The Cloud Security Alliance STAR Registry is an assurance program and documentation registry or cloud provider assessments based on the CSA Cloud Controls Matrix and Consensus Assessments Initiative Questionnaire. Some providers also disclose documentation for additional certifications and assessments(including self-assessments).
Ref: Security Guidance v4.0 Copyright2017, Cloud Security Alliance(used for educational purpose here)

Cloud computing changes the responsibilities and mechanisms for implementing and managing governance. Responsibilities and mechanisms for governance are defined in the contract. as with any business relationship. If the area of concern isnt in the contract. there are no mechanisms available to enforce. and there is a governance gap. Governance gaps dont necessarily exclude using the provider, but they do require the customer to adjust their own processes to close the gaps or accept the associated risks.
Ref: Security Guidance v4.0 Copyright2017, Cloud Security Alliance (used for educational purpose here)

Definitions:
Due diligence is the act of investigating and understanding the risks a company faces.
Due care is the development and implementation of policies and procedures to aid in protecting the company, its assets, and its people from threats

As with security. compliance in the cloud is a shared responsibility model. Both the cloud provider and customer have responsibilities. But the customer is always ultimately responsible for their own compliance. These responsibilities are defined through contracts, audits/assessments. and specifics of the compliance requirements.
Reference: CSA Security Guidelines V.4(reproduced here for the educational purpose)