Employees rights as full time

Absent any violation of applicable laws against discrimination (such as on account of sex, age, race, religion, military status, etc.) and absent any contractual right (such as under a collective bargaining agreement or individual employment contract) if your employer is a private business, you generally are an employee at will, and if so the employer can terminate you or change your rate of compensation at any time and does not need cause or any specific reason to do so (although employers often give reasons to help them avoid claims of illegal discrimination).

However, if there is a substantial adverse change in working conditions, as this would be, for purposes of the right to unemployment insurance you could elect to regard the move to part-time as a constructive termination.

Contact the California EDD and discuss the situation with them to confirm this and to get some guidance if you want unemployment coverage, as you'll want to know if you must actually quit after the change in hours, and if so how to phrase any resignation letter, or if you can still apply for unemployment provisionally if you work half-time. 

By the way, the employer is probably saving not only half the salary but also the costs of health insurance. Unfortunately many employers are doing this to stay in business or avoid firing half the staff.