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The Gita says Brahman/Atman is simply a witness and its the Maya or prakriti that acts. The Advaitins also holds the same belief.
If Maya is not a separate entity, i.e. if Brahman and Maya are inseparable, just like the sun and its rays are inseparable, then isn't it Brahman who's the doer of all deeds?
This is a very important question and has been subject of fierce debate. Let me first lay out what the critics of Advaita Vedanta's inert Brahman say about the issue.
It is because Sankara Vedanta considers Brahman to be only prakasa or jnana (light or illumination) without any vimarsa or activity that it has to invoke the help of Maya for the manifestation of the universe. Brahman is devoid of any activity; it is, therefore, impotent to create. It is, only Isvara or mayopahita caitanya that can manifest the universe. But whence does this maya drop in? If it is some power extraneous to Brahman or Isvara, then Sankara Vedanta is reduced to dualism. If Maya is only an expression of the power of Brahman, then Brahman cannot be divested of activity. Both Sankhya and Vedanta consider the Purusa or Atma to be niskriya or inactive, because they take the word 'activity' in a very crude sense. Surely, Brahman or Atma does not work like a potter or watch-maker. The very Vimarsa, the very Iccha (will) of the Divine is spiritual energy of incalculable force that can proliferate into any form from the subtlest to the grossest.
Introduction by Jaideva Singh to Siva Sutras, The Yoga of Supreme Identity, Commentary Vimarsini of Ksemraja
The above criticism is from the point of view of Kashmiri Saivism and pretty accurately sums up the puzzle of Maya. Is Maya a part of Brahman? Basically who ordered it? The question about whether Brahman is inert boils down to what is the nature of Maya.
The Advaita Vedanta answer is that the question regarding Maya is ill-posed. One cannot know the nature of Maya while we are inside it. The question ceases to exist once we attain Moksha and we find that Brahman is all.
From the Advaita Vedanta point of view Kashmiri Saivism's 'Parama Siva with will' has mental state associated with it and hence cannot be Impersonal. Brahman as the Impersonal Reality cannot by definition have any mental state associated with it and must be inert.
From the point of view of Advaita Vedanta, maya and avidya are the same. Just as avidya or ignorance has the powers, as it were, to hide Brahman and project something else, such as the world, on it, so also maya, as it were, can cover up Brahman and project the world onto it.
Maya cannot really cover the Reality that is Brahman. A patch of cloud can never cover the sun. It only covers the eyes of those who are looking at the sun. In the same way maya covers our knowing ability and thus obstructs our knowledge of Brahman.
In any act of knowing, the ‘knower’ and the object known’ have to be separate from each other. They cannot be the same. In order for us to know maya, it has to be separate from us. Individuals like us are the products of maya and we are all posited within the domain of maya. Maya and we individuals are not separate from each other, just as chocolate and a bear-shaped chocolate candy are not different from each other.
As we are identical with maya, we cannot know maya. For this reason, maya is not only unknowable; it is inexplicable as well (anirvachaniya).
Journey from one to many essentials of Advaita Vedanta by Swami Bhaskarananda