Co-Toeplitz operators and their associated quantization

Abstract

We define co-Toeplitz operators, a new class of Hilbert space operators, in order to define a co-Toeplitz quantization scheme that is dual to the Toeplitz quantization scheme introduced by the author in the setting of symbols that come from a possibly non-commutative algebra with unit. In the present dual setting the symbols come from a possibly non-co-commutative co-algebra with co-unit. However, this co-Toeplitz quantization is a usual quantization scheme in the sense that to each symbol we assign a densely defined linear operator acting in a fixed Hilbert space. Creation and annihilation operators are also introduced as certain types of co-Toeplitz operators, and then their commutation relations provide the way for introducing Planck's constant into this theory. The domain of the co-Toeplitz quantization is then extended as well to a set of co-symbols, which are the linear functionals defined on the co-algebra. A detailed example based on the quantum group (and hence co-algebra) SU q (2) as symbol space is presented.

Co-Toeplitz Quantization: A Simple Case

“…This is called the Manin quantum plane. The notation follows that used in [2]. We define the co-multiplication ∆ to be the algebra morphism determined by ∆(a) = a ⊗ a and ∆(c) = c ⊗ a.…”

“…In [2] I have defined co-Toeplitz operators in a dual way in terms of category theory to Toeplitz operators. The structures needed for this definition are a co-algebra C (see [1]) together with a sesqui-linear form ·, · defined on it.…”

“…The linear map π g : P ⊗ C → P, where g ∈ C is called the symbol of the co-Toeplitz operator C g , is defined in [2] by…”

“…The anti-linear map g → C g is called the co-Toeplitz quantization of C. This in general does not involve measure theory. For more details see [2]. The definition (1.1) is the dual diagram to that for a Toeplitz operator which is defined for a symbol g ∈ A, an algebra, as the composition (right to left)…”

“…If the sesqui-linear form is positive definite, then C g acts in a pre-Hilbert space and so may be considered as a densely defined operator acting in the Hilbert space completion of C. Thus we can construct models for quantum physics, including creation and annihilation co-Toeplitz operators. (See [2].) All objects in this paper are vector spaces over the complex numbers, and all arrows are linear maps, except as noted.…”