  We derive the quantum Teichm\"uller space, previously constructed by Kashaev
and by Fock and Chekhov, from tensor products of a single canonical
representation of the modular double of the quantum plane. We show that the
quantum dilogarithm function appears naturally in the decomposition of the
tensor square, the quantum mutation operator arises from the tensor cube, the
pentagon identity from the tensor fourth power of the canonical representation,
and an operator of order three from isomorphisms between canonical
representation and its left and right duals. We also show that the quantum
universal Teichm\"uller space is realized in the infinite tensor power of the
canonical representation naturally indexed by rational numbers including the
infinity. This suggests a relation to the same index set in the classification
of projective modules over the quantum torus, the unitary counterpart of the
quantum plane, and points to a new quantization of the universal Teichm\"uller
space.
