Entire analytic element for a one-parameter automorphism group

Definition

Let $A$ be a $C^*$ -algebra and $\alpha:\mathbb R\to\operatorname{Aut}(A)$ a point-norm continuous one-parameter group of $*$ -automorphisms. An element $a\in A$ is entire analytic for $\alpha$ if the orbit map

t\longmapsto\alpha_t(a)

extends to an entire $A$ -valued function on $\mathbb C$ . The extension is unique and its value at $z\in\mathbb C$ is denoted $\alpha_z(a)$ . Thus an expression such as $\alpha_{-i}(a)$ is defined only for analytic $a$ , not by extending the automorphism group to all complex times.

Algebraic properties

Entire analytic elements form a dense $*$ -subalgebra $A_{\mathrm{an}}$ that is invariant under every $\alpha_z$ . For $a,b\in A_{\mathrm{an}}$ ,

\alpha_z(ab)=\alpha_z(a)\alpha_z(b),\qquad \alpha_z(a^*)=\alpha_{\overline z}(a)^*.

The maps $\alpha_z$ on $A_{\mathrm{an}}$ generally are not bounded for nonreal $z$ , which is why they need not extend to all of $A$ .

Gaussian analytic approximation

For $a\in A$ and $n>0$ , the norm-convergent Bochner integral

a_n=\sqrt{\frac n\pi}\int_{\mathbb R}e^{-nt^2}\alpha_t(a)\,dt

is entire analytic, and $a_n\to a$ in norm as $n\to\infty$ . This Gaussian smoothing proves density and gives the extension by replacing $e^{-nt^2}$ with $e^{-n(t-z)^2}$ Bratteli–Robinson, §2.5.3.

Modular-theory use and conventions

There is a parallel $W^*$ -algebra convention. If $\sigma:\mathbb R\to\operatorname{Aut}(M)$ is only point-ultraweakly continuous, an element $a\in M$ is entire analytic when its orbit has an entire extension $z\mapsto\sigma_z(a)$ in the ultraweak sense: for every normal functional $\omega\in M_*$ , the scalar function $z\mapsto\omega(\sigma_z(a))$ is entire. This is not a norm-entire extension on all of $M$ .

For a modular automorphism group $\sigma^\varphi$ , it is this $W^*$ -analytic convention that makes the imaginary-time term $\sigma^\varphi_{-i}(a)$ in the modular KMS condition meaningful. Some sources reserve “analytic element” for extension to a strip or neighborhood of zero; “entire analytic” here always means extension to all of $\mathbb C$ , in the topology appropriate to the stated action.

References

Ola Bratteli and Derek W. Robinson, Operator Algebras and Quantum Statistical Mechanics 1, 2nd ed., Springer, 1987. Publisher DOI record. Relevant: §2.5.3 on analytic elements for one-parameter groups.
Masamichi Takesaki, Theory of Operator Algebras II, Springer, 2003. Publisher DOI record. Relevant: Chapter VIII, §1 on analytic elements and modular automorphism groups.