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Hamas, as an offshoot of the Muslim Brotherhood, had close ties to the Morsi government.

At the same time, the Egyptian intelligence and security services have maintained close ties with Israel. They share common concerns including the rise of jihadist groups in Sinai.

“There have been differences between Israel and Egypt over the Palestinian issue,” said Gamal Soltan, an associate professor of political science at the American University of Cairo. “Yet Israel and Egypt have quietly and professionally maintained their ties.”

He added, “Egypt has a role to play regardless of the regime type and the ideological orientation of the ruling elite in Egypt. Egypt is working for a ceasefire despite the hostility between the regime and Hamas.”

Michele Dunne, of the Carnegie Endowment for International Peace in Washington, said, “In the past the Egyptian government was seen by both the Islamists and the secular Palestinian leadership as even handed enough to mediate between the two of them as well as between them and Israel, that’s not the case any more.

“This time Hamas wants to bring in Turkey or Qatar because there’s so little trust with Egypt.”

Ms. Dunne said that Egypt will have to be involved because one of the key Hamas demands is access to the Rafah crossing point. If Hamas succeeds in achieving an outlet for Gaza’s population, it could gain strength. That, however, does not seem likely.

“Both Israel and Egypt feel that Hamas cannot stay in power,” said former Israeli ambassador to Egypt Tzvi Mazel. “Hamas wants to impose an Islamic state like the jihadi movements in Syria and Iraq and that is dangerous for both Israel and Egypt.”