Several authorities apply, often to the same project:
Cal/OSHA Asbestos in Construction (8 CCR §1529): worker protection and work practices. The permissible exposure limit is 0.1 fiber per cubic centimeter (8-hour average), with a short-term excursion limit of 1.0 f/cc over 30 minutes.
Cal/OSHA DOSH Asbestos Contractors' Registration Unit (ACRU): any contractor or employer engaged in asbestos-related work involving 100 square feet or more of ACM must be registered with Cal/OSHA.
CSLB (Contractors State License Board): asbestos abatement requires the C-22 Asbestos Abatement classification. CSLB will not issue a C-22 unless the contractor is registered with Cal/OSHA DOSH (or has an application in process).
Certified Asbestos Consultant (CAC) / Certified Site Surveillance Technician (CSST): surveys, assessments, and oversight on projects involving 100 square feet or more must be done by consultants certified by Cal/OSHA DOSH.
Your regional Air Quality Management District: in Southern California, South Coast AQMD Rule 1403 governs asbestos emissions from demolition and renovation, requiring a survey and advance notification.
Federal: the EPA Asbestos NESHAP (40 CFR 61, Subpart M) and, for schools, AHERA (40 CFR 763).