Proposition

Lemma 7 ([Trace-null supports]). If B has no finite-dimensional irreducible representations and τ is a tracial state on B , then τ(r_h)=0 for every pure state h .

Prove 1Disprove 0Cited as proposition 1Chain graphAdd Your Proof

Add Your Proof1

Prove
Lemma 2. Equivalent pure states of a unital C^* -algebra B are conjugate by a…Lemma 6 ([Cross-excision]). Let h∼k be pure states of B , and let (x_λ) and…Proposition
The space H_h is infinite-dimensional. Choose orthonormal unit vectors η_j and unitaries u_j∈B with π_h(u_j)ξ_h=η_j , as in Lemma 2. The projections u_jr_hu_j^* are pairwise orthogonal, since r_hu_i^*u_jr_h=h(u_i^*u_j)r_h=0 (i≠j). They all have trace τ(r_h) , so mτ(r_h)≤1 for every positive integer m .
Show More →