I have 2 young daughters who like many their age are currently obsessed by "Frozen". I watched it for the first time last night and actually ended up googling to work out whether Elsa's sister is name Anna, or Arne. It seems the former is correct but there is no way I could pronounce it that way. Is this a general trend in N.Am English or does it only apply in certain cases?
As other people have mentioned in the comments, this does not in any way reflect a general process of vowel-backing in American English.
In fact, almost the reverse is true: the phoneme /æ/ has actually been moving further back in many forms of British English (some transcriptions use the IPA symbol [a] to represent the British pronunciation of this vowel) and there are regional varieties of North American English where /æ/ is fronted to something like [ɛə] or [eə] (this is a feature of the Northern Cities Vowel Shift that keshlam mentions). While /æ/ may be backed to some degree in other regional varieties of North American English (a more back realization of /æ/ is supposed to be common for speakers of Californian English, for example), it is quite unlikely that a British English speaker would hear any North American English speaker's /æ/ phoneme as the British English /ɑː/ phoneme.
So the difference here is not just a detail of pronunciation; it's a difference in the actual phonemic identity of the vowel (whether it belongs to the set of words with the "PALM/SPA" vowel, or the set of words with the "TRAP" vowel).
In most cases, I would not say this is due to a sound change; it's due to different strategies for adapting loanwords to English phonology. Many loanwords show this kind of variation between the TRAP phoneme (/æ/) and the PALM phoneme (/ɑː/). In some cases, the choice between the two variant forms basically occurs at the individual level. For example, as far as I know, both British and North American speakers may pronounce the vowel in the second syllable of "Iraq" as either /æ/ or /ɑː/.
In other cases, the choice is basically constrained by the regional variety of English that the individual speaks. Most British English speakers can only have /æ/ in "pasta," while most American English speakers can only have /ɑː/ in this same word.
In loanwords from European languages such as Italian , it seems that British speakers are more likely to adopt "a" as /æ/, while North American speakers are more likely to adopt "a" as /ɑː/. (In part, this may be due to the generally backer realization of /æ/ in British English that I mentioned earlier: the stereotypical realization of the /a/ phoneme in European languages is a neutral IPA [ä], and British English /æ/ may be closer to this than American English /æ/).
For that specific example, I'd check what the source culture is for the name. It probably isn't of North American origin, which would make it orthogonal to the rest of your question.
However, I'm going to cite http://en.wikipedia.org/wiki/Northern_Cities_Vowel_Shift... there is an apparent vowel shift in progress in some parts of the US. I know a few linguists who are absolutely delighted by the opportunity to actually be able to watch a vowel shift occur.