# Toeplitz Quantization of a Free $ * $-Algebra

## TL;DR

This paper introduces a novel Toeplitz quantization method for free *-algebras, defining operators with non-commutative symbols and exploring creation and annihilation operators within this framework.

## Contribution

It presents the first application of Toeplitz quantization to free *-algebras, expanding the theory to non-commutative algebraic structures.

## Key findings

- Defined Toeplitz operators with non-commutative symbols.
- Established properties of creation and annihilation operators.
- Provided a new example of Toeplitz quantization in non-commutative setting.

## Abstract

In this note we quantize the free $ * $-algebra generated by finitely many variables, which is a new example of the theory of Toeplitz quantization of $ * $-algebras as developed previously by the author. This is achieved by defining Toeplitz operators with symbols in that non-commutative free $ * $-algebra. These are densely defined operators acting in a Hilbert space. Then creation and annihilation operators are introduced as special cases of Toeplitz operators, and their properties are studied.