The invitation arrived by email, bearing the seal of the State of New Jersey and the name of its new governor, Chris Christie. It asked a select group of Muslim leaders to break the daily Ramadan fast at Mr. Christie’s home, and began with a traditional Muslim salutation.

“Assalamu Alaikum (Peace be with you),” the greeting, from summer 2010, read. “Wishing you a happy and blessed Ramadan.”

With the gathering, at an evening meal known as Iftar, Mr. Christie opened what Muslim leaders recall as a period of exceptional warmth between the state’s sizable Muslim community and a prominent Republican. The governor became a fierce defender of local Muslims, rebuking his party in forceful terms for its hostility to a proposed Islamic center in Manhattan, and denouncing what he called “the crazies” on the right for attacking a Muslim lawyer Mr. Christie had selected for a judgeship.

But as he campaigns for the Republican presidential nomination half a decade later, Mr. Christie’s ties to Muslim leaders in New Jersey have grown deeply strained. The governor has recast himself as a relentless warrior against terrorism, with little patience for what he calls “politically correct” national security policy. Among some community leaders, who saw Mr. Christie as a rare Republican who rejected alarmist, broad-brush rhetoric about Islam, a sense of betrayal has set in.