Miura allegedly killed his wife back in 1981 while visiting Los Angeles.
Miura's wife was taken back to Japan after the shooting, but never regained consciousness. She died a year later.
Miura was arrested on a warrant six years after his wife's death.
In 1994, Miura was found guilty in Japan of his wife's murder. However, a high court in Japan reversed the conviction and acquitted Miura.
The Los Angeles District Attorney wants to try Miura for murder and conspiracy in the United States.
Mark Geragos, the defense attorney for Miura, argued that it would be double jeopardy to try Miura in this country.
During Friday's hearing, a college professor, an expert in Japanese law, was on the stand. The defense is trying to make the argument that conspiracy does not exist in Japanese law. In addition, the defense says Miura was tried for and acquitted of conspiracy, which is barred by the principles of double jeopardy.
Miura's wife was shot near the Los Angeles Civic Center. Miura is not suspected of shooting his wife himself. However, there are allegations that he gave a hand signal to someone to shoot his wife in the head.
Another hearing is yet to be scheduled.
Members of the Japanese media say this trial is, in Japan, the equivalent to the Scott Peterson case in the United States.
City News Service contributed to this report.