  In the theory of Toeplitz quantization of algebras, as developed by the
second author, coherent states are defined as eigenvectors of a Toeplitz
annihilation operator. These coherent states are studied in the case when the
algebra is the generically non-commutative Manin plane. In usual quantization
schemes one starts with a classical phase space, then quantizes it in order to
produce annihilation operators and then their eigenvectors and eigenvalues. But
we do this in the opposite order, namely the set of the eigenvalues of the
previously defined annihilation operator is identified as a generalization of a
classical mechanical phase space. We introduce the resolution of the identity,
upper and lower symbols as well as a coherent state quantization, which in turn
quantizes the Toeplitz quantization. We thereby have a curious composition of
quantization schemes. We proceed by identifying a generalized Segal-Bargmann
space SB of square-integrable, anti-holomorphic functions as the image of a
coherent state transform. Then SB has a reproducing kernel function which
allows us to define a secondary Toeplitz quantization, whose symbols are
functions. Finally, this is compared with the coherent states of the Toeplitz
quantization of a closely related non-commutative space known as the
paragrassmann algebra.
