Positive rescaling of an upper-strip W*-KMS flow

Assume the Axiom of Choice. Let $H$ be a complex Hilbert space and let $M\subseteq \mathcal{B}(H)$ be a nonzero unital $\ast$-subalgebra with $M=M^{\mathrm{\prime }\mathrm{\prime }}$, where $\mathcal{B}(H)$ is the bounded-operator algebra and $M^{\mathrm{\prime }}$ is its commutant. Let $\omega$ be a normal state, meaning a bounded positive linear functional with $\omega (I)=1$ that preserves suprema of bounded increasing nets of positive elements. Let $t\mapsto {\alpha }_t$ be a one-parameter group of bijective unital $\ast$-automorphisms of $M$, with dimensionless parameter $t$, such that $t\mapsto \phi ({\alpha }_t(X))$ is continuous for every $X\in M$ and every normal linear functional $\phi$ on $M$ (an element of the predual). Suppose that for some dimensionless $\theta >0$, for every $A,B\in M$ there is a bounded continuous function $F_{A,B}$ on $S_{\theta }=\{z\in \mathbb{C}:0\le \mathrm{Im}z\le \theta \}$, holomorphic inside, with $F_{A,B}(t)=\omega (A{\alpha }_t(B)),F_{A,B}(t+i\theta )=\omega ({\alpha }_t(B)A)\mathrm{.}$ This is the upper-strip convention of the preceding von Neumann KMS definition. For any scalar $\lambda >0$, set ${\tilde{\alpha }}_{\tau }={\alpha }_{\tau \mathrm{/}\lambda }$ for $\tau \in \mathbb{R}$. Then $\tilde{\alpha }$ is again a point-ultraweakly continuous one-parameter group, and $\omega$ satisfies the same upper-strip boundary condition for $\tilde{\alpha }$ on the strip of width $\lambda \theta$.

The Axiom of Choice

Upper-strip KMS condition for von Neumann algebras