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You are going to have to open a probate of some type to pass title to the property. In CA this is done through the probate court whereas in NJ it is done through the Surrogate's court. Different name but they do the same thing.
Usually one state will accept a certified copy of another state's orders under the "Full Faith and Credit" clause of the Constitution. However, if they don't then you could end up having to do the extra step of registering the CA order as a "foreign judgment" in NJ and "domesticating the judgment". That then makes it an order of a NJ court and the bank will have to follow the order.
There are a lot of different type of probate ranging from the very informal to a very formal probate including a jury trial. Very few cases require the jury trial and many can get by with an informal probate. However, you will have to visit with a local lawyer that does Probate law (you can find one at www.lawyers.com) and have them do an in depth interview to see which kind will work best for your mother's estate.
If there is nothing more in the estate than the checking account and if it doesn't have much money in it then it is probably not going to be worth pursuing. However, if there is more property in the estate then a probate is the only way to pass title to that property.
Please ask any follow up questions in this thread.