By H. S Hall
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Additional info for A Short Introduction to Graphical Algebra
On the quotient A=N! If V : A ! A=N! is the canonical projection, then by de nition (V A V B )! 114) The Hilbert space H! is the closure of A=N! in this inner product. 3. The representation ! (A) is rstly de ned on A=N! H! by ! 115) it follows that ! is continuous. Hence ! (A) may be de ned on all of H! 110). 4. The cyclic vector is de ned by ! = V I, so that ( ! (A) ! 10 The Gel'fand-Neumark theorem 39 We now prove the various claims made here. First note that the null space N! of ( )0 can be de ned in two equivalent ways N!
A state that is not pure is called a mixed state. When K = S (A) is a state space of a C -algebra we write P (A), or simply P , for @e K , referred to as the pure state space of A. Hence the pure states on A = C C are the points 0 and 1 in 0 1], where 0 is identi ed with the functional mapping +_ to , whereas 1 maps it to . 109) for which x2 + y2 + z 2 = 1 these are the projections onto one-dimensional subspaces of C 2 . 10 that the state space of Mn (C ) consists of all positive matrices with unit trace the pure state space of Mn (C ) then consists of all one-dimensional projections.
The set of all 2 H with k k 1. Firstly, for any sequence (or net) An 2 Bf (H) we may choose a unit vector n 2 (An H)? Then (An ; I) = ; , so that k (An ; I) k= 1. Hence supk k=1 k (An ; I) k 1, hence k An ; I k! 3) in B(H) (hence in B0 (H)). Secondly, when A 2 Bf (H) and B 2 B(H) then AB 2 Bf (H), since AB H = AH. But since BA = (A B ) , and Bf (H) is a -algebra, one has A B 2 Bf (H) and hence BA 2 Bf (H). Hence Bf (H) is an ideal in B(H), save for the fact that it is not norm-closed (unless H has nite dimension).
A Short Introduction to Graphical Algebra by H. S Hall