The belief that public opposition to same-sex marriage has softened in recent years will face an important test this fall, when Washington voters decide whether to throw out a new state law legalizing such unions.

National groups on each side of the debate are expected to pour big money and political muscle into what is sure to be a nasty referendum fight here — as well in Maine and Maryland, where gay marriage also is up for a vote.

On this issue, there’s no question the nation remains divided.

But gay-rights supporters are buoyed by what they believe has been a significant shift in public attitudes in the three years since Maine voters repealed that state’s same-sex marriage law.

During that time, national polls have shown support reaching 50 percent or better. In states where gay marriage has been approved, conservatives have crossed their party to embrace a traditionally liberal position.

And President Obama’s public endorsement last month of gay marriage triggered new conversations on a topic sure to figure prominently in a contentious presidential election.

Approval of gay marriage by voters in any of these states would be a huge symbolic victory for the gay-rights movement.

“This is clearly a turning point year when it comes to marriage,” said Michael Cole-Schwartz, of the Human Rights Campaign, one of the major organizations in the country actively involved in defending Washington state’s new marriage law.

“There are a slew of polls showing majority support, the president of the United States is lending his endorsement to the cause and you have these ballot battles around the country taking place in an environment we’ve not seen before,” Cole-Schwartz said.

In Maine, gay-marriage supporters are so convinced that attitudes have changed in three years that they will ask voters to reconsider their 2009 vote.

In the years since that defeat, activists have gone door-to-door to gauge sentiment on this issue, talking to people who voted for or against the measure, were on the fence or didn’t vote at all.

“We saw attitudes had changed, we saw the results of education,” said David Farmer, with the gay-marriage advocacy group Mainers United for Marriage. “People were anxious

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