In recent times, expectations for lead female characters in British TV thrillers have been ratcheted up to dauntingly high levels. First there was the icily cool performance of Gillian Anderson, the sexually liberated police detective in The Fall. Then earlier this year in The Line of Duty, Keeley Hawes was dizzyingly good as a suspected corrupt police officer. Now it's the turn of the American actress Maggie Gyllenhaal to see if she can grip our attention with something equally complex and intriguing.  Gyllenhaal is the star of The Honourable Woman, Hugo Blick's eight-part conspiracy thriller centred on the historical triangle of Britain, Israel and Palestine. She plays the philanthropic Baroness Nessa Stein, the heir of a murdered Zionist arms dealer, who is attempting to use her family wealth to improve Israeli-Palestinian relations.  In the opening episode Gyllenhaal showed that she can handle an English accent without looking as if she's straining to remember vowel sounds. So effortless was her Kensington enunciation that it enabled the viewer to focus on a plot that required an awful lot of concentration.  Blick, who made 2011's bewildering The Shadow Line, likes to offer his actors plenty of opportunity for stagey silences and impassive expressions, but initially at least, Gyllenhaal plays Nessa as a morally earnest idealist, a carefully composed blue stocking in ermine.  The accuracy of the accent should come as no surprise, as Gyllenhaal has twice before played an Englishwoman – in Nanny McPhee Returns, under the tutelage of Emma Thompson, which is a bit like being schooled by the Queen, and in Hysteria, which did the admirable job of dramatising the invention of the vibrator.  Both those roles came in feature films and only a few years ago the presence of a Hollywood actress in a British TV series would have denoted a sharp downturn in her career opportunities. Nowadays, cinema is increasingly the preserve of teen comedies and the most ambitious drama is often be found on television.  Gyllenhaal herself admits to falling prey to the old system of ranking. "I think in my own mind there still is a hierarchy," she recently said. "But then I think, this is better work than I've done in my life. I feel more proud of this than anything."  That's no small self-praise, coming from a woman who was nominated for an Oscar for her part in Crazy Heart and who drew widespread critical plaudits with 2002's Secretary, playing a former psychiatric patient with a penchant for being spanked by her boss. When she walked on screen wearing a tight skirt, high heels and across her shoulders a rod to which her hands were manacled, it was as clear as a buttock welt that here was a genuinely idiosyncratic talent.  Not because the scene was sexually charged, but because the then 23-year-old Gyllenhaal brought to it a captivating quality somewhere between sweet innocence and crazed conviction.  For a brief moment, it was she, more than her younger brother Jake, who was the toast of the town. She has said that during the period after Secretary's acclamation, she was invited to a lot of parties, which she enjoyed until she realised that it was not her personality but her earning potential that people were interested in. However, at one dinner party she met the actor Peter Sarsgaard, best known here for his role as a creepy groomer in An Education. Sarsgaard and Gyllenhaal may sound like a Swedish law firm, but that didn't stop them forming a partnership. They got married in 2009 and have two daughters.  With her Betty Boop face and bookish intelligence – she studied literature at Columbia University – Gyllenhaal was never cut out to be paparazzi-fodder. Like Shirley MacLaine, she is a vivaciously attractive woman who happens to have a prettier brother. But as with MacLaine (sister of Warren Beatty), she has a strong claim to be the more gifted sibling, acting-wise.  Understandably, Gyllenhaal discourages speculation about sibling rivalry. "He's one of my best friends and I really adore him," is one of her stock replies to the repeated question about Jake. They are the children of the director-turned-academic Stephen Gyllenhaal and screenwriter Naomi Foner. They grew up in New York in a bohemian environment that featured a cast of famous actors – Paul Newman was Jake's godfather.  By Gyllenhaal's own account, her parents, who split up five years ago after 31 years of marriage, are "left of Trotsky". Her "very operatic and baroque" father was an early critic of Obama for what he perceived as a lack of radicalism. Gyllenhaal, having been a vocal supporter of the president, left her disillusionment to later. As she put it: "He's broken my heart in a lot of ways. I'm not clear what Obama believes in – and I wish I did."  She's one of those actors who thinks it's important to use whatever fame she's granted to ventilate her political concerns. "If I care about the state of the world then I should say pay attention to who the president is and what his policies are, pay attention to Iran and what's happening in Gaza," she says.  As a result, she has spoken out in support of Chelsea (formerly Bradley) Manning and, less controversially, Pussy Riot. In other words, Gyllenhaal is the kind of Hollywood liberal who can place Fox News presenters under threat of myocardial infarction. But what she's not prepared to express an opinion on, it seems, is the issue of Israel and Palestine, which is at the heart of The Honourable Woman.  When asked during promotion of the show what she thought of the situation vis-a-vis Israel, Gyllenhaal, whose mother is Jewish, declined. Instead, she said that the series "articulated so beautifully how incredibly complex it is". You can say that again. In later episodes, The Honourable Woman apparently takes a more intimate dramatic turn as Nessa, according to Gyllenhaal, "starts sleeping with strangers in stairwells" and "cracks up and becomes more human". She said in an interview that she "related" to that experience, meaning, we can safely assume, in the sense of becoming more human rather than sleeping with strangers in stairwells.  Yet although she speaks of the role as a mixture of career high and key stage in her personal development, she very nearly turned it down. She was concerned about uprooting her children, but more than that she took an instant dislike to Blick, the writer, producer, director.  "I was really put off by him," she said, "and I thought, 'What a drag, because I really like this script and I can't seem to communicate with this guy.'"  She says Blick invited himself to a disastrous dinner party at her house in Park Slope, Brooklyn, in which she made stinging nettle pasta and he made barbed jokes about going into anaphylactic shock. The hostess was not impressed. Relations obviously improved because she now says that she's never had "a more loving, inspiring collaboration", which is probably actor-speak for still being on speaking terms.  The other relationship that blossomed during filming was Gyllenhaal's with London. After graduating from Columbia, she had come to do a summer term at Rada and enjoyed herself. But having completed The Honourable Woman, she says that if she could persuade Sarsgaard she would like to move to London.  Her one doubt about the UK is the way she says that Jewishness is treated here. "It was a culture shock for me. In America, we don't expect there to be any social difference [between Jews and non-Jews]. I don't seem Jewish, I don't have a Jewish name. No one would ever know. But when I came here people started talking in a different way about what it meant to be Jewish. People would talk about specific areas of London being Jewish, about Jewish ways of behaving."  Nessa does have a Jewish name, but otherwise no one would ever know. Who she really is and what she really wants remain for the time being a mystery. Over the rest of the summer, all eyes will be on the enigmatic Gyllenhaal as the truth is slowly and no doubt perplexingly revealed.
Profile: One of Hollywood's most idiosyncratic stars is set to dominate our TVs this summer in a drama based on the Israel-Palestine conflict. But she's never been shy of embracing complexity