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The Reds right-back confirmed earlier this week he would not be signing a new contract when his current deal with the newly-crowned Premier League champions expires this summer.
Alexander-Arnold, who has been heavily linked with a move to Real Madrid, was subjected to audible jeers as he was introduced as a 67th-minute substitute during the 2-2 draw with Arsenal and that continued with most of his touches of the ball.
"I'm surprised how many.
When you're in a crowd of 60,000, there's no doubt there's a lot of unhappy people at Liverpool about the situation, and I've said that's understandable," Carragher told Sky Sports.
"But for me, I don't believe any player putting on that red shirt, on going out there to get them three points or winning titles, who goes on to win trophies should be being booed.
"I understand, there's a lot of ill-feeling, and some people outside of Liverpool won't be able to understand that.
All I can do is tell you how I feel about it," he told Sky Sports.
"First of all, I'm disappointed to lose my best mate.
It's an emotional decision.
"For Trent, it's not been an easy one.
But as I said, we can't tell people how to act.
"I can't tell you how I feel about it, I'm extremely proud of him.
He will be missed as one of my best friends in the game."
The England international, who is bringing to an end a 20-year association with the club he joined as a six-year-old, has won two domestic titles and the Champions League but that was forgotten by large sections of the crowd.
Head coach Arne Slot had said prior to the game he would not tell supporters how they should react to the player's announcement.
Alexander-Arnold had been on the bench as Slot looked to the future by starting with 21-year-old Conor Bradley on the right of his defence.
The highly-rated Northern Irishman has long been viewed as the natural successor in the position and the crowd sang his name both during the game and after he was replaced by Alexander-Arnold.
Carragher added: "Supporters of any club wouldn't want to hear this, but most players in that dressing room probably want to play for Real Madrid.
"I've been in that dressing room, with players who have left us for Real Madrid or Barcelona.
"It's sometimes really hard for supporters.
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