Is it permissible to admire a girl of complete faith and beauty without thinking of sex and carnal desires?

That love is not sinful if it is free from forbidden elements. If marriage to her is possible, then that is the desired outcome, as the Prophet (peace be upon him) said: "We have not seen for two who love one another anything like marriage." However, if marriage is not possible, then the relationship must be severed for fear of temptation.