Legal Question in Real Estate Law in California

Probably, but one of the duties of a trustee is to conduct trust business openly and without subterfuge or concealment, so especially if there are beneficiaries other than yourself, you should inform the lender of your becoming the successor trustee. Usually, the lenders find out anyway. If you are successor trustee under a living trust, you should at least consider having a short consultation with a local attorney with probate or trust administration experience, as there are a couple of duties a new trustee should carry out and some responsibilities of office of which you should be aware.

Full disclosure is required of trustees, but in this case you probably need only make the payments without saying anything, since the death doesn't change anything about the loan. The lender probably has no right to accelerate the loan, and shouldn't care even if they learn as long as the loan is paid. If you need legal help in the trust or probate, feel free to contact me.