Plasma Globe As An Air Ioniser?

Sure, except that the ionization occurs to the gas inside the globe, the outside of the globe is insulated from the two electrodes on the inside. The reason that the streamers "know" where your finger is is due to the capacitive coupling through the glass, not because of any direct contact.

There is typically only one electrode at the center of the globe.

The return is through ground. While the glass does act as the dielectric for the coupled capacitance, given the high frequency, it doesn't prevent current from flowing outside the globe.

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I believe Air ionizers are typically not AC, but instead DC. This makes me think that a plasma globe will not be very effective as an air ionizer.