Obama used his speech as a chance to continue his outreach to Muslims and to signal an approach to the region based more on pragmatism than ideology.

His words introduced what many here say is a sense of America's new vision throughout the world. "A sense that, finally, an American leader understands," wrote the New York Times yesterday.

According to the Washington Post-ABC News poll released this week, most Americans think Obama's pledge to "seek a new way forward" with the Muslim world is an important goal, even as nearly half hold negative views about Islam and a sizable number say that even mainstream adherents to the religion encourage violence against non-Muslims, 

An overwhelming 81 percent of Americans in the poll called it important for Obama to try to improve US relations with Muslim nations; 46 percent say it's "very important." 

While slightly more than one in five expressed concerns that he'll "go too far" in that effort, most by far, 65 percent, expect him to handle it about right. 

In a CNN/Opinion Research Poll also released this week, 51 percent of Americans said the US should trust Muslim allies as it would other allies. Forty-eight percent said the US should trust Muslim allies less.

A large majority, 79 percent, said people in other countries will have a more positive view of the US because of Obama, while only 19 percent said it would be more negative, according to the CNN poll.

Asked whether Obama has accomplished anything on his trip, 16 percent said that he had accomplished a great deal, while 45 percent thought he had accomplished a fair amount during his trip.

"Obama seems to have made a good first impression on the world stage, and Americans appear to have noticed," said CNN Polling Director Keating Holland

But if the US has a leader who understands Islam, the poll said most Americans do not.

Americans by 48-41 percent hold an unfavorable opinion of Islam - its highest unfavorable rating in ABC/Post polls since 2001. And 29 percent expressed the belief that mainstream Islam encourages violence against non-Muslims - double what it was early in 2002. 

Unfamiliarity is a central factor in these views. Fifty-five percent of Americans concede that they lack a good basic understanding of Islam; about as many, 53 percent, don't personally know a Muslim. 

Perceptions of Islam as a peaceful faith are the highest among non-religious Americans, with about two-thirds holding that view. Among Catholics, 60 percent see mainstream Islam as a peaceful faith; it is 55 percent among all Protestants, but drops to 48 percent among white evangelical Protestants. 

There are deep divisions in perceptions of Islam between younger and older Americans as well: More than six in 10 younger than 65 said Islam is a peaceful religion, but that drops to 39 percent among seniors. 

Republicans are also more apt than others to hold negative attitudes toward Islam, with six in 10 having unfavorable views, compared with about four in 10 for Democrats and independents. Among conservative Republicans, 65 percent view Islam unfavorably; liberal Democrats, in contrast, are 60 percent positive. 

This partisan divide is also apparent on the question of whether mainstream Islam encourages hostility toward non-Muslims, with Republicans about twice as likely as Democrats to say it does. Nearly half of conservative Republicans see centrist Islam as a promoter of violence. 

The polls found that those who profess an understanding of Islam, or know a Muslim, have much more positive views of the religion. 

This may be why President Obama said in Turkey: "The Untied States has been enriched by Muslim-Americans. Many other Americans have Muslims in their family, or have lived in a Muslim-majority country. I know, because I am one of them."

The line was a bold one for Obama, who has been falsely accused of being Muslim. The claim persists on some right-right Web sites, which now may try to twist his remarks as proof.

Obama's Kenyan father and grandfather were Muslims, and he spent time as a child in Indonesia, the country with the world's largest Muslim population.

But other factors also are at play, and negative views of Islam remain the same even amongst those who say they have become more familiar with the religion.