The England international completed a permanent move to the Hammers on Wednesday for a club record fee, signing a six-year deal after a successful loan spell last season which saw him bag seven goals.
Carroll, who joined the Reds for £35 million in January 2011, leaves Anfield having made just 58 appearances, scoring 11 times, but Allardyce feels that his new man is now looking to the future.
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"Andy decided to stay where he felt very comfortable. He's got more accustomed to life in London, more accustomed to the club and made some very good friends in the squad."
The 58-year-old also believes that Carroll is yet to reach his peak and is backing his new signing to improve his game at Upton Park.
"He's only 24 years old, so he's still got a long way to go to reach his best and hopefully we'’ll be the club to deliver a better player and a better Andy Carroll in the next few years," Allardyce added.
"The fans love Andy, they love to see players committed to the shirt, they want to see passionate players. It'll be great for him, for England and certainly for West Ham."