Upper-strip KMS condition for von Neumann algebras

Assume the Axiom of Choice. Let $M\mathrm{\ne }\{0\}$ be a concrete von Neumann algebra on a complex Hilbert space. Use the predual $M_{\ast }=Q$ of Trace-class quotient predual and normal functionals; its elements are the normal linear functionals, and the induced topology $\sigma (M,Q)$ is the ultraweak topology. A normal state on $M$ is a bounded complex-linear functional $\omega :M\to \mathbb{C}$ with $\omega (I)=1$, $\omega (a)\ge 0$ for every positive $a\in M$, and $\omega \text{\hspace{0.17em}⁣}\left({\sup }_i{a}_i\right)={\sup }_i\omega (a_i)$ for every increasing net $(a_i)$ of positive elements with $0\le a_i\le CI$ for a common finite $C$ and supremum taken in $M$. An automorphism of $M$ is normal when it preserves the suprema of all such bounded increasing positive nets. A point-ultraweakly continuous one-parameter automorphism group is a family $({\alpha }_t{)}_{t\in \mathbb{R}}$ of bijective unital complex-linear maps $M\to M$ preserving multiplication and adjoints, with ${\alpha }_0={\mathrm{id}}_M$, ${\alpha }_s{\alpha }_t={\alpha }_{s+t}$, and $t\mapsto \phi ({\alpha }_t(x))$ continuous for every $x\in M$ and every normal linear functional $\phi \in Q$. For dimensionless $t$ and dimensionless $\beta >0$, the normal state $\omega$ satisfies the upper-strip $(\alpha ,\beta )$ KMS condition when, for every $A,B\in M$, there is a bounded scalar function $F_{A,B}$ on $S_{\beta }=\{z\in \mathbb{C}:0\le \mathrm{Im}z\le \beta \}$ that is continuous on $S_{\beta }$, holomorphic in its interior, and satisfies $F_{A,B}(t)=\omega (A{\alpha }_t(B)),F_{A,B}(t+i\beta )=\omega ({\alpha }_t(B)A)(t\in \mathbb{R})\mathrm{.}$

Concrete von Neumann algebra and faithful vector

Trace-class quotient predual and normal functionals

The Axiom of Choice