Lawyers for Dundalk woman Lisa Smith have begun their defence before the Special Criminal Court.
The 40-year-old former member of the Defence Forces has pleaded not guilty to financing terrorism and membership of the terrorist organisation Isis.
This morning, defence counsel for Ms Smith called its first witness in the case.
Professor Hugh Kennedy is a professor of Arabic in the School of Oriental and African Studies in London and has written a book on the history of caliphates.
He outlined how a Caliph was selected in historical times, but said that by the end of the 20th century, the idea of choosing a new Caliph had been abandoned.
When asked if Isis was a caliphate, Prof Kennedy told Michael O'Higgins SC for Lisa Smith that there are different views about this.
He said it was important to understand that Muslims can believe very different things and that the criteria for appointing a caliphate are by no means clear and are not unanimously held.
Questioned by defence counsel, Prof Kennedy agreed that within a minority view, there were "respectable voices" among Islamic scholars arguing that this caliphate was legitimate.
He agreed that there might have been debates on both sides.
Prof Kennedy said that for some Muslims, this seemed to offer the opportunity of a revival of the power and prestige of the Muslim community in the world.
He said that people's attitude to the caliphate changed according to the growth and collapse of its political power.
He added that lots of people all over the world thought this could be a new beginning.
Under cross-examination by the prosecution, Prof Kennedy agreed that there is evidence that the history of the caliphate had been manipulated and distorted by Isis.
He said that Isis used history and its perceived command of history to present themselves as having a higher degree of learning and, therefore, others could follow.
He told the court that Isis and the caliphate are not necessarily the same thing and that you can believe in a caliphate without believing in the ideology of the Islamic State.
Prof Kennedy said that the Isis publication Dabiq made for uncomfortable reading and that it was a valuable tool for Isis to get is message out.
Asked by Seán Gillane, SC for the prosecution, whether the Islamic State representation of caliphate history was narrow and sectarian, Prof Kennedy agreed this was "probably all true" and that that is the way people always use history, finding what they want in it.
In a re-examination by the defence, Prof Kennedy said that the Isis publication Dabiq was a very consistent series of messages that was done like a "popular newspaper".
He said it told you "what you ought to do" and "how the world should be arranged".
He said that for "lots of people, it was very persuasive".
The trial is continuing before Mr Justice Tony Hunt, presiding, with Judge Gerard Griffin and Judge Cormac Dunne at the non-jury court.
It is expected that closing arguments in the trial will begin tomorrow.