The antagonist causes conflict for the protagonist. Since Madame Loisel is the protagonist, I would say that the necklace, itself, is the antagonist. It is at the center of the conflict that is created for the Loisels.
It is after the necklace is lost that all the trouble begins. So the necklace causes the misery.
The theme that stands out in this story for me is Madame Loisel's misconception about what makes life worthwhile. Before she went to the fancy party, wearing the dress and the necklace, she longed for this type of society. The only thing that she believes is worthwhile in life is to be able to attend fancy parties and wear party dresses. It is very superficial.
She comes to feel true misery in her life, as both she and her husband must dedicate themselves, like slaves, to earning money to pay off the debt created when they borrowed money to buy a new necklace.
Madame Loisel had a much better life before she went to the party.
I wouldn't say that the antagonist is an actual person, but rather a vice (weakness, bad habit) that a person has. The main character is Mathilde, and she is guilty of pride, of vanity, and of materialism. And, it is really those traits of hers that end up being the main force for evil in the story. It is because of her vanity and materialism that she wears the necklace, and it is pride that keeps her from telling Madame Forestier that she lost the necklace. If she would have told Madame Forestier from the beginning, then she would have known that the necklace was only worth 500 francs, and wouldn't have had to go through all of that labor and toil. However, in the end, hopefully Mathilde does learn a humbling lesson about vanity. So, the antagonist is Mathilde's own pride, even though she would try to say it was her poverty, or it was the wealth of others. She was alone left to blame for her circumstance.
There are several themes. One is that greed doesn't bring happiness. There is the theme of poverty and wealth, and of the impacts of both. I provided a link below that discusses the different themes in depth. I hope that helps!