Noncommutative Mathematics for Quantum Systems

178 Index

of a bialgebra, 31
of a dual (semi-)group, 100
corepresentation, 44
contragredient, 45
irreducible, 44
non-degenerate, 44
unitary, 44
counit
of a bialgebra, 31
of a dual (semi-)group, 100
Cuntz algebra, 126
Cuntz-Krieger family,

distribution
of a quantum random variable, 30
of a quantum stochastic process, 30
drift, 39
dual group, 100
dual semigroup, 100

Egorov Theorem, 166
Einstein-Podolski-Rosen
experiment, 15

endomorphism, 132
binary shift, 147
permutation, 132
shift, 132
shift, entropy of, 141
entropy, 134
Connes-Narnhofer-Thirring, 139
genuinely noncommutative, 147
noncommutative topological, 137
topological, 135
Voiculescu, 137
equivalence of quantum stochastic
processes, 30
ergodic theorem
almost uniform, 167
Birkhoff’s, 158
mean, noncommutative, 164

von Neumann’s, 158

multiparameter, 169

flip, 32

Fourier transform, 46

free

independence, 57

L ́evy process, 101

free product

of algebras, 84, 86–87

of states, 92

freeness, 57, 91, 101

Gaussian generating functional, 40

generating functional, 36

Poisson, 40

quadratic or Gaussian, 40

Gleason’s theorem, 20

GNS construction, 159–160

H-algebra, 100

Haar state, 44

Heisenberg’s uncertainty relation, 12

Hopf algebra, 32

Hunt’s formula, 27

increment

process, 33

property, 32, 102

independence

anti-monotone, 92, 101

boolean, 73–74, 92, 101

Bose or tensor, 31, 90

free, 57, 91, 101

in a category, 88

monotone, 60, 62, 92, 101

stochastic, 23, 89,

tensor, 31, 90, 101

infinitely divisible, 23

involutive

bialgebra, 31