By Jacobson N.
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106) In particular: 1. A state on a unital C -algebra is bounded, with norm 1. 2. An element ! 2 A for which k ! (I) = 1 is a state on A. When ! (I) k A k. (I)2 k B k2 : Hence k ! (I). 106). 66) may be copied, showing that ! is real on AR. (A) 0. Choose s > 0 small enough, so that k I ; sA k 1. Then (assuming ! (A) 0. We now pass to states on C -algebras without unit. Firstly, we look at a state in a more general context. 4 A positive map Q : A ! B between two C -algebras is a linear map with the property that A 0 implies Q(A) 0 in B.
Choose s > 0 small enough, so that k I ; sA k 1. Then (assuming ! (A) 0. We now pass to states on C -algebras without unit. Firstly, we look at a state in a more general context. 4 A positive map Q : A ! B between two C -algebras is a linear map with the property that A 0 implies Q(A) 0 in B. 5 A positive map between two C -algebras is bounded (continuous). Let us rst show that boundedness on A+ implies boundedness on A. 107) where A0+ etc. are positive. 5. Hence if k Q(B ) k C k B k for all B 2 A+ and some C > 0, then k Q(A) k k Q(A0+ ) k + k Q(A0; ) k + k Q(A00+ ) k + k Q(A00; ) k 4C k A k : Now assume that Q is not bounded by the previous argument it is not bounded on A+ , so that for each n 2 N there isP an1An 2 A+1 so that k Q(An ) k n3 (here A+1 consists of all A 2 A+ with k A k 1).
2 ELEMENTARY THEORY OF C -ALGEBRAS 48 The C -algebra B0 (H) of compact operators on H is the norm-closure of Bf (H) in B(H) in other words, it is the smallest C -algebra of B(H) containing Bf (H). 3). An operator A 2 B(H) lies in B0 (H) when it can be approximated in norm by nite-rank operators. It is clear that Bf (H) is a -algebra, since p = p for any projection p. The third item in the next proposition explains the use of the word `compact' in the present context. 2 1. The unit operator I lies in B0 (H) i H is nite-dimensional.