An offer letter could be construed as a contract depending on the language, but is not usually a binding agreement for purposes of your situation.
In CA, as in most states, you are an employee at will if working without a contract. Under the employment at will doctrine, you or the employer can terminate the employment relationship at any time without good cause.
This means that the employer did not have to provide you with a valid reason for your termination and does not have to justify it. Unless they discriminated against you or retaliated based on some protected status, which does not appear to be the case here, there would normally be no legal action you could take against them.
You should still be eligible for unemployment benefits though if they cannot prove you were terminated for good cause, so I would apply with the unemployment commission right away.
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