Contents
Chapter 1. Introduction 1
1.1. Noncommutative (nc) functions: examples and genesis 3
1.2. NC sets, nc functions, and nc difference-differential calculus 5
1.3. Applications of the Taylor–Taylor formula 9
1.4. An overview 11
Acknowledgements 14
Chapter 2. NC functions and their difference-differential calculus 15
2.1. Definition of nc functions 15
2.2. NC difference-differential operators 17
2.3. Basic rules of nc difference-differential calculus 22
2.4. First order difference formulae 25
2.5. Properties of ΔRf(X, Y ) and ΔLf(X, Y ) as functions of X and Y 27
2.6. Directional nc difference-differential operators 31
Chapter 3. Higher order nc functions and their difference-differential calculus 33
3.1. Higher order nc functions 33
3.2. Higher order nc difference-differential operators 41
3.3. First order difference formulae for higher order nc functions 54
3.4. NC integrability 56
3.5. Higher order directional nc difference-differential operators 57
Chapter 4. The Taylor–Taylor formula 61
Chapter 5. NC functions on nilpotent matrices 67
5.1. From nc functions to nc power series 67
5.2. From nc power series to nc functions 70
Chapter 6. NC polynomials vs. polynomials in matrix entries 77
Chapter 7. NC analyticity and convergence of TT series 83
7.1. Analytic nc functions 83
7.2. Uniformly-open topology over an operator space 94
7.3. Uniformly analytic nc functions 97
7.4. Analytic and uniformly analytic higher order nc functions 109
Chapter 8. Convergence of nc power series 125
8.1. Finitely open topology 125
8.2. Norm topology 130
8.3. Uniformly-open topology 133
Chapter 9. Direct summands extensions of nc sets and nc functions 155
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