Lyapunov theorems for operator algebras

Quantum measurable cardinals

“…Indeed, assuming ZFC + "a measurable cardinal exists" is consistent, [15] On the other hand, it is worth noting that this conclusion is not consistent with the continuum hypothesis. It follows from [1,Lemma 6.2] that if the continuum hypothesis holds and κ is smaller than the first measurable cardinal then there is no finitely additive regular probability measure on κ which vanishes on singletons. Thus, using Theorem 8.2 we can draw the following conclusion.…”

Geometry of the unit ball and representation theory for operator algebras

“…Given any path Γ (a) defined above, we now construct a pure state ω a as follows: start with any unit vector, i.e., normalized sum of diagonal matrix units, ξ and e (a) (1,2) , respectively. Let…”

Connectivity and components for $C^*$-algebras

“…In this case X is far from connected, but any continuous function defined on it has connected range if it vanishes at infinity. 1 : Condition (vii) is a Lyapunov theorem in the the language of [4], so the theorem above determines when a such a theorem holds true for all mappings a X SA 3 R. The set SA is weak à compact, and by definition, a is continuous in this topology. If A is not connected, by Theorem 1.7, we get aSA T aextSA for some a P A sa , and by the abstract Lyapunov theorem [4, 1.7] we conclude that the facial dimension of SA (see [4, p. 10]) s1ren eilers is one.…”

“…We have seen that PA is connected and locally connected. Combine [13,[3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] and [20, 4.3.2] Remark 5.10. The most obvious reason why a given connected C Ã -algebra A has a set of pure states that is not arcwise connected is that the underlying central structure of A, is not arcwise connected.…”