whether you have to inform your a-kasse or not depends on which A-kasse you currently are member of.
A-kasser only for certain proffessions
If you are a member in an A-kasse only for certain professions, then you can only keep your membership with that A-kasse, if you continue to work within the same industry/profession. If you continue to work within the A-kasse's proffessional field, you do not need to inform the A-kasse about your new job.
If your new job is not covered by your A-kasse, you need to change A-kasse.
A-kasser for anyone
If you currently are member in one of the interdiciplinary A-kasser, then you can keep this memberhsip regardless of what kind of job-change you make. In this case you do not need to inform your A-kasse.
This is something you have to change with your bank. You need to delete the "old" payment agreement in your "old" bank. And make a new payment agreement (BS-payment) in your new bank. But be sure that you dont get a gap in your payments because it cause your membership to stop.
Anders Weber, www.a-kasser.dk