Quantum Calculus and Ideals in the Algebra of Compact Operators

One of the goals of noncommutative geometry is to translate the basic notions of analysis into the language of Banach algebras. This translation is based on the quantization procedure. The arising operator calculus is called, following Connes, the quantum calculus. In this paper we give several assertions from this calculus concerning the interpretation of Schatten ideals of compact operators in a Hilbert space in terms of function theory. The main focus is on the case of Hilbert–Schmidt operators.