more important (in an era of declining forests).

--Thank you vaery much.

C: 2005 witnessed the rise in importance of other European markets like those of the United Kingdom, Portugal, Italy and even that of France.

Reworded:
European markets such as those of the United Kingdom, Portugal, Italy and even that of France became more important in 2005.

Edit:
Oops! I didn't realize that Nona had already answered.

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Nona The BritA - yes, your interpretation is correct.

B is not grammatically correct, due to the use of 'an'. It should read - In the era of declining forests, the importance of green properties such as biodegradability and favorable CO2 balance grows.

This means that green properties are growing more important, now that we are in the era of declining forests.

I think the phrase, "an importance of green properties" doesn't have a structure that
"green properties are important". - this is the meaning.
I think "an importance of green properties" can be paraphrased to "an importance that green properties has". no.

C: 2005 witnessed the rise in importance of other European markets like those of the United Kingdom, Portugal, Italy and even that of France.

I think "importance of other European markets" can be paraphrased to "an importance that other European markets have".
Am I right?
No. European markets are becoming more important.

---Thank you very much.