Can they be put into check once they're granted administratorship? For example, if someone suspected them of stealing or misappraising items
Let's make sure we are talking about the same thing. A person who has been appointed by a court as Independent Executor of an estate (where the decedent had a Will) functions independently of court supervision with a few exceptions. But, they can still be held accountable by the court. A person who has been appointed by a court as a Dependent Administrator of an estate (where the decedent did not have a Will) must either report to the court or get permission from the court for almost every act. They put up a bond and they can be held accountable. Both of these types of offices must file or provide an Inventory of the estate assets. The method of disputing the values used for items in the Inventory is to file an Objection to the Inventory. You should consult with an experienced probate lawyer if you have any suspicions about the conduct of the personal representative of the estate.
DISCLAIMER: This is not specific legal advice and does not establish an attorney/client relationship.
Once the court appoints someone to act as the administrator for an estate, that person has a fiduciary duty to the estate. If you believe that they are not adequately performing their duties or that they are self-dealing, you should advise the court and seek to have them replaced as administrator. I would strongly suggest that you consult with local probate counsel by exploring all of your allegations before going to the court.
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I agree with Attorney Paxton, who has given his customary excellent answer. I write only to address your two examples. Stealing is not only a crime, but a very serious allegation. You would not want to accuse someone of this, unless you had VERY solid evidence to back you up. Otherwise, you could be looking at a defamation lawsuit, among other things. "Mis-appraising items" is a lot trickier. If you are suggesting that the executor/administrator is deliberately making up valuations, in order to benefit himself/herself, (and if you can prove this), then this would likely be self-dealing and a breach of fiduciary duties. If you are saying that you do not agree with the appraiser that was used or that you think the price that the house is listed for is too low, then you are probably going to have a tough road.
You have recourse to the court, as a last resort, if there is serious wrongdoing. If you just are not happy with the way that the executor/administrator is doing the work, then that is something you may just need to live with. Administering an estate is often a difficult and thankless task. Much of the time, the person in charge was hand-picked by the deceased to do this. I would trust them to handle it, unless you have very clear proof that something is amiss.
*** LEGAL DISCLAIMER I am licensed to practice law in the State of Michigan and have offices in Wayne and Ingham Counties. My practice is focused in the areas of estate planning and probate administration. I am ethically required to state that the above answer does not create an attorney/client relationship. These responses should be considered general legal education and are intended to provide general information about the question asked. Frequently, the question does not include important facts that, if known, could significantly change the answer. Information provided on this site should not be used as a substitute for competent legal advice from a licensed attorney that practices in your state. The law changes frequently and varies from state to state. If I refer to your state's laws, you should not rely on what I say; I just did a quick Internet search and found something that looked relevant that I hoped you would find helpful. You should verify and confirm any information provided with an attorney licensed in your state.