And while there is now no Democrat going up against Ms. Murkowski in the race, it is not clear whether she can attract enough support from liberal voters to offset the conservatives who have been alienated by her stance against Mr. Trump. Many liberals have been angered by Ms. Murkowski’s opposition to sweeping climate change policies, as well as her support in 2017 for the $1.5 trillion Republican tax law that also allowed drilling in the Arctic National Wildlife Refuge.

So Ms. Murkowski has been reminding voters of her flair for pursuing bipartisan initiatives, such as the $1 trillion infrastructure law that is expected to send more than $1 billion to her state, and promoting her strong relationships with Democrats. At an Arctic policy event in the Dena’ina Civic and Convention Center, she appeared with Senator Joe Manchin III, the centrist West Virginia Democrat, who was wearing an “I’m on Team Lisa” button and proclaimed, “I’m endorsing her 1,000 percent.”…

“On one hand, had she chosen not to run, I would have been completely supportive because it’s just been like, ‘Damn girl, this has been a long haul,’” said Anne Gore, Ms. Murkowski’s cousin. “But on the other hand, you’re like, ‘Oh, sweet mother of Jesus, God on a bicycle — thank God you’re running’ because, you know, we can’t lose any more moderates.”

While Ms. Murkowski has never secured more than 50 percent of the vote in a general election, this year she could stand to benefit from the new election rules, which advantage candidates with the broadest appeal in a state where most voters are unaffiliated.