The legal issue at hand pertains to determining whether the Tata Memorial Centre functions under the authority of the Central Government or operates as an independent entity, which directly impacts which government is considered the "appropriate government" for the purposes of industrial dispute resolution and trade union recognition.

The key considerations involve the nature of the control and management of the Centre, the vesting of property, and the operational autonomy of the institution. The facts indicate that although the property was initially transferred to the Central Government and the Centre is listed under the rules for allocation of government business, the actual management and control are exercised by the Governing Council of the Society, which functions independently. The evidence shows that the Society manages day-to-day operations without interference from the Central Government, and the employees are not government servants.

Furthermore, the property is deemed to be vested in the Governing Council, and the institution’s functions are carried out with significant autonomy. The agreement and legal arrangements suggest that the Centre was established as an independent society with its own management structure, rather than as a direct arm of the Central Government.

Therefore, based on the legal principles governing the interpretation of "industry" and "appropriate government," and considering the specific facts related to property vesting, control, and operational independence, the appropriate government for the Tata Memorial Centre is the State Government. This conclusion aligns with the criteria that the industry is not carried on under the authority of the Central Government, but rather functions independently under its own governance.

This determination affects the maintainability of certain applications and recognition of trade unions under relevant statutes, with the final ruling affirming that the State Government is the appropriate authority for the Centre, and that the applications filed under the relevant labour laws are maintainable.