Just as akasha — space — being sarva-gata, all-pervading, is not stained by anything by virtue of saukshmya, extreme subtlety; so the atman, though everywhere stationed in the body, is not stained. The image is one of the chapter's most luminous: space contacts every object yet is colored by none. The atman's relation to body is the same.
Shankara reads precisely. The paramatman is anadi — what has a beginning perishes; this has none, hence avyaya. Being nirguna, it undergoes no deterioration through quality-loss as conditioned things do. Though said to dwell in the body because it is cognized there, it neither acts nor is touched by the fruits of action. The akasha-image grounds the structural argument: subtlety is the reason untouchability follows; the witness is structurally finer than what it witnesses.
Madhusudana addresses the worry directly: lest one suppose that even an inherently non-acting Self acquires aupadhika — conditional, superimposed — agency through its connection with a body, this verse removes that objection. Vyaya is twofold: decay of the dharmin itself, or decay through loss of its dharmas. The first is blocked by anaditva — the atman has no prior cause and so no decay-condition. The second is blocked by nirgunatva — there are no dharmas to lose. The akasha-image makes the structural argument intuitive.
Ramanuja reads akasha, though all-pervading and in contact with every object, is not stained because of ati-saukshma. Likewise the atman, though residing in every body — deva, human, the rest — is not stained by the distinct natures of each such body. The Self retains its own unblemished svarupa in every bodily vehicle. The image is for the bhakta a description of what the Lord's antaryamic presence is like: He inhabits without being inhabited.
Madhva reads on the bheda-frame: the paramatman is anadi — what originates is subject to vyaya. Hari, having no origin, has no decay. What bears gunas decays through their loss; being nirguna, Hari admits no such decay. The verse confirms avyayatva through two independent grounds. The statement that He does not act refers to the absence of dependent agency: Hari is svatantra, not paratantra — His actions are svecha, His own will, not driven by external karma. The akasha-image holds for the jiva too in its proper mode: paratantra but akarta in the witness-sense.
Vallabha reads the antarah-purusha — the inner Person — the akshara-svarupa atman-jiva, naturally free from all prakrita-sambandha by virtue of anadi and nirguna nature. Though Bhagavan's iccha causes the jiva to appear separately manifested, this is adhyasa only — in truth the jiva is amsha of the Lord. Akasha-saukshma is the bhakta's working image: presence without staining, intimacy without contamination.
Shridhara reads the question explicitly: if the Lord dwells in the body during samsara, how can He remain unaffected by the unevenness of sukha-duhkha? The answer: what has an origin perishes, what has gunas decays. This paramatman is anadi and nirguna, therefore avyaya. Akasha-saukshma names the precise mode of presence-without-contact.