  We introduce the notion of a quantum locally compact metric space, which is
the noncommutative analogue of a locally compact metric space, and generalize
to the nonunital setting the notion of quantum metric spaces introduced by
Rieffel. We then provide several examples of such structures, including the
Moyal plane, as well as compact quantum metric spaces and locally compact
metric spaces. This paper provides an answer to the question raised in the
literature about the proper notion of a quantum metric space in the nonunital
setup and offers important insights into noncommutative geometry for non
compact quantum spaces.
