Is there a word for "without any reason" but a more formal one

Your action in banning me was arbitrary and capricious, you'll be hearing from my solicitors shortly.

Consider gratuitously and unwarrantedly for "without any reason," and "unexplainedly" for without providing a reason.

gratuitous: being without apparent reason, cause, or justification.

unwarranted: without a basis for reason or fact; unjustified.

unexplainedly: without explanation.

^{Sources: Random House Dictionary, The Free Dictionary, Wiktionary}

Unjustifiably comes to mind: a reason is needed to justify an action. Or possibly perversely. Or unreasonably.

unjustifiable : Not able to be shown to be right or reasonable

perverse : Showing a deliberate and obstinate desire to behave in a way that is unreasonable or unacceptable

unreasonable : Not guided by or based on good sense

^{_{Oxford Dictionaries Online}}

However, without providing a reason doesn't mean that there isn't one and the ban cannot be justified.

I would say that your banishment was -

• groundless
• baseless
• reasonless
• prejudiced
• irrational
• nonsensical
• unsubstantiated
• and without consideration

Sorry if I copied any words from other answers, I was getting into my Jackie Chiles zone.

You may write:

• On what account did you ban me?