Gibbs States, Algebraic Dynamics and Generalized Riesz Systems

Abstract

In PT-quantum mechanics the generator of the dynamics of a physical system is not necessarily a self-adjoint Hamiltonian. It is now clear that this choice does not prevent to get a unitary time evolution and a real spectrum of the Hamiltonian, even if, most of the times, one is forced to deal with biorthogonal sets rather than with on orthonormal basis of eigenvectors. In this paper we consider some extended versions of the Heisenberg algebraic dynamics and we relate this analysis to some generalized version of Gibbs states and to their related KMS-like conditions. We also discuss some preliminary aspects of the Tomita–Takesaki theory in our context.

1 Introduction

In the past 25 years or so it has become clearer and clearer that the role of self-adjointness of the observables of some given microscopic system can be, sometimes, relaxed, without modifying the essential benefits of dealing with, for instance, a self-adjoint Hamiltonian. In fact, we can still find real eigenvalues, a unitary time evolution and a preserved probability even if the requirement of the Hamiltonian being self-adjoint is replaced by some milder assumption, like in PT- or in pseudo-hermitian quantum mechanics. We refer to [1,2,3,4,5] for some references on these approaches, both from a more physical point of view and from their mathematical consequences.

Considering a non-selfadjoint Hamiltonian $H\ne H^*$ may lead to the appearance of new and often unpleasant features; for instance, the set $\{\varphi _n\}$ of eigenstates of H, if any, in general is no longer an orthonormal system, but this set $\{\varphi _n\}$ and the set $\{\psi _n\}$ of the eigenstates of $H^*$ turn out to be biorthogonal i.e., $(\varphi _n|\psi _m)=\delta _{n,m}$. Also, in concrete examples they are not bases for the Hilbert space $\mathcal H$ where the model is defined, but they may still be complete in $\mathcal H$. This is the reason why the notion of ${\mathcal D}$-quasi bases was proposed in [6].

This concept can be thought as a suitable extension of Riesz biorthogonal bases, and similar biorthogonal sets are found in several concrete physical applications, playing often the role that in the traditional setup is played by orthonormal bases (ONB). In recent papers many other extensions of Riesz bases, mostly involving unbounded operators, have also been considered. In particular we mention generalized Riesz systems introduced by one of us (H.I) and analyzed in a series of papers [7,8,9,10,11,12,13,14]). For other studies on extensions of Riesz bases or on generalizations to different environments (Krein spaces, Rigged Hilbert spaces) we refer to [15,16,17].

In [18] the role of similar biorthogonal sets, in particular Riesz bases, in the analysis of Gibbs states, KMS condition and algebraic Heisenberg dynamics was first considered. More recently a similar analysis has been carried out by other authors (see, e.g. [19]). Here we want to give our contribution to this line of research, by using the biorthogonal sets originated by generalized Riesz systems.

The paper is organized as follows: in the next section we give some preliminaries. In Sect. 3 we propose our definition of Gibbs state defined by generalized Riesz systems, when the dynamics is driven by a self-adjoint operator $H_0$. The natural settings which we will adopt is the $O^*$-algebra ${\mathcal {L}}^\dagger ({\mathcal D})$, where ${\mathcal D}$ is a dense subspace of $\mathcal H$, [20,21,22]. This will appear to be a good choice, due to the fact that the operators appearing in our analysis are mostly unbounded. In Sect. 4 we will consider possible definitions of the algebraic dynamics for non self-adjoint Hamiltonians, and then we will consider how these dynamics are related to the generalized Gibbs states introduced first, and the KMS-like relations which arise from this construction. In Sect. 5 we will propose a preliminary analysis of the Tomita–Takesaki modular theory in our context, while our conclusions are given in Sect. 6.

2 Preliminaries

In this section we review the basic definitions and results on generalized Riesz systems and $O^*$-algebras needed in this paper.

Definition 2.1

A sequence $\{ \varphi _n \}$ in a Hilbert space $\mathcal H$ with inner product $( \cdot | \cdot )$ is called a generalized Riesz system if there exist an ONB $\{ f_n \}$ in $\mathcal H$ and a densely defined closed operator T in $\mathcal H$ with densely defined inverse, such that $\{ f_n \} \subset D(T) \cap D((T^{-1})^*)$ and $Tf_n =\varphi _n$, $n=0,1, \ldots$. Such a $(\{ f_n \}, T)$ is called a constructing pair for $\{ \varphi _n \}$ and T is called a constructing operator for $\{ \varphi _n \}$.

Suppose that $(\{ \varphi _n \},\{ \psi _n \})$ is a biorthogonal pair such that $\{ \varphi _n \}$ be a generalized Riesz system with a constructing pair $(\{ f_n \},T)$. Then putting $\psi _n^T = (T^{-1})^*f_n$, $n=0,1, \ldots$, $\{ \varphi _n \}$ and $\{ \psi _n^T \}$ are biorthogonal sequences, that is, $(\varphi _n |\psi _m^T)=\delta _{nm}$, $n,m=0,1, \ldots$. If $\psi _n^T=\psi _n$, $n=0,1, \ldots$, then $\{ \psi _n \}$ is a generalized Riesz system with a constructing pair $(\{ f_n\} ,(T^{-1})^*)$. But, the equality $\psi _n^T=\psi _n$, $n=0,1, \ldots$ does not necessarily hold. If this equality holds, then T is called natural and $(\{ f_n\},T)$ is called natural constructing pair.

Let ${\mathcal D}$ be a dense subspace in $\mathcal H$. We denote by ${\mathcal {L}}^\dagger ({\mathcal D},\mathcal H)$ the set of all closable linear operators X in $\mathcal H$ such that $D(X)={\mathcal D}$ and $D(X^*) \supset {\mathcal D}$. As usual we put, for $X\in {\mathcal {L}}^\dagger ({\mathcal D},\mathcal H)$, $X^\dagger := X^*\lceil _{\mathcal D}$. Let

$$\begin{aligned} {\mathcal {L}}({\mathcal D})= & {} \{ X\in {\mathcal {L}}^\dagger ({\mathcal D},\mathcal H); \; X{\mathcal D}\subset {\mathcal D}\} , \\ {\mathcal {L}}^\dagger ({\mathcal D})= & {} \{ X\in {\mathcal {L}}({\mathcal D}); \; X^*{\mathcal D}\subset {\mathcal D}\}. \end{aligned}$$

Then ${\mathcal {L}}({\mathcal D})$ is an algebra with the usual operations: $X+Y$, $\alpha X$ and XY, and ${\mathcal {L}}^\dagger ({\mathcal D})$ is a $*$-algebra with the involution $X\rightarrow X^\dagger := X^*\lceil _{\mathcal D}$, inherited by ${\mathcal {L}}^\dagger ({\mathcal D},\mathcal H)$. A $*$-subalgebra ${\mathcal {M}}$ of ${\mathcal {L}}^\dagger ({\mathcal D})$ is said to be an $O^*$-algebra on ${\mathcal D}$ in $\mathcal H$. Here we assume that ${\mathcal {M}}$ has the identity operator I. A locally convex topology defined by a family $\{ \Vert \cdot \Vert _X ; \; X\in {\mathcal {M}} \}$ of seminorms: $\Vert \xi \Vert _X :=\Vert X\xi \Vert$, $\xi \in {\mathcal D}$ is called the graph topology on ${\mathcal D}$ and denoted by $t_{{\mathcal {M}}}$. If the locally convex space ${\mathcal D}[t_{{\mathcal {M}}}]$ is complete, then ${\mathcal {M}}$ is called closed and it is shown that ${\mathcal {M}}$ is closed if and only if ${\mathcal D}= \bigcap _{X\in {\mathcal {M}}} D({\bar{X}})$. If ${\mathcal D}=\bigcap _{X\in {\mathcal {M}}}D(X^*)$, then ${\mathcal {M}}$ is called self-adjoint. Next we define a weak commutant of ${\mathcal {M}}$ as follows:

$$\begin{aligned} {\mathcal {M}}_w^\prime := \{ C\in B(\mathcal H); \; (CX \xi |\eta )=(C\xi |X^\dagger \eta ) \;\mathrm{for \; all}\; X\in {\mathcal {M}} \;\mathrm{and} \; \xi ,\eta \in {\mathcal D}\} , \end{aligned}$$

where $B(\mathcal H)$ is the $C^*$-algebra of all bounded linear operators on $\mathcal H$. Then ${\mathcal {M}}_w^\prime$ is a weakly closed $*$-invariant subspace of $B(\mathcal H)$, but it is not necessarily an algebra. If ${\mathcal {M}}$ is self-adjoint, then ${\mathcal {M}}_w^\prime$ is a von Neumann algebra on $\mathcal H$ satisfying ${\mathcal {M}}_w^\prime {\mathcal D}\subset {\mathcal D}$. Furthermore, we see that ${\mathcal {L}}^\dagger ({\mathcal D})_w^\prime ={\mathbb {C}}I$. We define some topologies on ${\mathcal {M}}$. For any $\xi , \eta \in {\mathcal D}$ we put $p_{\xi ,\eta }(X) :=|(X\xi |\eta )|$, $p_\xi (X):= \Vert X\xi \Vert$, $X\in {\mathcal {L}}^\dagger ({\mathcal D})$. The locally convex topology on ${\mathcal {L}}^\dagger ({\mathcal D})$ defined by the family $\{ p_{\xi ,\eta }(\cdot ); \; \xi ,\eta \in {\mathcal D}\}$ (resp. $\{ p_\xi (\cdot ); \; \xi \in {\mathcal D}\}$) of seminorms on ${\mathcal {L}}^\dagger ({\mathcal D})$ is called the weak (resp. strong) topology, and the induced topology of the weak (resp. strong) topology on ${\mathcal {M}}$ is called the weak (resp. strong) topology on ${\mathcal {M}}$. For any $Y\in {\mathcal {M}}$ and $\xi \in {\mathcal D}$ we define a seminorm on ${\mathcal {M}}$ by

$$\begin{aligned} p_{\xi ,Y}(X) := \Vert YX\xi \Vert , \quad X\in {\mathcal {M}} . \end{aligned}$$

The locally convex topology on ${\mathcal {M}}$ defined by the family $\{ P_{\xi ,Y}(\cdot ); \; \xi \in {\mathcal D}, Y\in {\mathcal {M}} \}$ is called the quasi-strong topology on ${\mathcal {M}}$. A linear functional $\omega$ on ${\mathcal {M}}$ is called positive if $\omega (X^\dagger X) \geqq 0$ for all $X\in {\mathcal {M}}$, and a positive linear functional $\omega$ on ${\mathcal {M}}$ is a state if $\omega (I)=1$. A $(*)$-isomorphism of ${\mathcal {M}}$ onto ${\mathcal {M}}$ is called a $(*)$-automorphism of ${\mathcal {M}}$ and $\{ \alpha _t\}_{t\in {\mathbb {R}}}$ is called a one-parameter group of $(*)$-automorphisms of ${\mathcal {M}}$ if $\alpha _0(X)=X$ and $\alpha _{s+t}(X)=\alpha _s(\alpha _t(X))$ for all $X\in {\mathcal {M}}$. A one-parameter group $\{ \alpha _t \}_{t\in {\mathbb {R}}}$ of automorphisms of ${\mathcal {M}}$ is weakly (resp. strongly, quasi-strongly) continuous if $\lim _{t\rightarrow 0}\alpha _t(X)=X$ for any $X\in {\mathcal {M}}$ under the weak (resp. strong, quasi-strong) topology. An operator H in ${\mathcal {L}}^\dagger ({\mathcal D})$ is called a weak (resp. strong, quasi-strong) generator for $\{ \alpha _t\}_{t\in {\mathbb {R}}}$ if $\lim _{t\rightarrow 0} \frac{\alpha _t(X)-X}{t}=i[H,X]$ under the weak (resp. strong, quasi-strong) topology. For $O^*$-algebras refer to [23].

3 Gibbs States Defined by Generalized Riesz Systems

Throughout this section let $\{ \varphi _n \}$ be a generalized Riesz system in a Hilbert space $\mathcal H$ with a constructing pair $(\{ f_n \}, T)$ and $\lambda _n>0, \; n=0,1, \ldots$, such that $\{e^{-\frac{\beta }{2} \lambda _n}\}\in \ell ^1$, for some $\beta >0$. In this section we shall define and investigate a Gibbs state $\omega _\varphi ^\beta$ defined through $\{ \varphi _n \}$ on the maximal $O^*$-algebra ${\mathcal {L}}^\dagger ({\mathcal D})$ on a dense subspace ${\mathcal D}$ in $\mathcal H$. We put

$$\begin{aligned} H_0 := \sum _{n=0}^\infty \lambda _n f_n \otimes {\bar{f}}_n , \end{aligned}$$

where, for $x,y \in \mathcal H$, the operator $x\otimes {\bar{y}}$ is defined by

$$\begin{aligned} (x\otimes {\bar{y}})\xi =(\xi |y)x, \quad \xi \in \mathcal H. \end{aligned}$$

Then $H_0$ is a non-singular positive self-adjoint operator in $\mathcal H$ such that

$$\begin{aligned} H_0f_n=\lambda _n f_n, \quad n\in {\mathbb {N}}_0 := {\mathbb {N}} \cup \{ 0 \} , \end{aligned}$$

and it is called a standard hamiltonian for $\{ f_n \}$.

Before entering in the main matter of the paper some comments are in order. Once $H_0$ and a generalized Riesz system $\{ \varphi _n \}$ with constructing pair $(\{ f_n \}, T)$ are given, one can define an operator H on the linear span $D_\varphi$ of $\{ \varphi _n \}$ by putting $H\varphi _n=\lambda _n \varphi _n; \; n\in {\mathbb {N}}_0$ and extending by linearity to $D_\varphi$. Since $D_\varphi$ needs not be dense in $\mathcal H$, it is natural to consider H as an operator acting in $\mathcal H_\varphi$, the closure of $D_\varphi$ in $\mathcal H$. It is then natural to write $H\varphi _n= HTf_n = \lambda _n \varphi _n = TH_0f_n, \; n\in {\mathbb {N}}$, which looks like an intertwining (or, better, when T is invertible, a similarity) condition for H and $\mathcal H_0$, as discussed in [18] for Riesz bases. Similarity is a quite strong condition in particular when considering the spectrum of the involved operators or trying to get a functional calculus. We will not pursue this approach here because it doesn’t fit with the general situation we are considering.

Lemma 3.1

Let ${\mathcal D}$ be a dense subspace in $\mathcal H$. Suppose that

$$\begin{aligned} e^{-\frac{\beta }{2}H_0} \mathcal H\subset {\mathcal D}. \end{aligned}$$

(3.1)

Then $Xe^{-\beta H_0}$ is trace class on $\mathcal H$, for all $X\in {\mathcal {L}}^\dagger ({\mathcal D},\mathcal H)$.

Proof

Take an arbitrary $X\in {\mathcal {L}}^\dagger ({\mathcal D},\mathcal H)$. Since $e^{-\frac{\beta }{2}H_0}\mathcal H\subset {\mathcal D}$, we have

$$\begin{aligned} D(Xe^{-\frac{\beta }{2}H_0} )=\mathcal H. \end{aligned}$$

Thus, $Xe^{-\frac{\beta }{2}H_0}$ is an everywhere defined operator on $\mathcal H$ and it is simple to show that it is closable. Therefore $Xe^{-\frac{\beta }{2}H_0}$ is a closed operator in $\mathcal H$. By the closed graph theorem $Xe^{-\frac{\beta }{2}H_0}$ is a bounded operator on $\mathcal H$ and since

$$\begin{aligned} Xe^{-\beta H_0} = \left( Xe^{-\frac{\beta }{2}H_0} \right) e^{-\frac{\beta }{2}H_0}, \end{aligned}$$

we have $Xe^{-\beta H_0}$ is trace class. This completes the proof.$\square$

We remark that any subspace ${\mathcal D}$ in $\mathcal H$ satisfying (3.1) is dense in $\mathcal H$ since it contains the ONB $\{ f_n \}$ in $\mathcal H$, and $D^\infty (H_0) := \bigcap _{n\in {\mathbb {N}}} D(H_0^n)$ is a subspace in $\mathcal H$ satisfying (3.1).

Under assumption (3.1) we can introduce a state on ${\mathcal {L}}^\dagger ({\mathcal D})$ by

$$\begin{aligned} \omega _f^\beta (X) := \frac{1}{Z_f} \sum _{n=0}^\infty e^{-\beta \lambda _n} (Xf_n|f_n), \quad X\in {\mathcal {L}}^\dagger ({\mathcal D}), \end{aligned}$$

where $Z_f := \sum _{n=0}^\infty e^{-\beta \lambda _n }$. Indeed, by Lemma 3.1 we have

$$\begin{aligned} \mathrm{tr}\; (Xe^{-\beta H_0})= & {} \sum _{n=0}^\infty \left( Xe^{\beta H_0}f_n |f_n \right) \\= & {} \sum _{n=0}^\infty e^{-\beta \lambda _n}(Xf_n|f_n) , \end{aligned}$$

for all $X\in {\mathcal {L}}^\dagger ({\mathcal D})$, and hence $\omega _f^\beta$ is a state on ${\mathcal {L}}^\dagger ({\mathcal D})$, and it is called a Gibbs state on ${\mathcal {L}}^\dagger ({\mathcal D})$ for the ONB $\{ f_n \}$. We formally define a Gibbs state $\omega _\varphi ^\beta$ on ${\mathcal {L}}^\dagger ({\mathcal D})$ for the generalized Riesz system $\{ \varphi _n \}$ by

$$\begin{aligned} \omega _\varphi ^\beta (X) := \frac{1}{Z_\varphi } \sum _{n=0}^\infty e^{-\beta \lambda _n}(X\varphi _n|\varphi _n), \;\; X\in {\mathcal {L}}^\dagger ({\mathcal D}), \end{aligned}$$

where $Z_\varphi := \sum _{n=0}^\infty e^{-\beta \lambda _n} \Vert \varphi _n \Vert ^2$. Conditions for that are discussed in [18]. In what follows we will consider only generalized Riesz system $\{ \varphi _n\}$ for which $Z_\varphi <\infty$.

We do not know whether $\omega _\varphi ^\beta$ is a state on ${\mathcal {L}}^\dagger ({\mathcal D})$, namely, in particular, $|\omega _\varphi ^\beta (X)|<\infty$ for all $X\in {\mathcal {L}}^\dagger ({\mathcal D})$. For that, we assume that a constructing pair $(\{f_n\},T)$ for a generalized Riesz system $\{ \varphi _n\}$ satisfies the following

Assumption 1

There exists a dense subspace ${\mathcal D}$ in $\mathcal H$ satisfying

• (i) $e^{-\frac{\beta }{2}H_0}\mathcal H\subset {\mathcal D}$,

• (ii) ${\mathcal D}\subset D(T)\cap D(T^*)$,

• (iii) $T\lceil _{\mathcal D}$ (the restriction of T to ${\mathcal D}$) $\in {\mathcal {L}}({\mathcal D})$.

By (ii) in Assumption 1, $T\lceil _{\mathcal D}\in {\mathcal {L}}^\dagger ({\mathcal D},\mathcal H)$. In the rest of the paper we will use the same symbol for T, $e^{-\frac{\beta }{2}H_0}$ and $e^{-\beta H_0}$, and for their restrictions to ${\mathcal D}$. Then we have the following

Theorem 3.2

Under Assumption 1, $\omega _\varphi ^\beta$ is a faithful state on ${\mathcal {L}}^\dagger ({\mathcal D})$ and

$$\begin{aligned} \omega _\varphi ^\beta (X)= & {} \frac{1}{Z_\varphi } \;\mathrm{tr}\; \left( T^*XT e^{-\beta H_0} \right) \\= & {} \frac{1}{Z_\varphi } \;\mathrm{tr}\; \left( \left( Te^{-\frac{\beta }{2} H_0} \right) ^*X \left( Te^{-\frac{\beta }{2}H_0}\right) \right) , \end{aligned}$$

for all $X\in {\mathcal {L}}^\dagger ({\mathcal D})$.

Here a state $\omega$ on ${\mathcal {L}}^\dagger ({\mathcal D})$ is said to be faithful if $\omega (X^\dagger X)=0$, $X\in {\mathcal {L}}^\dagger ({\mathcal D})$, then $X=0$.

Proof

Take an arbitrary $X\in {\mathcal {L}}^\dagger ({\mathcal D})$. Then, by Assumption 1, $T^*XT\in {\mathcal {L}}^\dagger ({\mathcal D},\mathcal H)$ and by Lemma 3.1$(T^*XT)e^{-\beta H_0}$ is trace class, which implies that

$$\begin{aligned} \frac{1}{Z_\varphi }\mathrm{tr}\; \left( T^*X Te^{-\beta H_0} \right)= & {} \frac{1}{Z_\varphi } \sum _{n=0}^\infty (T^*XT e^{-\beta H_0}f_n|f_n). \nonumber \\= & {} \frac{1}{Z_\varphi }\sum _{n=0}^\infty e^{-\beta \lambda _n} (T^*XTf_n|f_n) \nonumber \\= & {} \sum _{n=0}^\infty e^{-\beta \lambda _n} (X\varphi _n|\varphi _n), \end{aligned}$$

(3.2)

for all $X\in {\mathcal {L}}^\dagger ({\mathcal D})$. Hence $\omega _\varphi ^\beta (X) = \frac{1}{Z_\varphi } \;\mathrm{tr}\; \left( T^*XT e^{-\beta H_0} \right)$ and it is a state on ${\mathcal {L}}^\dagger ({\mathcal D})$. Since T and $T^*$ are non-singular; that is, $T^{-1}$ and $(T^*)^{-1}$ exist, we see that $\omega _\varphi ^\beta$ is faithful. Furthermore, we have

$$\begin{aligned} \omega _\varphi (X)= & {} \frac{1}{Z_\varphi }\; \mathrm{tr}\; \left( T^*XT e^{-\beta H_0} \right) \\= & {} \frac{1}{Z_\varphi }\;\mathrm{tr}\; \left( \left( T^*XT e^{-\frac{\beta }{2} H_0}\right) e^{-\frac{\beta }{2} H_0} \right) \\= & {} \frac{1}{Z_\varphi } \;\mathrm{tr}\; \left( e^{-\frac{\beta }{2} H_0}(T^*XT)e^{-\frac{\beta }{2} H_0} \right) \\= & {} \frac{1}{Z_\varphi } \;\mathrm{tr}\; \left( \left( Te^{-\frac{\beta }{2} H_0}\right) ^*X \left( Te^{-\frac{\beta }{2} H_0} \right) \right) , \end{aligned}$$

for all $X\in {\mathcal {L}}^\dagger ({\mathcal D})$. This completes the proof. $\square$

Remark

Clearly, $\omega _\varphi (X)= \frac{Z_0}{Z_\varphi } \omega _0 (T^*XT),$ for every $X\in {\mathcal {L}}^\dagger ({\mathcal D})$, where $\omega _0(X)= \frac{1}{Z_0}\mathrm{tr}(Xe^{-\beta H_0})$ and $Z_0=\sum _{n=0}^\infty e^{-\beta \lambda _n}$, as usually introduced in the literature when in presence of a self-adjoint Hamiltonian $H_0$.

Now we put

$$\begin{aligned} \psi _n := (T^{-1})^*f_n , \quad n\in {\mathbb {N}}_0 . \end{aligned}$$

Then $\{ \psi _n \}$ is a generalized Riesz system with a constructing pair $(\{ f_n\} ,(T^{-1})^*)$ and $\{ \varphi _n \}$ and $\{ \psi _n \}$ are biorthogonal sequences. For the constructing operator $(T^{-1})^*$ for $\{ \psi _n \}$, we assume the following, which is completely analogous to what stated in Assumption 1 above.

Assumption 2

Assume that there exists a dense subspace $\mathcal{E}$ in $\mathcal H$ satisfying

• (i) $e^{-\frac{\beta }{2}H_0}\mathcal H\subset \mathcal{E}$,

• (ii) $\mathcal{E}\subset D(T^{-1})\cap D((T^{-1})^*)$,

• (iii) $(T^{-1})^*\lceil _\mathcal{E}\in {\mathcal {L}}(\mathcal{E})$.

As before, we use the same symbol for the operators $(T^{-1})^*$, $e^{-\frac{\beta }{2}H_0}$ and $e^{-\beta H_0}$ and for their restrictions to $\mathcal{E}$. Now we define put

$$\begin{aligned} \omega _\psi ^\beta (X) := \frac{1}{Z_\psi } \sum _{n=0}^\infty e^{-\beta \lambda _n} (\psi _n|\psi _n), \quad X\in {\mathcal {L}}^\dagger (\mathcal{E}), \end{aligned}$$

where $Z_\psi := \sum _{n=0}^\infty e^{-\beta \lambda _n} \Vert \psi _n\Vert ^2$, which is assumed to exist finite, see [18]. Then we have the following

Theorem 3.3

Under Assumption 2, $\omega _\psi ^\beta$ is a faithful state on ${\mathcal {L}}^\dagger (\mathcal{E})$ and

$$\begin{aligned} \omega _\psi (X)= & {} \frac{1}{Z_\psi } \; \mathrm{tr}\; \left( T^{-1} X(T^{-1})^*e^{-\beta H_0} \right) \\= & {} \frac{1}{Z_\psi } \; \mathrm{tr}\; \left( \left( (T^{-1})^*e^{-\frac{\beta }{2} H_0} \right) ^*X(T^{-1})^*e^{-\frac{\beta }{2} H_0} \right) , \end{aligned}$$

for all $X\in {\mathcal {L}}^\dagger (\mathcal{E})$.

Proof

It is proved similarly to Theorem 3.2. $\square$

By Theorems 3.2 and 3.3 we have the following

Corollary 3.4

Let $\{ \varphi _n \}$ and $\{ \psi _n \}$ be biorhotogonal sequences and $\{ \varphi _n \}$ be generalized Riesz system with natural constructing pair $(\{ f_n \},T)$. Suppose that Assumptions 1 and 2 are satisfied, with ${\mathcal D}=\mathcal{E}$. Then the Gibbs states $\omega _\varphi ^\beta$ and $\omega _\psi ^\beta$ are faithful states on ${\mathcal {L}}^\dagger ({\mathcal D})$ satisfying

$$\begin{aligned} \omega _\varphi ^\beta (X)= & {} \frac{1}{Z_\varphi }\;\mathrm{tr}\; \left( (T^*XT)e^{-\beta H_0} \right) \\= & {} \frac{1}{Z_\varphi } \;\mathrm{tr}\; \left( \left( Te^{-\frac{\beta }{2} H_0} \right) ^*X \left( Te^{-\frac{\beta }{2} H_0} \right) \right) \end{aligned}$$

and

$$\begin{aligned} \omega _\psi ^\beta (X)= & {} \frac{1}{Z_\psi }\;\mathrm{tr}\; \left( T^{-1} X(T^{-1})^*e^{-\beta H_0} \right) \\= & {} \frac{1}{Z_\psi } \;\mathrm{tr}\; \left( \left( (T^{-1})^*e^{-\frac{\beta }{2} H_0}\right) ^*X \left( (T^{-1})^*e^{-\frac{\beta }{2} H_0} \right) \right) , \end{aligned}$$

for all $X\in {\mathcal {L}}^\dagger ({\mathcal D})$.

Corollary 3.5

Let $\{ \varphi _n \}$ be a Riesz basis with a constructing pair $(\{ f_n\} ,T)$. Suppose that there exists a dense subspace ${\mathcal D}$ in $\mathcal H$ such that

• (i) $e^{-\frac{\beta }{2} H_0} \mathcal H\subset {\mathcal D}$.

• (ii) $T{\mathcal D}\subset {\mathcal D}$.

• (iii) $(T^{-1})^*{\mathcal D}\subset {\mathcal D}$.

Then the Gibbs states $\omega _\varphi ^\beta$ and $\omega _\psi ^\beta$ are faithful states on ${\mathcal {L}}^\dagger ({\mathcal D})$.

4 Dynamics and KMS-Like Condition

4.1 Standard Heisenberg Time Evolution

Let $H_0$ be a non-singular positive self-adjoint operator in $\mathcal H$ satisfying $H_0=\sum _{n=0}^\infty \lambda _n f_n \otimes f_n$, where $\{ f_n\}$ is an ONB in a Hilbert space $\mathcal H$ and $\{ \lambda _n \}$ is a sequence of strictly positive numbers satisfying $\sum _{n=0}^\infty e^{-\frac{1}{2}\lambda _n}<\infty$, and ${\mathcal D}$ be a dense subspace in $\mathcal H$ such that

$$\begin{aligned} H_0 {\mathcal D}\subset {\mathcal D}\quad \mathrm{and}\quad e^{itH_0}{\mathcal D}\subset {\mathcal D}\quad \mathrm{for \; all} \;\; t \in {\mathbb {R}} . \end{aligned}$$

(4.1)

For example, ${\mathcal D}=D^\infty (H_0):= \cap _{n\in {\mathbb {N}}_0} D(H_0^n)$ satisfies (4.1). Indeed, since $H^n_0 e^{itH_0}x=e^{itH_0} H_0^n x$, $x\in D^\infty (H_0)$ we have $e^{itH_0}x \in D(H_0^n)$, for all $n\in {\mathbb {N}}_0$. Here we put

$$\begin{aligned} \alpha ^0_t(X) := e^{i tH_0}Xe^{-i tH_0}, \quad X\in {\mathcal {L}}^\dagger ({\mathcal D}), \; t\in {\mathbb {R}}. \end{aligned}$$

Then, $\{ \alpha ^0_t \}_{t\in {\mathbb {R}}}$ is a one-parameter group of $*$-automorphisms of ${\mathcal {L}}^\dagger ({\mathcal D})$, and we have the following

Lemma 4.1.1

Suppose that (4.1) is satisfied and that the one-parameter unitary group $\{ e^{i tH_0}\}_{t\in {\mathbb {R}}}$ is quasi-strongly continuous on ${\mathcal {L}}^\dagger ({\mathcal D})$.

Then, $\{ \alpha ^0_t\}_{t\in {\mathbb {R}}}$ is strongly continuous and its weak generator is $H_0$. In particular, if ${\mathcal D}=D^\infty (H_0)$, then $\{ \alpha ^0_t\}_{t\in {\mathbb {R}}}$ is a quasi-strongly continuous one-parameter group of $*$-automorphisms of ${\mathcal {L}}^\dagger (D^\infty (H_0))$ and its quasi-strong generator is $H_0$.

Proof

First, we show that $\{ \alpha ^0_t \}$ is strongly continuous. Take arbitrary $X\in {\mathcal {L}}^\dagger ({\mathcal D})$ and $\xi \in {\mathcal D}$. Then by assumption we have

$$\begin{aligned} \lim _{t\rightarrow 0} \Vert \alpha ^0_t (X)\xi -X\xi \Vert= & {} \lim _{t\rightarrow 0} \Vert e^{i tH_0} Xe^{-i tH_0}\xi -X\xi \Vert \\\leqq & {} \lim _{t\rightarrow 0} \left\{ \Vert e^{i tH_0}Xe^{-i tH_0}\xi -e^{i tH_0 }X\xi \Vert +\Vert e^{i tH_0}X\xi -X\xi \Vert \right\} \\\leqq & {} \lim _{t\rightarrow 0} \Vert Xe^{-i tH_0}\xi -X\xi \Vert +\lim _{t\rightarrow 0} \Vert e^{i tH_0}X\xi -X\xi \Vert \\= & {} 0 . \end{aligned}$$

Thus, $\{ \alpha ^0_t \}_{t\in {\mathbb {R}}}$ is strongly continuous.

Next, we show that $H_0$ is a weak generator of $\{ \alpha ^0_t\}_{t\in {\mathbb {R}}}$. Take arbitrary $X\in {\mathcal {L}}^\dagger ({\mathcal D})$ and $\xi ,\eta \in {\mathcal D}$. Then it follows from our assumptions that

$$\begin{aligned} \left( \frac{\alpha _0^t(X)-X}{t}\xi |\eta \right)= & {} \left( \frac{e^{i tH_0} X e^{-i tH_0}\xi -e^{i tH_0}X\xi +e^{i tH_0}X\xi -X\xi }{t}| \eta \right) \\= & {} \left( \frac{e^{-i tH_0}\xi -\xi }{t}| X^\dagger e^{-i tH_0} \eta \right) +\left( \frac{ e^{ tH_0}X\xi -X\xi }{t}| \eta \right) \\\rightarrow & {} (-iH_0 \xi |X^\dagger \eta )+(iH_0X\xi | \eta ) \quad \mathrm{as}\;\; t\rightarrow 0 \\= & {} (i[H_0,X]\xi |\eta ), \end{aligned}$$

which yields that $H_0$ is a weak generator of $\{ \alpha ^0_t\}_{t\in {\mathbb {R}}}$.

Let ${\mathcal D}=D^\infty (H_0)$ and $t_{H_0}$ be a locally convex topology on ${\mathcal D}$ defined by a sequence $\{ \Vert \cdot \Vert _{H_0^n} ; \; n\in {\mathbb {N}}_0 \}$ of norms on ${\mathcal D}$. Since $H_0^n \in {\mathcal {L}}^\dagger ({\mathcal D})$, for all $n\in {\mathbb {N}}$, we have $t_{H_0} \prec t_{{\mathcal {L}}^\dagger ({\mathcal D})}$. Conversely we show that $t_{{\mathcal {L}}^\dagger ({\mathcal D})} \prec t_{H_0}$. Take an arbitrary $X\in {\mathcal {L}}^\dagger ({\mathcal D})$. Since the identity $\iota$ is a closed map of the Fréchet space ${\mathcal D}[t_{H_0}]$ into the Hilbert space ${\mathcal D}(\Vert \cdot \Vert _{{\bar{X}}})$ with the graph norm $\Vert \cdot \Vert _{{\bar{X}}} :=\Vert \cdot \Vert +\Vert {\bar{X}} \cdot \Vert$, it follows from the closed graph theorem that it is continuous, which implies that $t_{{\mathcal {L}}^\dagger ({\mathcal D})} \prec t_{H_0}$. Thus we have

$$\begin{aligned} t_{H_0}=t_{{\mathcal {L}}^\dagger ({\mathcal D})} \end{aligned}$$

(4.2)

and for any $X\in {\mathcal {L}}^\dagger ({\mathcal D})$ there exist $n\in {\mathbb {N}}$ and $\gamma >0$ such that

$$\begin{aligned} \Vert X\xi \Vert \leqq \gamma \Vert \xi \Vert _{H_0^n} \quad \mathrm{for \; all} \quad \xi \in {\mathcal D}. \end{aligned}$$

(4.3)

Then, for any $X,Y\in {\mathcal {L}}^\dagger ({\mathcal D})$ and $\xi \in {\mathcal D}$ it follows from $H_0^nX\in {\mathcal {L}}^\dagger ({\mathcal D})$ and by our assumptions that

$$\begin{aligned}&\Vert Y\alpha ^0_t(X)\xi -YX\xi \Vert \\&\quad \leqq \gamma \Vert H_0^n \alpha ^0_t(X)\xi -H_0^n X\xi \Vert \\&\quad = \gamma \Vert H_0^n e^{itH_0} Xe^{-itH_0}\xi -H_0^n X\xi \Vert \\&\quad \leqq \gamma \left\{ \Vert H_0^n e^{itH_0}X e^{-itH_0}\xi -e^{itH_0}H_0^n X\xi \Vert +\Vert e^{itH_0}H_0^n X\xi -H_0^n X\xi \Vert \right\} \\&\quad = \gamma \left\{ \Vert H_0^n X(e^{-itH_0}\xi -\xi )\Vert +\Vert e^{itH_0}H_0^n X\xi -H_0^n X\xi \Vert \right\} \\&\qquad \rightarrow 0 \quad \mathrm{as} \;\; t\rightarrow 0, \end{aligned}$$

which implies that $\alpha ^0_t$ is quasi-strongly continuous. Furthermore, we have

$$\begin{aligned}&H_0^m \left( \frac{\alpha _0^t(X)\xi -X\xi }{t}-i[H_0,X]\xi \right) \nonumber \\&\quad = H_0^m \left( \frac{e^{itH_0}Xe^{-itH_0}\xi -e^{itH_0}X\xi +e^{itH_0}X\xi -X\xi }{t}-i[H_0,X]\xi \right) \nonumber \\&\quad = H_0^m \left\{ \left( e^{itH_0}\frac{Xe^{-itH_0}\xi -X\xi }{t}+iXH_0\xi \right) +\left( \frac{e^{itH_0}X\xi -X\xi }{t}-iH_0 X\xi \right) \right\} \nonumber \\&\quad = \left( e^{itH_0}H_0^m \frac{Xe^{-itH_0}\xi -X\xi }{t}+iH_0^mXH_0\xi \right) +H_0^m\left( \frac{e^{itH_0}X\xi -X\xi }{t}-iH_0X\xi \right) . \nonumber \\&\quad =e^{it H_0} \left( H_0^m \frac{Xe^{-itH_0}\xi -X\xi }{t} +iH_0^m XH_0\xi \right) -iH_0^m \left( e^{itH_0}XH_0\xi -XH_0\xi \right) \nonumber \\&\qquad +\,H_0^m\left( \frac{e^{itH_0}X\xi -X\xi }{t}-iH_0X\xi \right) . \end{aligned}$$

(4.4)

Then, it follows from $H_0^m X\in {\mathcal {L}}^\dagger ({\mathcal D})$ and (4.3) that

$$\begin{aligned} \left\| H_0^mX\left( \frac{e^{-itH_0}\xi -\xi }{t}+iH_0\xi \right) \right\|\leqq & {} \gamma ^\prime \left\| H_0^{n^\prime } \left( \frac{e^{-itH_0}\xi -\xi }{t}+iH_0\xi \right) \right\| \\= & {} \gamma ^\prime \left\| \frac{e^{-itH_0}H_0^{n^\prime }\xi -H_0^{n^\prime }\xi }{t}+iH_0H_0^{n^\prime }\xi \right\| \\&\rightarrow \gamma ^\prime \left\| -i H_0H_0^{n^\prime }\xi +iH_0H_0^{n^\prime }\xi \right\| =0 \quad \mathrm{as}\;\; t\rightarrow 0 \end{aligned}$$

and from (ii) that

$$\begin{aligned} H_0^m \left( e^{itH_0}XH_0 \xi -XH_0\xi \right) \rightarrow 0 \quad \mathrm{as}\;\; t\rightarrow 0, \end{aligned}$$

which implies by (4.2) and (4.4) that $\lim _{t\rightarrow 0}\frac{\alpha ^0_t(X)-X}{t}=i[H_0,X]$ under the quasi-strong topology. This completes the proof. $\square$

4.2 The Heisenberg Time Evolution for Generalized Riesz Systems

Let $\{ \varphi _n \}$ be a generalized Riesz system with a constructing pair $(\{ f_n\} , T)$. We assume the following

Assumption 3

There exists a dense subspace ${\mathcal D}$ in $\mathcal H$ such that

• (i) $\{ f_n\} \subset {\mathcal D}$, for all $n\in {\mathbb {N}}_0$,

• (ii) $H_0{\mathcal D}\subset {\mathcal D}$ and $e^{itH_0}{\mathcal D}\subset {\mathcal D}$, for all $t\in {\mathbb {R}}$,

• (iii) $T{\mathcal D}={\mathcal D}$ and $T^*{\mathcal D}={\mathcal D}$,

• (iv) $\{ e^{itH_0} \}_{t\in {\mathbb {R}}}$ is quasi-strongly continuous.

Henceforth we denote an operator $A\lceil _{\mathcal D}\in {\mathcal {L}}^\dagger ({\mathcal D})$ by A for simplicity. Then, we have $\varphi _n$, $\psi _n := (T^\dagger )^{-1} f_n \in {\mathcal D}$, for all $n\in {\mathbb {N}}_0$, and we can define a non-self-adjoint operator H by $H:= TH_0T^{-1}$. Then $H\in {\mathcal {L}}^\dagger ({\mathcal D})$ with $H^\dagger =(T^\dagger )^{-1} H_0 T^\dagger$ and $H\varphi _n =\lambda _n \varphi _n$ and $H^\dagger \psi _n=\lambda _n \psi _n$, $n\in {\mathbb {N}}_0$ (we notice that (iii) implies that $(T^\dagger )^{-1}=(T^*)^{-1}\lceil _{\mathcal D}$ and then $(T^\dagger )^{-1}=(T^{-1})^\dagger$ ). Hence H and $H^\dagger$ can be considered as non-self-adjoint hamiltonians for $\{ \varphi _n \}$ and $\{ \psi _n \}$, respectively. Furthermore, take arbitrary $\xi ,\eta \in {\mathcal D}$ and $t\in {\mathbb {R}}$. By Assumption 3, (iii) there exists a element $\zeta \in {\mathcal D}$ such that $\xi =T\zeta$. Then it follows that

$$\begin{aligned} \left( \left( \sum _{k=0}^n \frac{1}{k!}(it)^k H^k \right) \xi |\eta \right)= & {} \left( T \left( \sum _{k=0}^n \frac{1}{k!} (it)^k H_0^k \right) T^{-1}T \zeta |\eta \right) \\= & {} \left( \sum _{k=0}^n \frac{1}{k!}(it)^k H_0^k \zeta |T^\dagger \eta \right) \\&\rightarrow \left( e^{itH_0}\zeta |T^\dagger \eta \right) \quad \mathrm{as}\;\; n\rightarrow \infty \\= & {} \left( Te^{itH_0}T^{-1}\xi |\eta \right) . \end{aligned}$$

Hence, $T(\sum _{k=0}^n \frac{1}{k!}(it)^k H_0^k)T^{-1}$ converges weakly to $Te^{itH_0}T^{-1}$ on ${\mathcal D}$.

Similarly, $(T^\dagger )^{-1} (\sum _{k=0}^n \frac{1}{k !} (it)^k H_0^k)T^\dagger$ converges weakly to $(T^\dagger )^{-1}e^{itH_0}T^\dagger$ on ${\mathcal D}$. Thus, it is natural to define $e^{itH}$ and $e^{itH^\dagger }$ by

$$\begin{aligned} e^{itH}:= Te^{itH_0}T^{-1} \quad \mathrm{and} \quad e^{itH^\dagger }:= (T^\dagger )^{-1}e^{itH_0}T^\dagger \quad t\in {\mathbb {R}} . \end{aligned}$$

(4.5)

Then we have the following

Lemma 4.2.1

$\{ e^{itH}\}_{t\in {\mathbb {R}}}$ and $\{ e^{itH^\dagger } \}_{t\in {\mathbb {R}}}$ are quasi-strongly continuous one-parameter groups of ${\mathcal {L}}^\dagger ({\mathcal D})$ satisfying $(e^{itH})^\dagger =e^{-itH^\dagger }$, for all $t\in {\mathbb {R}}$.

Proof

By (4.5) it is immediately shown that $\{ e^{itH} \}$ and $\{ e^{itH^\dagger } \}$ are one-parameter groups of ${\mathcal {L}}^\dagger ({\mathcal D})$ satisfying $(e^{itH})^\dagger =e^{-itH^\dagger }$, for all $t\in {\mathbb {R}}$. We show that they are quasi-strongly continuous. Indeed, it follows from Assumption 3, (iv) that for any $X\in {\mathcal {L}}^\dagger ({\mathcal D})$ and $\xi \in {\mathcal D}$

$$\begin{aligned} \lim _{t\rightarrow 0} \Vert Xe^{itH}\xi -X\xi \Vert= & {} \lim _{t\rightarrow 0} \Vert XTe^{itH_0}T^{-1}\xi -X\xi \Vert \\= & {} \lim _{t\rightarrow 0} \Vert XT(e^{itH_0}\eta -\eta )\Vert \\= & {} 0, \end{aligned}$$

where $\eta \in {\mathcal D}$ with $T\eta =\xi$.

Similarly, we can show that $\{ e^{itH^\dagger } \}$ is quasi-strongly continuous. $\square$

We now define what we call the Heisenberg time evolution for $\{ \varphi _n \}$ and $\{ \psi _n \}$ as follows:

$$\begin{aligned} \alpha ^\varphi _t(X) := e^{itH}Xe^{-itH} \quad \mathrm{and}\quad \alpha ^\psi _t(X):= e^{itH^\dagger } Xe^{-itH^\dagger }, \quad X\in {\mathcal {L}}^\dagger ({\mathcal D}), \; t\in {\mathbb {R}} . \end{aligned}$$

By (4.5) we see that

$$\begin{aligned} \alpha ^\varphi _t(X) = e^{itH}Xe^{-itH}=Te^{itH_0}T^{-1}XTe^{-itH_0}T^{-1}=T\alpha ^0_t(T^{-1}XT) T^{-1}, \end{aligned}$$

where $\alpha ^0_t$ was defined before. This is in complete agreement with what originally proposed in [18]. Analogously,

$$\begin{aligned} \alpha ^\psi _t(X)=(T^\dagger )^{-1}\alpha _t^0(T^\dagger X(T^\dagger )^{-1})T^\dagger . \end{aligned}$$

Then we have the following

Theorem 4.2.2

$\{ \alpha ^\varphi _t \}_{t\in {\mathbb {R}}}$ and $\{ \alpha ^\psi _t \}_{t\in {\mathbb {R}}}$ are weakly continuous one-parameter groups of automorphisms of ${\mathcal {L}}^\dagger ({\mathcal D})$ satisfying $\alpha ^\varphi _t(X)^\dagger =\alpha ^\psi _t(X^\dagger )$, for all $X\in {\mathcal {L}}^\dagger ({\mathcal D})$ and $t\in {\mathbb {R}}$. Furthermore their weak generators are H and $H^\dagger$, respectively. Moreover, in particular, if $T\in B(\mathcal H)$ (resp. $T^{-1}\in B(\mathcal H)$), then $\{ \alpha ^\varphi _t\}$ (resp. $\{ \alpha ^\psi _t \}$) is strongly continuous.

Proof

By Lemma 4.2.1, $\{ \alpha ^\varphi _t \}_{t\in {\mathbb {R}}}$ and $\{ \alpha ^\psi _t \}_{t\in {\mathbb {R}}}$ are one-parameter groups of automorphisms of ${\mathcal {L}}^\dagger ({\mathcal D})$ satisfying $\alpha ^\varphi _t(X)^\dagger =\alpha ^\psi _t(X^\dagger )$, for all $X\in {\mathcal {L}}^\dagger ({\mathcal D})$ and $t\in {\mathbb {R}}$. Let us now show that $\{ \alpha ^\varphi _t\}_{t\in {\mathbb {R}}}$ and $\{ \alpha ^\psi _t \}_{t\in {\mathbb {R}}}$ are weakly continuous. Take arbitrary $X\in {\mathcal {L}}^\dagger ({\mathcal D})$ and $\xi ,\eta \in {\mathcal D}$. Since $\alpha ^\varphi _t(X)=T\alpha ^0_t (T^{-1}XT)T^{-1}$, for all $t\in {\mathbb {R}}$, it follows from Lemma 4.1.1 that

$$\begin{aligned} \left( \alpha ^\varphi _t (X)\xi |\eta \right)= & {} \left( T\alpha ^0_t (T^{-1}XT)T^{-1}T \zeta |\eta \right) \\= & {} \left( \alpha ^0_t (T^{-1}XT)\zeta |T^\dagger \eta \right) \\&\rightarrow \left( T^{-1}XT \zeta |T^\dagger \eta \right) \quad \mathrm{as}\;\; t\rightarrow 0 \\= & {} \left( X \xi | \eta \right) , \end{aligned}$$

which yields that $\{ \alpha ^\varphi _t \}_{t\in {\mathbb {R}}}$ is weakly continuous. Next we show that H is a weak generator of $\{ \alpha ^\varphi _t \}_{t\in {\mathbb {R}}}$. By Lemma 4.1.1, it follows that

$$\begin{aligned} \left( \left( \frac{\alpha ^\varphi _t(X)-X}{t} \right) \xi | \eta \right)= & {} \left( \frac{T\alpha ^0_t (T^{-1} X T) T^{-1}-X}{t} T \zeta | \eta \right) \\= & {} \left( \frac{\alpha ^0_t (T^{-1}XT)-T^{-1}XT}{t}\zeta | T^\dagger \eta \right) \\&\rightarrow \left( i[H_0,T^{-1}XT] \zeta | T^\dagger \eta \right) \quad \mathrm{as}\;\; t\rightarrow 0 \\= & {} i \left( T(H_0T^{-1}XT-T^{-1}XTH_0)\zeta | \eta \right) \\= & {} i \left( (HX-XH)T\zeta | \eta \right) \\= & {} i \left( [H,X] \xi |\eta \right) . \end{aligned}$$

Thus, H is a weak generator of $\{ \alpha ^\varphi _t \}_{t\in {\mathbb {R}}}$. Similarly we can show that $\{ \alpha ^\psi _t \}_{t\in {\mathbb {R}}}$ is weakly continuous and its weak generator is $H^\dagger$. Finally, we show that if $T\in B(\mathcal H)$, then $\{ \alpha ^\varphi _t \}_{t\in {\mathbb {R}}}$ is strongly continuous. Take arbitrary $X\in {\mathcal {L}}^\dagger ({\mathcal D})$ and $\xi \in {\mathcal D}$. Then, as usual, there exists an element $\zeta \in {\mathcal D}$ such that $\xi =T\zeta$ and by Assumption 3, (iii) we have

$$\begin{aligned} \Vert \alpha ^\varphi _t(X)\xi -X\xi \Vert= & {} \Vert Te^{itH_0}T^{-1} XTe^{-itH_0}T^{-1}T \zeta -T^{-1}TXT\zeta \Vert \\\leqq & {} \Vert Te^{itH_0}T^{-1}XTe^{-itH_0}\zeta -Te^{itH_0}T^{-1}XT\zeta \Vert \\&+\,\Vert Te^{itH_0}T^{-1}XT \zeta -TT^{-1}XT\zeta \Vert \\\leqq & {} \Vert T\Vert \Vert T^{-1}XTe^{-itH_0}\zeta -T^{-1}XT \zeta \Vert \\&+\,\Vert T\Vert \Vert e^{itH_0}T^{-1}XT \zeta -T^{-1}XT\zeta \Vert \\&\rightarrow 0 \quad \mathrm{as}\;\; t\rightarrow 0 . \end{aligned}$$

Similarly, if $T^{-1}\in B(\mathcal H)$, then we can show that $\{ \alpha ^\psi _t \}_{t\in {\mathbb {R}}}$ is strongly continuous. This completes the proof. $\square$

Next, let us consider the case of ${\mathcal D}=D^\infty (H_0)$. Then, Assumption 3, (i) and (ii) hold automatically, and (iv) holds from (4.1.2). Therefore, the following result easily follows.

Corollary 4.2.3

Suppose that

$$\begin{aligned} TD^\infty (H_0)=D^\infty (H_0) \quad \mathrm{and}\quad T^*D^\infty (H_0)=D^\infty (H_0) . \end{aligned}$$

Then $\{ \alpha ^\varphi _t \}_{t\in {\mathbb {R}}}$ and $\{ \alpha ^\psi _t \}_{t\in {\mathbb {R}}}$ are quasi-strongly continuous and their quasi strong generators are H and $H^\dagger$, respectively.

Proof

Since $t_{{\mathcal {L}}^\dagger ({\mathcal D})}=t_{H_0}$ by (4.2), for any $X\in {\mathcal {L}}^\dagger ({\mathcal D})$ there exist $n\in {\mathbb {N}}$ and $r>0$ such that

$$\begin{aligned} \Vert X\xi \Vert \leqq r\Vert \xi \Vert _{H_0^n} \quad \mathrm{for \; all}\;\; \xi \in {\mathcal D}. \end{aligned}$$

(4.6)

For any $X,Y\in {\mathcal {L}}^\dagger ({\mathcal D})$ and $\xi \in {\mathcal D}$ with $\xi =T\zeta$ for some $\zeta \in {\mathcal D}$, it follows from (4.6) and Assumption 3, (iv) that

$$\begin{aligned} \Vert Y\alpha ^\varphi _t(X)\xi -YX\xi \Vert= & {} \Vert YTe^{itH_0}T^{-1}XTe^{-itH_0}T^{-1}T\zeta -YTT^{-1}XT\zeta \Vert \\\leqq & {} \Vert YT(e^{itH_0}T^{-1}XTe^{-itH_0}\zeta -e^{itH_0}T^{-1}XT\zeta )\Vert \\&+\Vert YT(e^{itH_0}T^{-1}XT\zeta -T^{-1}XT\zeta )\Vert \\\leqq & {} r_1 \{ \Vert H_0^{n_1}e^{itH_0} \left( T^{-1}XTe^{-itH_0} \eta -T^{-1}XT\eta \right) \Vert \\&+\,\Vert H_0^{n_1}( e^{itH_0}T^{-1}XT\eta -T^{-1}XT)\eta \Vert \} \\= & {} r_1 \{ \Vert H_0^{n_1}T^{-1}XT(e^{-itH_0}\zeta -\zeta )\Vert \\&+\,\Vert H_0^{n_1}(e^{itH_0}T^{-1}XT\zeta -T^{-1}XT)\zeta \Vert \} \\\leqq & {} r_1r_2 \Vert H_0^{n_2}(e^{-itH_0}\zeta -\zeta )\Vert \\&+\,r_1\Vert H_0^{n_1}(e^{itH_0}T^{-1}XT\zeta -T^{-1}XT\zeta )\Vert \\= & {} r_1r_2 \Vert (e^{-itH_0}-I)H_0^{n_2}\zeta \Vert \\&+\,r_1\Vert (e^{itH_0}-I)(H_0^{n_1}T^{-1}XT\zeta )\Vert \\&\rightarrow 0 \quad \mathrm{as} \;\; t\rightarrow 0 . \end{aligned}$$

Thus $\{ \alpha ^\varphi _t \}$ is quasi-strongly continuous. We show that the quasi-strong generator of $\{ \alpha ^\varphi _t \}$ equals H. Indeed, take arbitrary $X,Y\in {\mathcal {L}}^\dagger ({\mathcal D})$ and $\xi \in {\mathcal D}$. Then $\xi =T\zeta$ for some $\zeta \in {\mathcal D}$ and by Lemma 4.1.1 the generator of $\{ \alpha ^0_t \}$ equals $H_0$, which yields that

$$\begin{aligned}&\left\| Y \left( \frac{\alpha ^\varphi _t(X)-X}{t} \right) \xi -Y\left( i[H,X] \right) \xi \right\| \\&\quad = \left\| Y\left( \frac{T\alpha ^0_t(T^{-1}XT)T^{-1}-X}{t} \right) T \zeta -iYT[H_0,T^{-1}XT] \zeta \right\| \\&\quad = \left\| YT\left( \frac{\alpha ^0_t(T^{-1}XT)-T^{-1}XT}{t} \zeta -i[H_0,T^{-1}XT]\zeta \right) \right\| \\&\qquad \rightarrow 0 \quad \mathrm{as}\;\; t\rightarrow 0 . \end{aligned}$$

Thus, the quasi-strong generator of $\{ \alpha _t^\varphi \}$is H. Similarly, we can show that $\{ \alpha ^\psi _t \}$ is quasi-strongly continuous and its quasi-strong generator of $\{ \alpha _t^\psi \}$ is $H^\dagger$. This completes the proof.$\square$

4.3 Few Words on Generalized Von Neumann Entropy

In this section we briefly show how what is done with the dynamics can be repeated for the von Neumann entropy. We work here under a slightly generalized version of Assumption 3. In particular, we assume (i) and (iii) hold as in Assumption 3, and that t in (ii) can be complex-valued, $t=t_r+it_i$, with $t_i>0$. More explicitly we assume that $e^{itH_0}{\mathcal D}\subset {\mathcal D}$, for all $t\in {\mathbb {C}}$, with $\mathrm{Im} \; t>0$. Assumption (3.iv) is not relevant for us here, and will not be considered. Our original assumption on the eigenvalues $\lambda _n$, $\sum _{n=0}^\infty e^{-\frac{1}{2}\lambda _n}<\infty$, is here replaced by the stronger assumptions

$$\begin{aligned} \sum _{n=0}^\infty e^{-\gamma \lambda _n}<\infty , \qquad \sum _{n=0}^\infty \lambda _ne^{-\gamma \lambda _n}<\infty , \end{aligned}$$

(4.7)

for all $\gamma >0$. Therefore, in particular we have $Z_0(\beta )= \sum _{n=0}^\infty e^{-\beta \lambda _n}<\infty$. To simplify our treatment, from now on we will assume the following normalization: $Z_0(\beta )=1$. Here $\beta$ is just a positive parameter which, in the following section, will acquire an explicit physical meaning, the inverse temperature of a given system.

The von Neumann entropy connected to the self-adjoint Hamiltonian $H_0$ is defined as

$$\begin{aligned} S_{\rho _0}=-{\mathrm{tr}}\left( \rho _0\log \rho _0\right) , \end{aligned}$$

where, with our normalization, $\rho _0=e^{-\beta H_0}$. A straightforward computation of $S_{\rho _0}$ produces $S_{\rho _0}=\beta \sum _{n=0}^\infty \lambda _ne^{-\gamma \lambda _n}$, which is finite because of our assumption (4.7).

With the same steps as in the definition of $e^{itH}$ and $e^{itH^\dagger }$, using our stronger assumptions, we conclude that $\sum _{k=0}^n \frac{1}{k!}(-\beta )^k H^k$ converges weakly to $Te^{-\beta H_0}T^{-1}=T\rho _0T^{-1}$ on ${\mathcal D}$, and $\sum _{k=0}^n \frac{1}{k !} (-\beta )^k {H^\dagger }^k$ converges weakly to $(T^\dagger )^{-1}e^{-\beta H_0}T^\dagger =(T^\dagger )^{-1}\rho _0T^\dagger$ on ${\mathcal D}$. This suggests to define, in analogy with (4.5),

$$\begin{aligned} \rho = T\rho _0T^{-1}, \qquad \rho ^\dagger =(T^\dagger )^{-1}\rho _0T^\dagger . \end{aligned}$$

Notice now that $(\rho - 1 \!\! 1)^k=T(\rho _0- 1 \!\! 1)^kT^{-1}$, for all $k=0,1,2,\ldots$. Therefore, using the same argument bringing to definitions (4.5), we can check that $\sum _{k=1}^n (-1)^{k-1}\frac{1}{k}(\rho - 1 \!\! 1)^k$ converges weakly to $T\log (\rho _0)T^{-1}$ on ${\mathcal D}$, and $\sum _{k=0}^n (-1)^{k-1}\frac{1}{k}(\rho ^\dagger - 1 \!\! 1)^k$ converges weakly to $(T^\dagger )^{-1}\log (\rho _0)T^\dagger$ on ${\mathcal D}$. Hence we put

$$\begin{aligned} \log \rho :=T\log (\rho _0)T^{-1}, \qquad \log \rho ^\dagger :=(T^\dagger )^{-1}\log (\rho _0)T^\dagger , \end{aligned}$$

and we define a new von Neumann-like entropy as follows:

$$\begin{aligned} S_{\rho }=-\sum _{n}(\psi _n|\rho \log \rho \varphi _n) =-\sum _{n}(\rho ^\dagger \psi _n|(\log \rho )\varphi _n), \end{aligned}$$

under our working assumptions, and in particular the fact that $\rho ^\dagger \psi _n\in {\mathcal D}$ and $(\log \rho )\varphi _n\in {\mathcal D}$, we easily conclude that $S_{\rho }=S_{\rho _0}$.

Remark

It is worth pointing out that even in the cases when $H_0{\mathcal D}\subset {\mathcal D}$, we cannot say that $\log (\rho _0)\in {{\mathcal {L}}}^\dagger (D)$ as it happens if ${\mathcal D}=D^\infty (H_0)$; due to the assumptions on T ($T{\mathcal D}={\mathcal D}; T^*D=D$) this would imply that also $\log (\rho )$ maps ${\mathcal D}$ into ${\mathcal D}$. Nevertheless in the above computations only the action the $\{\varphi _n\}$’s is involved, were everything goes in the appropriate way.

4.4 KMS-Like Condition

In this section we investigate whether the Gibbs state $\omega _\varphi ^\beta$ satisfies the KMS-condition with respect to $\{ \alpha ^\varphi _t \}$, that is, for any $X,Y\in {\mathcal {L}}^\dagger ({\mathcal D})$ there exists a bounded continuous function $f_{X,Y}$ on the strip ${\mathcal {S}}_\beta := \{ z\in {\mathbb {C}}; \; 0 \leqq \mathrm{Im}\; z \leqq \beta \}$ such that

$$\begin{aligned} f_{X,Y}(t)= & {} \omega _\varphi ^\beta (X\alpha ^\varphi _t(Y)), \\ f_{X,Y}(t+\beta i)= & {} \omega _\varphi ^\beta (\alpha ^\varphi _t(Y)X), \end{aligned}$$

for all $t\in {\mathbb {R}}$.

Throughout this section let $\{ \varphi _n \}$ be a generalized Riesz system in $\mathcal H$ with a constructing pair $(\{ f_n\},T)$ satisfying $TD^\infty (H_0)=D^\infty (H_0)$ and $T^*D^\infty (H_0)=D^\infty (H_0)$ and $\beta >0$. Here we put ${\mathcal D}:=D^\infty (H_0)$. Then, since $e^{-\delta H_0}\mathcal H\subset {\mathcal D}$ for any $\delta >0$, it follows from Lemma 3.1 that

$$\begin{aligned} Xe^{-\delta H_0} \; \mathrm{is \; trace \; class \; for \; each} \; \delta >0 \;\mathrm{and} \; X\in {\mathcal {L}}^\dagger ({\mathcal D},\mathcal H) , \end{aligned}$$

(4.8)

and from Corollarys 3.4 and 4.2.3 that $\omega _\varphi ^\beta$ and $\omega _\psi ^\beta$ are faithful states on ${\mathcal {L}}^\dagger ({\mathcal D})$ and $\{ \alpha ^\varphi _t\}_{t\in {\mathbb {R}}}$ and $\{ \alpha ^\psi _t \}_{t\in {\mathbb {R}}}$ are quasi-strongly continuous one-parameter groups of automorphisms of ${\mathcal {L}}^\dagger ({\mathcal D})$. We have the following

Theorem 4.4.1

For any $X,Y\in {\mathcal {L}}^\dagger ({\mathcal D})$ there exists a bounded continuous function $f_{X,Y}$ on the strip ${\mathcal {S}}_\beta$ in ${\mathbb {C}}$ which is analytic on $0< \mathrm{Im} \; z<\beta$ such that

$$\begin{aligned} f_{X,Y}(t)= & {} \omega _\varphi ^\beta (X\alpha ^\varphi _t(Y)) , \\ f_{X,Y}(t+\beta i)= & {} \omega _\varphi ^\beta \left( (TT^\dagger )^{-1}\alpha ^\varphi _t(Y) TT^\dagger X \right) , \end{aligned}$$

for all $t\in {\mathbb {R}}$.

Proof

By Theorem 3.2 we have

$$\begin{aligned} \omega _\varphi ^\beta (X)= & {} \frac{1}{Z_\varphi } \;\mathrm{tr}\; \left( T^\dagger XTe^{-\beta H_0} \right) \nonumber \\= & {} \frac{1}{Z_\varphi } \;\mathrm{tr}\; \left( Te^{-\beta H_0}T^\dagger X \right) \nonumber \\= & {} \frac{1}{Z_\varphi }\;\mathrm{tr}\; \left( e^{-\beta H}TT^\dagger X\right) , \end{aligned}$$

(4.9)

for all $X\in {\mathcal {L}}^\dagger ({\mathcal D})$. In order to define a function $f_{X,Y}$ on the strip ${\mathcal {S}}_\beta$ in ${\mathbb {C}}$, we extend $\alpha _t^\varphi$ to the strip ${\mathcal {S}}_\beta$ as follows:

$$\begin{aligned} \alpha ^\varphi _z(Y)= & {} Te^{izH_0}T^{-1}YTe^{-izH_0}T^{-1} \\= & {} Te^{-sH_0}e^{itH_0}T^{-1}YTe^{-itH_0}e^{sH_0}T^{-1} , \quad z=t+is \in {\mathcal {S}}_\beta \;\;\mathrm{and}\;\; Y\in {\mathcal {L}}^\dagger ({\mathcal D}). \end{aligned}$$

Then, $\alpha _z^\varphi (Y)$ is not necessarily contained in ${\mathcal {L}}^\dagger ({\mathcal D})$. However, we have

$$\begin{aligned} T^\dagger X\alpha ^\varphi _z (Y)Te^{-\beta H_0}= & {} T^\dagger XT e^{-sH_0}e^{itH_0}T^{-1}YTe^{-itH_0}e^{sH_0}e^{-\beta H_0} \\= & {} T^\dagger XTe^{-sH_0}\alpha ^0_t(T^{-1}YT)e^{-(\beta -s)H_0} . \end{aligned}$$

Hence, because of (4.8), taking into account that $T^\dagger XTe^{-sH_0}\alpha ^0_t(T^{-1}YT)\in {\mathcal {L}}^\dagger ({\mathcal D})$), we conclude that $T^\dagger X\alpha ^\varphi _z (Y)Te^{-\beta H_0}$ is trace class. Now we put

$$\begin{aligned} f_{X,Y}(z) := \frac{1}{Z_\varphi }\;\mathrm{tr}\; \left( T^\dagger X\alpha ^\varphi _z (Y)Te^{-\beta H_0} \right) , \quad z\in {\mathcal {S}}_\beta . \end{aligned}$$

(4.10)

Then $f_{X,Y}(z)$ is analytic on $z\in {\mathcal {S}}_\beta$ with $0< \mathrm{Im}\; z<\beta$. Indeed, take arbitrary a sufficient small constant $\delta >0$ ($0<\delta <\beta$). Then we have

$$\begin{aligned} f_{X,Y}(z)= & {} \frac{1}{Z_\varphi }\;\mathrm{tr}\; \left( T^\dagger XTe^{izH_0}T^{-1}YTe^{-izH_0}T^{-1}Te^{-\beta H_0} \right) \\= & {} \frac{1}{Z_\varphi }\;\mathrm{tr}\; \left( T^\dagger XTe^{izH_0}T^{-1}YTe^{-izH_0}T^{-1}Te^{-(\beta -\delta ) H_0}e^{-\delta H_0} \right) \\= & {} \frac{1}{Z_\varphi }\;\mathrm{tr}\; \left( e^{-(\beta -\delta )H_0} \left( e^{-\frac{\delta }{2}H_0}T^\dagger XT \right) e^{izH_0} \left( T^{-1}YTe^{-\frac{\delta }{2}H_0} \right) e^{-izH_0} \right) \\= & {} \frac{1}{Z_\varphi }\;\mathrm{tr}\; \left( e^{-(\beta -\delta )H_0} \left( e^{-\frac{\delta }{2}H_0}T^{-1}XT \right) \alpha _z^0 \left( T^{-1}YT e^{-\frac{\delta }{2}H_0} \right) \right) , \\= & {} \frac{1}{Z_\varphi }\;\mathrm{tr}\; \left( e^{-(\beta -\delta )H_0} A \alpha _z^0 (B) \right) , \end{aligned}$$

where $A:=\left( e^{-\frac{\delta }{2}H_0}T^{-1}XT \right)$ and $B:=\left( T^{-1}YT e^{-\frac{\delta }{2}H_0} \right)$. By (4.8), A and B are trace class. Hence it is known that

$$\begin{aligned} z \rightarrow \mathrm{tr}\; \left( e^{-(\beta -\delta )H_0}A \alpha _0^z (B) \right) \;\;\mathrm{is \; analytic \; on} \; {\mathcal {S}}_{\beta -\delta } \;\mathrm{with}\; 0< \mathrm{Im}z <\beta -\delta \end{aligned}$$

(4.11)

(see 4.3 in [24]). Then for any $z_0 \in {\mathcal {S}}_\beta$ with $0< \mathrm{Im}\; z_0 <\beta$ there exists a constant $\delta >0$ such that $\mathrm{Im}\; z_0 <\beta -\delta$. By (4.11), $f_{X,Y}$ is analytic at $z_0$. Thus $f_{X,Y}$ is analytic on ${\mathcal {S}}_\beta$ with $0< \mathrm{Im}\; z < \beta$. Furthermore, by (4.9) we have

$$\begin{aligned} f_{X,Y}(t)= & {} \frac{1}{Z_\varphi }\;\mathrm{tr}\; \left( T^\dagger X\alpha ^\varphi _t(Y)Te^{-\beta H_0} \right) \\= & {} \frac{1}{Z_\varphi }\;\mathrm{tr}\; \left( Te^{-\beta H_0}T^\dagger X\alpha ^\varphi _t(Y) \right) \\= & {} \omega _\varphi ^\beta \left( X\alpha ^\varphi _t(Y)\right) \end{aligned}$$

and

$$\begin{aligned}&f_{X,Y}(t+\beta i)\\&\quad =\frac{1}{Z_\varphi }\;\mathrm{tr}\; \left( Te^{-\beta H_0}T^{-1} TT^\dagger X \left( Te^{-\beta H_0}T^{-1} \right) \right. \\&\qquad \left. \left( Te^{itH_0} \right) T^{-1} Y \left( Te^{-itH_0}T^{-1} \right) \left( Te^{-\beta H_0}T^{-1} \right) \right) \\&\quad = \frac{1}{Z_\varphi } \;\mathrm{tr}\; \left( TT^\dagger X \left( Te^{-\beta H_0}T^{-1} \right) T\alpha _t^0 \left( T^{-1}YT \right) T^{-1} \right) \\&\quad = \frac{1}{Z_\varphi }\;\mathrm{tr}\; \left( e^{-\beta H} T\alpha _t^0 (T^{-1}YT)T^\dagger X \right) \\&\quad = \frac{1}{Z_\varphi } \;\mathrm{tr}\; \left( e^{-\beta H} \alpha _t^\varphi (Y)TT^\dagger X \right) \\&\quad = \frac{1}{Z_\varphi }\;\mathrm{tr}\; \left( e^{-\beta H} TT^\dagger (TT^\dagger )^{-1} \alpha _t^\varphi (Y)TT^\dagger X \right) \\&\quad = \omega _\varphi ^\beta \left( (TT^\dagger )^{-1} \alpha _t^\varphi (Y)TT^\dagger X \right) . \end{aligned}$$

This completes the proof. $\square$

Thus $\omega _\varphi ^\beta$ does not satisfy the KMS-condition with respect to $\{ \alpha ^\varphi _t \}$, but still it satisfies the KMS-like condition with respect to $\{ \alpha ^\varphi _t \}$, as Theorem 4.4.1 shows. Furthermore, we have a similar result for the Gibbs state $\omega _\psi ^\beta$ as follows:

Theorem 4.4.2

For any $X,Y\in {\mathcal {L}}^\dagger ({\mathcal D})$ there exists a bounded continuous function $F_{X,Y}$ on the strip ${\mathcal {S}}_\beta$ in ${\mathbb {C}}$ which is analytic on $0< \mathrm{Im}\; z<\beta$ such that

$$\begin{aligned} F_{X,Y}(t)= & {} \omega _\psi ^\beta (X\alpha ^\psi _t(Y)) , \\ F_{X,Y}(t+\beta i)= & {} {\omega _\psi ^\beta } \left( (TT^\dagger )\alpha ^\psi _t (Y)(TT^\dagger )^{-1}X \right) , \end{aligned}$$

for all $t\in {\mathbb {R}}$.

Remark

We do not know whether Theorems 4.4.1 and 4.4.2 hold for a general subspace ${\mathcal D}$ satisfying Assumption 3. This is because we do not know whether (4.10) holds for unbounded operators $T^\dagger X\alpha _z^\varphi (Y)T$.

5 Gibbs States and Unbounded Tomita–Takesaki Theory

5.1 Unbounded Tomita–Takesaki Theory in Hilbert Space of Hilbert–Schmidt Operators

In this subsection we review the basic definitions and results of unbounded Tomita–Takesaki theory in the Hilbert space of Hilbert–Schmidt operators. For details refer to [25]. Let $\mathcal H$ be a separable Hilbert space and $\mathcal H\otimes {\bar{\mathcal H}}$ be the Hilbert space of all Hilbert–Schmidt operators on $\mathcal H$ with the inner product

$$\begin{aligned} (S|T):= \;\mathrm{tr}\; (T^*S), \quad S,T\in \mathcal H\otimes {\bar{\mathcal H}}. \end{aligned}$$

Let ${\mathcal D}$ be a dense subspace in $\mathcal H$ such that ${\mathcal {L}}^\dagger ({\mathcal D})$ is closed, namely ${\mathcal D}=\cap _{X\in {\mathcal {L}}^\dagger ({\mathcal D})}D({\bar{X}})$. We define a dense subspace $\sigma _2({\mathcal D})$ of $\mathcal H\otimes {\bar{\mathcal H}}$ by

$$\begin{aligned} \sigma _2({\mathcal D}) := \{ T\in \mathcal H\otimes {\bar{\mathcal H}} ; \; T\mathcal H\subset {\mathcal D}\;\mathrm{and} \; XT\in \mathcal H\otimes {\bar{\mathcal H}} \;\mathrm{for \; all}\; X\in {\mathcal {L}}^\dagger ({\mathcal D}) \} \end{aligned}$$

and an operator $\pi (X)$ on $\sigma _2({\mathcal D})$ by

$$\begin{aligned} \pi (X):=XT, \quad X\in {\mathcal {L}}^\dagger ({\mathcal D}),\; T\in \sigma _2({\mathcal D}). \end{aligned}$$

Then $\pi$ is a $*$-homomorphism of the $O^*$-algebra ${\mathcal {L}}^\dagger ({\mathcal D})$ into the $O^*$-algebra ${\mathcal {L}}^\dagger ( \sigma _2({\mathcal D}))$, and hence $\pi ({\mathcal {L}}^\dagger ({\mathcal D}))$ is an $O^*$-algebra on $\sigma _2({\mathcal D})$ in $\mathcal H\otimes {\bar{\mathcal H}}$. We can also define a bounded $*$-homomorphism $\pi ^{\prime \prime }$ and an anti $*$-homomorphism $\pi ^\prime$ of $B(\mathcal H)$ into the $C^*$-algebra $B(\mathcal H\otimes {\bar{\mathcal H}})$ by

$$\begin{aligned} \pi ^{\prime \prime }(A)T=AT \quad \mathrm{and}\quad \pi ^\prime (A)T=TA, \quad A\in B(\mathcal H) , \; T\in \mathcal H\otimes {\bar{\mathcal H}}, \end{aligned}$$

and $\pi ^{\prime \prime }(B(\mathcal H))$ and $\pi ^\prime (B(\mathcal H))$ are von Neumann algebras on $\mathcal H\otimes {\bar{\mathcal H}}$ satisfying $\pi ^\prime (B(\mathcal H))=\pi ^{\prime \prime }(B(\mathcal H))^\prime =J\pi ^{\prime \prime }(B(\mathcal H))J$, where $JT=T^*$ for any $T\in \mathcal H\otimes {\bar{\mathcal H}}$. Then it follows from Lemma 2.4.14 in [25] that

$$\begin{aligned} \pi ({\mathcal {L}}^\dagger ({\mathcal D}))_w^\prime =\pi ^\prime (B(\mathcal H)) \;\;\mathrm{and}\;\; \left( \pi ({\mathcal {L}}^\dagger ({\mathcal D}))^\prime _w \right) ^\prime =\pi ^{\prime \prime }(B(\mathcal H)). \end{aligned}$$

(5.1)

Suppose that $\Omega$ is a non-singular positive self-adjoint operator on $\mathcal H$ belonging to $\sigma _2({\mathcal D})$. Then, it follows from Lemma 2.4.16 in [25] that $\Omega$ is a strongly cyclic vector for the $O^*$-algebra $\pi ({\mathcal {L}}^\dagger ({\mathcal D}))$ (namely, $\pi ({\mathcal {L}}^\dagger ({\mathcal D}))\Omega$ is $t_{\pi ({\mathcal {L}}^\dagger ({\mathcal D}))}$-dense in $\mathcal H\otimes {\bar{\mathcal H}}$) and $\pi ({\mathcal {L}}^\dagger ({\mathcal D}))^\prime _w \Omega$ is dense in $\mathcal H\otimes {\bar{\mathcal H}}$, and hence it is a cyclic and separating vector for the von Neumann algebra $\pi ^{\prime \prime }(B(\mathcal H))$, which implies that $\pi ^{\prime \prime }(B(\mathcal H))\Omega$ is a left Hilbert algebra in $\mathcal H\otimes {\bar{\mathcal H}}$ under the following multiplication and involution:

$$\begin{aligned} \left( \pi ^{\prime \prime }(A)\Omega \right) \left( \pi ^{\prime \prime }(B)\Omega \right):= & {} \pi ^{\prime \prime }(AB)\Omega , \\ \left( \pi ^{\prime \prime }(A)\Omega \right) ^\sharp:= & {} \pi ^{\prime \prime }(A^*)\Omega , \quad A,B\in B(\mathcal H) . \end{aligned}$$

Let $S_{{\mathfrak {A}}}^{\prime \prime }=J_\Omega ^{\prime \prime }\bigtriangleup _\Omega ^{\prime \prime \frac{1}{2}}$ be the polar decomposition of the conjugate linear closed operator $S_\Omega ^{\prime \prime }$ which is the closure of the involution $\pi ^{\prime \prime }(A)\Omega \rightarrow \pi ^{\prime \prime }(A^*)\Omega$. Then $J_\Omega ^{\prime \prime }$ is a conjugate linear isometry on $\mathcal H\otimes {\bar{\mathcal H}}$ and $\bigtriangleup _\Omega ^{\prime \prime }$ is a non-singular positive self-adjoint operator in $\mathcal H\otimes {\bar{\mathcal H}}$ and they are called the modular conjugation and the modular operator of the left Hilbert algebra $\pi ^{\prime \prime }(B(\mathcal H))\Omega$. By the Tomita theorem a strongly continuous one-parameter group $\{ (\delta _t^\Omega )^{\prime \prime } \}_{t\in {\mathbb {R}}}$ of the von Neumann algebra $\pi ^{\prime \prime }(B(\mathcal H))$ is defined by

$$\begin{aligned} (\delta _t^\Omega )^{\prime \prime } (\pi ^{\prime \prime }(A)) = \bigtriangleup _\Omega ^{\prime \prime \; it} \pi ^{\prime \prime }(A)\bigtriangleup _\Omega ^{\prime \prime \; -it}, \quad A\in B(\mathcal H),\; t\in {\mathbb {R}}, \end{aligned}$$

and it is called the modular automorphism group of $\pi ^{\prime \prime }(B(\mathcal H))$. For the Tomita–Takesaki theory we refer to [26]. Then it follows from Theorem 2.4.18 in [25] that

$$\begin{aligned} J^{\prime \prime }_\Omega =J \quad \mathrm{and}\quad \bigtriangleup ^{\prime \prime }_\Omega =\pi ^\prime (\Omega ^{-2} ) \pi ^{\prime \prime }(\Omega ^2), \end{aligned}$$

(5.2)

where the positive self-adjoint operator $\pi ^\prime (\Omega ^{-2})$ is defined by

$$\begin{aligned} \left\{ \begin{array}{ccc} &{}D(\pi ^\prime (\Omega ^{-2})) = \{ T\in \mathcal H\otimes {\bar{\mathcal H}}; T\Omega ^{-2} \;\mathrm{is \; closable \; and} \; \overline{T\Omega ^{-2}} \in \mathcal H\otimes {\bar{\mathcal H}} \} \\ &{}\pi ^\prime (\Omega ^{-2})T = \overline{T\Omega ^{-2}} , T\in D(\pi ^\prime (\Omega ^{-2})). \\ \end{array} \right. \end{aligned}$$

By (5.1) we have

$$\begin{aligned} \pi ({\mathcal {L}}^\dagger ({\mathcal D}))^\prime _w \Omega= & {} \pi ^\prime (B(\mathcal H))\Omega , \\ (\pi ({\mathcal {L}}^\dagger ({\mathcal D}))^\prime _w)^\prime \Omega= & {} \pi ^{\prime \prime }(B(\mathcal H))\Omega , \end{aligned}$$

and so the involution: $\pi (X)\Omega \rightarrow \pi (X^\dagger )\Omega$, $X\in {\mathcal {L}}^\dagger ({\mathcal D})$ is a conjugate linear closable operator in $\mathcal H\otimes {\bar{\mathcal H}}$ and its closure is denoted by $S_{{\mathfrak {A}}}$. Let $S_{{\mathfrak {A}}}=J_{{\mathfrak {A}}}\bigtriangleup _{{\mathfrak {A}}}^{\frac{1}{2}}$ be the polar decomposition of $S_{{\mathfrak {A}}}$. Then we can show that $S_{{\mathfrak {A}}}=S_{{\mathfrak {A}}}^{\prime \prime }$, and so $J_{{\mathfrak {A}}}=J_{{\mathfrak {A}}}^{\prime \prime }$ and $\bigtriangleup _{{\mathfrak {A}}}=\bigtriangleup _{{\mathfrak {A}}}^{\prime \prime }$. Hereafter, we use $S_{{\mathfrak {A}}}$, $J_{{\mathfrak {A}}}$, $\bigtriangleup _{{\mathfrak {A}}}$ and $\{\delta _t^\Omega \}_{t\in {\mathbb {R}}}$. Suppose that $\Omega ^{it} {\mathcal D}\subset {\mathcal D}$, for all $t\in {\mathbb {R}}$, namely $\Omega ^{it}\in {\mathcal {L}}^\dagger ({\mathcal D})$. Then since

$$\begin{aligned} \bigtriangleup _\Omega ^{it} =\pi ^\prime (\Omega ^{-2it})\pi ^{\prime \prime }(\Omega ^{2it}) = \pi ^\prime (\Omega ^{-2it})\pi (\Omega ^{2it})\in {\mathcal {L}}^\dagger ({\mathcal D}), \;\; t\in {\mathbb {R}} \end{aligned}$$

by (5.2), we can define a one-parameter group $\{\sigma _t^\Omega \}_{t\in {\mathbb {R}}}$ of the $O^*$-algebra $\pi ({\mathcal {L}}^\dagger ({\mathcal D}))$ by

$$\begin{aligned} \sigma _t^\Omega (\pi (X)) := \bigtriangleup ^{it}_\Omega \pi (X) \bigtriangleup _\Omega ^{-it}, \quad X\in {\mathcal {L}}^\dagger ({\mathcal D}), \; t\in {\mathbb {R}}, \end{aligned}$$

and we see that

$$\begin{aligned} \sigma _t^\Omega (\pi (X))= & {} \pi ^\prime (\Omega ^{-2it})\pi (\Omega ^{2it})\pi (X) \pi ^\prime (\Omega ^{2it})\pi (\Omega ^{-2it}) \\= & {} \pi (\Omega ^{2it})\pi (X) \pi (\Omega ^{-2it}) \\= & {} \pi (\Omega ^{2it}X\Omega ^{-2it}), \end{aligned}$$

for all $X\in {\mathcal {L}}^\dagger ({\mathcal D})$ and $t\in {\mathbb {R}}$. This $\{ \sigma _t^\Omega \}$ is called the modular automorphism group of $\pi ({\mathcal {L}}^\dagger ({\mathcal D}))$. Thus we have the following

Proposition 5.1.1

Suppose that $\Omega$ is a non-singular positive self-adjoint operator on $\mathcal H$ belonging to $\sigma _2({\mathcal D})$ and $\Omega ^{it} {\mathcal D}\subset {\mathcal D}$, for all $t\in {\mathbb {R}}$. Then

$$\begin{aligned} \sigma _t^\Omega (X) := \Omega ^{it}X\Omega ^{-it}, \quad X\in {\mathcal {L}}^\dagger ({\mathcal D}), \; t\in {\mathbb {R}} \end{aligned}$$

is a one-parameter group of $*$-automorphisms of ${\mathcal {L}}^\dagger ({\mathcal D})$, which is induced by the modular automorphism group $\{ \sigma _t^\Omega \}_{t\in {\mathbb {R}}}$ of $\pi ({\mathcal {L}}^\dagger ({\mathcal D}))$.

5.2 Modular Automorphism Group Defined by the Gibbs State $\omega _\varphi ^\beta$

Let $\{ \varphi _n \}$ be a generalized Riesz system with a constructing pair $(\{ f_n \} ,T)$ and $H_0$ be a standard Hamiltonian. We assume the following

Assumption 4

There exists a dense subspace ${\mathcal D}$ in $\mathcal H$ such that

• (i) $e^{-\frac{\beta }{2}H_0}\mathcal H\subset {\mathcal D}\subset D(T)\cap D(T^*)$,

• (ii) $T\lceil _{\mathcal D}\in {\mathcal {L}}({\mathcal D})$,

• (iii) ${\mathcal {L}}^\dagger ({\mathcal D})$ is self-adjoint, namely ${\mathcal D}=\cap _{X\in {\mathcal {L}}^\dagger ({\mathcal D})}D(X^*)$.

As seen in Sect. 3, the Gibbs state $\omega _0$ on ${\mathcal {L}}^\dagger ({\mathcal D})$ is defined by

$$\begin{aligned} \omega _0(X) =\frac{1}{Z_0}\;\mathrm{tr}\; \left( e^{-\frac{\beta }{2}H_0}Xe^{-\frac{\beta }{2}H_0} \right) , \quad X\in {\mathcal {L}}^\dagger ({\mathcal D}). \end{aligned}$$

Then we see that

$$\begin{aligned} \Omega _0 := \frac{1}{\sqrt{Z_0}}e^{-\frac{\beta }{2}H_0} \in \left( \sigma _2({\mathcal {L}}^\dagger ({\mathcal D})) \right) _+ \end{aligned}$$

and

$$\begin{aligned} \omega _0(X) = ( \pi (X)\Omega _0 | \Omega _0) , \quad X\in {\mathcal {L}}^\dagger ({\mathcal D}) . \end{aligned}$$

By Proposition 5.1.1, the modular automorphism group $\{ \sigma _t^{\Omega _0}\}_{t\in {\mathbb {R}}}$ of $\pi ({\mathcal {L}}^\dagger ({\mathcal D}))$ defined by $\omega _0$ coincides with $\{ \alpha _t^0 \}_{t\in {\mathbb {R}}}$. By Theorem 3.2, the Gibbs state $\omega _\varphi ^\beta$ for $\{ \varphi _n \}$ is defined by

$$\begin{aligned} \omega _\varphi ^\beta (X) = \frac{1}{Z_\varphi } \;\mathrm{tr}\; \left( \left( Te^{-\frac{\beta }{2} H_0} \right) ^*X \left( Te^{-\frac{\beta }{2}H_0}\right) \right) , \end{aligned}$$

for all $X\in {\mathcal {L}}^\dagger ({\mathcal D})$. Here we shall extend results in Sect. 3 for the Gibbs state $\omega _\varphi ^\beta$ on ${\mathcal {L}}^\dagger ({\mathcal D})$.

Let $(Te^{-\frac{\beta }{2}H_0})^*=U|(Te^{-\frac{\beta }{2}H_0})^*|$ be the polar decomposition of $(Te^{-\frac{\beta }{2}H_0})^*$. By Lemma 3.1 and Assumption 4, (i) and (ii), we have

$$\begin{aligned} \left( Te^{-\frac{\beta }{2}H_0} \right) \left( Te^{-\frac{\beta }{2}H_0} \right) ^*\mathcal H\subset T \left( e^{-\frac{\beta }{2}H_0} \left( Te^{-\frac{\beta }{2}H_0} \right) ^*\right) {\mathcal H} \subset T{\mathcal D}\subset {\mathcal D}\end{aligned}$$

and since ${\mathcal {L}}^\dagger ({\mathcal D})$ is a self-adjoint $O^*$-algebra on ${\mathcal D}$, it follows by Lemma 2.4 in [27] that

$$\begin{aligned} \left| \left( Te^{-\frac{\beta }{2}H_0} \right) ^*\right| \mathcal H= \left( \left( Te^{-\frac{\beta }{2}H_0} \right) \left( Te^{-\frac{\beta }{2}H_0} \right) ^*\right) ^{\frac{1}{2}} \mathcal H\subset {\mathcal D}. \end{aligned}$$

(5.3)

From the above, we put

$$\begin{aligned} \Omega _\varphi := \frac{1}{\sqrt{Z_\varphi }} \left| (Te^{-\frac{\beta }{2}H_0})^*\right| . \end{aligned}$$

Since $XT\in {\mathcal {L}}^\dagger ({\mathcal D},\mathcal H)$ for all $X\in {\mathcal {L}}^\dagger ({\mathcal D})$ by Assumption 4, (i), it follows from Lemma 3.1 that

$$\begin{aligned} X\Omega _\varphi= & {} \frac{1}{\sqrt{Z_\varphi }} X \left| (Te^{-\frac{\beta }{2}H_0})^*\right| \\= & {} \frac{1}{\sqrt{Z_\varphi }}X Te^{-\frac{\beta }{2}H_0}U \in \mathcal H\otimes {\bar{\mathcal H}}, \end{aligned}$$

for all $X\in {\mathcal {L}}^\dagger ({\mathcal D})$, which implies by (5.3) that $\Omega _\varphi \in \sigma _2({\mathcal D})$. Thus, $\Omega _\varphi$ is a non-singular positive self-adjoint Hilbert Schmidt operator on $\mathcal H$ contained in $\sigma _2 ({\mathcal D})$. Hence, $\Omega _\varphi$ is a strongly cyclic and separating vector for $\pi ({\mathcal {L}}^\dagger ({\mathcal D}))$ and

$$\begin{aligned} \omega _\varphi ^\beta (X)= & {} \;\mathrm{tr}\; \left( U\Omega _\varphi X\Omega _\varphi U^*\right) \\= & {} \;\mathrm{tr}\; (\Omega _\varphi X \Omega _\varphi ) \\= & {} (\pi (X)\Omega _\varphi | \Omega _\varphi ), \quad X\in {\mathcal {L}}^\dagger ({\mathcal D}) . \end{aligned}$$

Remark

The previous expression for $\omega _\varphi$ is, of course, the GNS representation (up to unitary equivalences) and the cyclic and separating vector $\Omega _\varphi$ which is actually an operator in $\sigma _2({\mathcal D})$ helps with identifying the density operator $\rho$ for which one can write $\omega _\varphi ^\beta (X) = \mathrm{tr}\;(X\rho )$.

By Proposition 5.1.1, we have the following

Theorem 5.2.1

Suppose that $\{ \varphi _n \}$ be a generalized Riesz system with a constructing pair $(\{ f_n \} ,T)$ and there exists a dense subspace ${\mathcal D}$ in $\mathcal H$ satisfying Assumption 4. Then $\Omega _\varphi :=\frac{1}{\sqrt{Z_\varphi }} |(Te^{-\frac{\beta }{2}H_0})^*|$ is a non-singular strongly cyclic and separating vector for $\pi ({\mathcal {L}}^\dagger ({\mathcal D}))$ contained in $\sigma _2({\mathcal D})$ and the Gibbs state $\omega _\varphi ^\beta$ is represented as

$$\begin{aligned} \omega _\varphi ^\beta (X) =(\pi (X)\Omega _\varphi |\Omega _\varphi ), \quad X\in {\mathcal {L}}^\dagger ({\mathcal D}). \end{aligned}$$

Furthermore, if $\Omega _\varphi ^{it} {\mathcal D}\subset {\mathcal D}$ for all $t\in {\mathbb {R}}$, then $\sigma _t^{\Omega _\varphi }:=\Omega _\varphi ^{it}X\Omega _\varphi ^{it}$, $X\in {\mathcal {L}}^\dagger ({\mathcal D})$, $t\in {\mathbb {R}}$ is a one-parameter group of $*$-automorphisms of ${\mathcal {L}}^\dagger ({\mathcal D})$, which is induced by the modular automorphism group

$$\begin{aligned} \sigma _t^{\Omega _\varphi }(\pi (X)) := \bigtriangleup _{\Omega _\varphi }^{it}\pi (X)\bigtriangleup _{\Omega _\varphi }^{-it}, \quad t\in {\mathbb {R}} \end{aligned}$$

of $\pi ({\mathcal {L}}^\dagger ({\mathcal D}))$.

Remark

If ${\bar{T}}$ commutes to $e^{-H_0}$, that is $e^{-H_0} {\bar{T}} \subset {\bar{T}}e^{-H_0}$, then $\alpha _{ t}^\varphi (X) =|T^*|^{it} \sigma _{2t}^\varphi (X)|T^*|^{-it}$, for all $X\in {\mathcal {L}}^\dagger ({\mathcal D})$ and $t\in {\mathbb {R}}$. Since $\sigma _t^\varphi$ is a $*$-automorphism of ${\mathcal {L}}^\dagger ({\mathcal D})$, but $\alpha _t^\varphi$ is not a $*$-automorphism, these two one-parameter groups $\{ \sigma _t^\varphi \}$ and $\{ \alpha _t^\varphi \}$ of automorphisms of ${\mathcal {L}}^\dagger ({\mathcal D})$ have no relation in general.

For the Gibbs state $\omega _\psi ^\beta$ on ${\mathcal {L}}^\dagger ({\mathcal D})$ we similarly have the following

Theorem 5.2.2

Let $\{ \varphi _n \}$ be a generalized Riesz system with a constructing pair $(\{ f_n\} ,T)$, $n\in {\mathbb {N}}_0$. Suppose that there exists a dense subspace ${\mathcal D}$ in $\mathcal H$ satisfying

• (i) $e^{-\frac{\beta }{2}H_0}\mathcal H\subset {\mathcal D}\subset D(T^{-1})\cap D((T^{-1})^*)$,

• (ii) $(T^{-1})^*\lceil _{\mathcal D}\in {\mathcal {L}}({\mathcal D})$,

• (iii) ${\mathcal {L}}^\dagger ({\mathcal D})$ is self-adjoint.

Then $\Omega _\psi := \frac{1}{\sqrt{Z_\psi }} \left| \left( (T^{-1})^*e^{-\frac{\beta }{2}H_0}\right) ^*\right|$ is a non-singular strongly cyclic and separating vector for $\pi ({\mathcal {L}}^\dagger ({\mathcal D}))$ contained in $\sigma _2({\mathcal D})$ and the Gibbs state $\omega _\psi ^\beta$ is represented as

$$\begin{aligned} \omega _\psi ^\beta (X) =(\pi (X)\Omega _\psi | \Omega _\psi ),\quad X\in {\mathcal {L}}^\dagger ({\mathcal D}) . \end{aligned}$$

Furthermore, if $\Omega _\psi ^{it}{\mathcal D}\subset {\mathcal D}$ for all $t\in {\mathbb {R}}$, then

$$\begin{aligned} \sigma _t^{\Omega _\psi }(X):= \Omega _\psi ^{it}X\Omega _\psi ^{-it}, \quad X\in {\mathcal {L}}^\dagger ({\mathcal D}), \; t\in {\mathbb {R}} \end{aligned}$$

is a one-parameter group of $*$-automorphisms of ${\mathcal {L}}^\dagger ({\mathcal D})$, which is induced by the modular automorphism group

$$\begin{aligned} \sigma _t^{\Omega _\psi }(\pi (X)) := \bigtriangleup _{\Omega _\psi }^{it} \pi (X) \bigtriangleup _{\Omega _\psi }^{-it},\quad t\in {\mathbb {R}} \end{aligned}$$

of $\pi ({\mathcal {L}}^\dagger ({\mathcal D}))$.

6 Conclusions

In this paper we have discussed how to generalize the standard notions of Heisenberg dynamics, Gibbs states, KMS- condition and Tomita–Takesaki theory to the case in which the dynamics is driven by a non self-adjoint Hamiltonian, as it often happens in PT- and in pseudo-hermitian quantum mechanics and we have chosen to consider observables as elements of $\mathcal{L}^\dagger ({\mathcal D})$. We have also seen how generalized Riesz systems can be used in this context, and how the results deduced here differ from the standard ones. We have also discussed some preliminary results on entropy and on the Tomita–Takesaki theory in our settings.

Of course, many other aspects could be considered in future, from the use of Gibbs states defined by generalized Riesz systems in the analysis of concrete physical systems to more mathematical aspects. For instance, since it is often difficult or even impossible to find a common invariant dense domain ${\mathcal D}$ for the observables, one could try to enlarge the setting to some other relevant subset of $\mathcal{L}^\dagger ({\mathcal D}, \mathcal H)$. We plan to work on these and other aspects of our framework soon.