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Uma Thurman

Actress Uma Thurman has lost her bid to keep parts of her custody battle with ex-fiance Arpad Busson blocked from the media.

The Kill Bill star and her ex-partner have been embroiled in a legal wrangle over daughter Rosalind since 2014, when the businessman went to court to demand regular access to the youngster, who is nicknamed Luna.

While an initial deal was agreed in September, 2015, sources previously told the New York Post that "renewed tensions" between the pair had pushed the former couple to continue the fight.

Arpad had previously requested his daughter remain in New York while Uma filmed overseas last year, and on Friday (13Jan17) the custody trial begins in Manhattan Supreme Court.

Uma previously requested a press ban from the hearings, because she did not want the public to hear testimony from a court-appointed psychologist.

"I think it's a lot different now for children because 30 years ago you couldn't Google it. Now it follows the child for the rest of her life," Thurman's attorney, Eleanor Alter, said.

However, Justice Matthew Cooper disagreed, stating, "(The report) only says good things about the child."

"Let's face it, the only problem in this case is that the parents don't like each other," he continued.

Busson's attorney also argued the custody trial is important for foreign parents fighting for the right to be with children who live in America.

"I have a responsibly to make sure that my client - who is not American, who is a French citizen, who has a home in London, and a home in the Bahamas, and has an American child - is able to see his child," Peter Bronstein argued.

However, the judge also reminded Bronstein the public would be able to see information about his client's finances, because child support is also an issue in the trial.

"He is not happy about that," Bronstein said. "The question here is about his love for the child. The question is, do foreign dads have rights? If he's going to be denied his rights they're going to be denied in public."

Uma and Arpad split in 2009 but gave their romance another go and then broke up once and for all in April, 2014. She gave birth to Luna in 2012.