Spectral Derivatives in C^*-algebras

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These are parts of a working paper on spectral derivatives. The concept of spectral derivative is derived from the idea of the of a quantum mechanical momentum operator algebraically defined by the Canonical Commutation Relations and represented in the Schödinger representation as a derivative. In the quantum mechanical case, the derivative is taken, of functions defined on the spectrum of the position operator with respect to the variable that ranges over that spectrum.

In FCCR, presented in the physics pages, the idea that physically suitable Canonical Commutation Relations can be given in terms of nxn complex matricies that connect the Canonical Commutation Relations (CCR) for n unbounded with the Canonical Anticommutation Relations (CAR) for n=2. A Spectral Derivative is supposed to adapt the formal recipe for a derivative to the case of finite dimensional spaces, and therefore to discrete operator spectra.

CONTENTS

• Abstract

• I. Introduction and Preliminaries

• II. (Not Yet)

• III. (Not Yet)

• IV. (Not Yet)

• V. (Not Yet)

• Appendix A:
  Some Topological and Algebraic Definitions

• Notes on the Lax Equation

• References

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