E$_0$-semigroups and product systems of W$^*$-bimodules

TL;DR

This paper extends the classification and dilation theory of E$_0$-semigroups and CP$_0$-semigroups from type I factors to general von Neumann algebras using W$^*$-bimodules, introducing a new product system framework.

Contribution

It develops a new theory of product systems of W$^*$-bimodules and establishes a correspondence with CP$_0$-semigroups, generalizing previous results to broader algebraic settings.

Findings

Established a one-to-one correspondence between CP$_0$-semigroups and units of W$^*$-bimodule product systems

Constructed dilations of CP$_0$-semigroups using the new product system framework

Classified E$_0$-semigroups on von Neumann algebras up to cocycle equivalence

Abstract

Product systems have been originally introduced to classify E$_0$-semigroups on type I factors by Arveson. We develop the classification theory of E$_0$-semigroups on a general von Neumann algebra and the dilation theory of CP$_0$-semigroups in terms of W$^*$-bimodules. For this, we provide a notion of product system of W$^*$-bimodules. This is a W$^*$-bimodule version of Arveson's and Bhat-Skeide's product systems. There exists a one-to-one correspondence between CP$_0$-semigroups and units of product systems of W$^*$-bimodules. The correspondence implies a construction of a dilation of a given CP$_0$-semigroup, a classification of E$_0$-semigroups on a von Neumann algebra up to cocycle equivalence and a relationship between Bhat-Skeide's and Muhly-Solel's constructions of minimal dilations of CP$_0$-semigroups.

Equations

$$⟨{\xi }_1{\varphi }^{-\frac{1}{2}}{\eta }_1,{\xi }_2{\varphi }^{-\frac{1}{2}}{\eta }_2⟩=⟨{\eta }_1,{\pi }_{\varphi }({\xi }_1{)}^{\ast }{\pi }_{\varphi }({\xi }_2){\eta }_2⟩,$$

$$(U_{t}f)(x)=e^{itx}f(x),(V_{t}f)(x)=f(x+t)$$

$$W_{{\mathbf{x}}_1}{W}_{{\mathbf{x}}_2}=\exp \left(\frac{i}{2}(x_2{y}_2-x_1{y}_2)\right)W_{{\mathbf{x}}_1+{\mathbf{x}}_2}$$

$$⟨x\otimes \xi ,y\otimes \eta ⟩=⟨\xi ,T(x^{\ast }y)\eta ⟩$$

$${\pi }_{\varphi }(x_1\otimes y_1{\varphi }^{\frac{1}{2}}{)}^{\ast }{\pi }_{\varphi }(x_2\otimes y_2{\varphi }^{\frac{1}{2}})=y_1^{\ast }T(x_1^{\ast }{x}_2)y_2$$

$$⟨{\pi }_{\varphi }(x_1\otimes y_1{\varphi }^{\frac{1}{2}}){\varphi }^{\frac{1}{2}}z,x^{\prime }\otimes y^{\prime }{\varphi }^{\frac{1}{2}}{z}^{\prime }⟩=⟨x_1\otimes y_1{\varphi }^{\frac{1}{2}}z,x^{\prime }\otimes y^{\prime }{\varphi }^{\frac{1}{2}}{z}^{\prime }⟩=⟨{\varphi }^{\frac{1}{2}}z,y_1^{\ast }T(x_1^{\ast }{x}^{\prime })y^{\prime }{\varphi }^{\frac{1}{2}}{z}^{\prime }⟩,$$

$$U_{r,s+t}({\mathrm{id}}_{{\mathcal{H}}_r}{\otimes }^M{U}_{s,t})=U_{r+s,t}(U_{r,s}{\otimes }^M{\mathrm{id}}_{{\mathcal{H}}_t})$$

$${\mathcal{H}}_{r+s+t}$$

$$U_{s,t}(\xi (s){\varphi }^{-\frac{1}{2}}\xi (t))=\xi (s+t)$$

$$V_{s,t}(u_s{\otimes }^M{u}_t)=u_{s+t}{U}_{s,t}$$

$$U(\mathfrak{p})=U_{t_1,t_1^{\prime }}({\mathrm{id}}_{t_1}\otimes U_{t_2,t_2^{\prime }})({\mathrm{id}}_{t_1,t_2}\otimes U_{t_3,t_3^{\prime }})\cdots ({\mathrm{id}}_{t_1,\cdots ,t_{n-2}}\otimes U_{t_{n-1},t_{n-1}^{\prime }})$$

$$\{U(\mathfrak{p})(x_1\xi (t_1){\varphi }^{-\frac{1}{2}}\cdots {\varphi }^{-\frac{1}{2}}{x}_{n-1}\xi (t_{n-1}){\varphi }^{-\frac{1}{2}}{x}_n\xi (t_n)y)\mid \mathfrak{p}\in {\mathfrak{P}}_t,x_1,\cdots ,x_n,y\in M\}$$

$$b_{t,s}(\xi )=U_{t-s,s}(\xi (t-s){\varphi }^{-\frac{1}{2}}\xi )$$

$$U_t(({\kappa }_s{\xi }_s){\varphi }^{-\frac{1}{2}}{\eta }_t)={\kappa }_{s+t}{U}_{s,t}({\xi }_s{\varphi }^{-\frac{1}{2}}{\eta }_t).$$

$$⟨U_t(({\kappa }_s{\xi }_s){\varphi }^{-\frac{1}{2}}{\eta }_t),U_t(({\kappa }_s{\xi }_s^{\prime }){\varphi }^{-\frac{1}{2}}{\eta }_t^{\prime })⟩=⟨{\xi }_s{\varphi }^{-\frac{1}{2}}{\eta }_t,{\xi }_s^{\prime }{\varphi }^{-\frac{1}{2}}{\eta }_t^{\prime }⟩$$

$$=⟨{\eta }_t,{\pi }_{\varphi }({\xi }_s{)}^{\ast }{\pi }_{\varphi }({\xi }_s^{\prime }){\eta }_t^{\prime }⟩=⟨{\eta }_t,{\pi }_{\varphi }({\kappa }_s{\xi }_s{)}^{\ast }{\pi }_{\varphi }({\kappa }_s{\xi }_s^{\prime }){\eta }_t^{\prime }⟩=⟨({\kappa }_s{\xi }_s){\varphi }^{-\frac{1}{2}}{\eta }_t,({\kappa }_s{\xi }_s^{\prime }){\varphi }^{-\frac{1}{2}}{\eta }_t^{\prime }⟩.$$

$$⟨U_t(({\kappa }_s{\xi }_s){\varphi }^{-\frac{1}{2}}{\eta }_t),U_t(({\kappa }_r{\zeta }_r){\varphi }^{-\frac{1}{2}}{\eta }_t^{\prime })⟩$$

$$=⟨U_t(({\kappa }_{s+t}{b}_{s+r.s}{\xi }_s){\varphi }^{-\frac{1}{2}}{\eta }_t),U_t(({\kappa }_{s+r}{b}_{s+r,r}{\zeta }_r){\varphi }^{-\frac{1}{2}}{\eta }_t^{\prime })⟩$$

$$=⟨({\kappa }_{s+t}{b}_{s+r.s}{\xi }_s){\varphi }^{-\frac{1}{2}}{\eta }_t,({\kappa }_{s+r}{b}_{s+r,r}{\zeta }_r){\varphi }^{-\frac{1}{2}}{\eta }_t^{\prime }⟩=⟨({\kappa }_s{\xi }_s){\varphi }^{-\frac{1}{2}}{\eta }_t,({\kappa }_r{\zeta }_r){\varphi }^{-\frac{1}{2}}{\eta }_t^{\prime }⟩.$$

$$\pi (x)\xi ={\kappa }_0(x({\kappa }_0^{\ast }\xi ))$$

$${\theta }_t(a)=U_t(a{\otimes }^M{\mathrm{id}}_{{\mathcal{H}}_t})U_t^{\ast }$$

$$U_t(\xi {\varphi }^{-\frac{1}{2}}\xi (t))\to \xi (t\to +0)$$

$${\theta }_t(a)\xi -a\xi$$

$${\pi }_{\varphi }(\lambda (t){)}^{\ast }{\pi }_{\varphi }(\lambda (t)){\varphi }^{\frac{1}{2}}x={\kappa }_0^{\ast }{w}_t^{\ast }{w}_t{\kappa }_0{\varphi }^{\frac{1}{2}}x$$

$$\lambda (s+t)$$

$$w_t\xi ={\kappa }_t{U}_{t,0}(\lambda (t){\varphi }^{-\frac{1}{2}}{\kappa }_0^{\ast }\xi )$$

$$w_t({\kappa }_0{\varphi }^{\frac{1}{2}}x)={\kappa }_t\lambda (t)x$$

$${\theta }_t(w_s)w_t({\kappa }_0{\varphi }^{\frac{1}{2}}x)$$

$${\kappa }_t{\kappa }_t^{\ast }{w}_t\xi ={\kappa }_t{\kappa }_t^{\ast }{w}_t{\kappa }_0{\kappa }_0^{\ast }\xi ={\kappa }_t{\kappa }_t^{\ast }{\kappa }_t{U}_{t,0}(\lambda (t){\varphi }^{-\frac{1}{2}}({\kappa }_0^{\ast }\xi ))=w_t{\kappa }_0{\kappa }_0^{\ast }\xi =w_t\xi .$$

$$\begin{gathered} w_t{\kappa }_t{\kappa }_t^{\ast }\xi ={\kappa }_t{U}_{t,0}(\lambda (t){\varphi }^{-\frac{1}{2}}{\kappa }_0^{\ast }{\kappa }_t{\kappa }_t^{\ast }\xi ) \\ ={\kappa }_t{U}_{t,0}(\lambda (t){\varphi }^{-\frac{1}{2}}{\kappa }_0^{\ast }\xi )=w_t\xi . \end{gathered}$$

$$T_t^{\Xi }(x)={\pi }_{\varphi }(\xi (t){)}^{\ast }{\pi }_{\varphi }(x\xi (t))\in M$$

$$⟨T_s^{\Xi }(T_t^{\Xi }(x)){\varphi }^{\frac{1}{2}}y,{\varphi }^{\frac{1}{2}}z⟩=⟨{\pi }_{\varphi }(\xi (s){)}^{\ast }{\pi }_{\varphi }({\pi }_{\varphi }(\xi (t){)}^{\ast }\pi (x\xi (t))\xi (s)){\varphi }^{\frac{1}{2}}y,{\varphi }^{\frac{1}{2}}z⟩$$

Full text

1. Introduction

E0-semigroups naturally arise in the quantum field theory. An E0-semigroup is a semigroup of normal $*$-endomorphisms on a von Neumann algebra with $\sigma$-weak continuity, and the study of E0-semigroups have been initiated by Powers in [15]. In [2], Arveson has provided the notion of product system and associated a product system with an E0-semigroup on a type I factor. A product system $\{\mathcal{H}_{t}\}_{t>0}$ is a measurable family of Hilbert spaces $\mathcal{H}_{t}$ parameterized by positive real numbers equipped with isomorphisms $\mathcal{H}_{s}\otimes\mathcal{H}_{t}\cong\mathcal{H}_{s+t}$ with the associativity. Note that this is not his original definition, however they are the essentially same (see [11]). He also classified E0-semigroups on type I factors by product systems up to cocycle conjugacy. E0-semigroups on type I factors are roughly divided into type I, II and III by units of associated product systems. Every product systems have a numerical index and type I E0-semigroups and product systems are completely classified by their indexes. We refer the reader to his monograph [4] for the physical background of E0-semigroups and the theory of product systems. The theory of E0-semigroups on von Neumann algebras which are not type I factors, has often been developed in terms of Hilbert modules. A Bhat-Skeide’s product system introduced in [8] is a family $\{E_{t}\}_{t\geq 0}$ of Hilbert bimodules over a C∗-algebra satisfying a similar property with Arveson’s one with respect to tensor products of Hilbert bimodules. They classified E0-semigroups on a C∗-algebra by their product systems up to cocycle equivalence. In [1], Alevras has associated a product system of Hilbert bimodules with each E0-semigroup on a II1 factor by a different way from Bhat-Skeide’s one and they form a complete invariant. In Skeide’s monograph [23], we have the classification theory of E0-semigroups on the algebra $\mathcal{B}^{a}(E)$ of all adjointable right $A$-linear maps on a Hilbert (von Neumann) $A$-module $E$. On the other hand, Margetts-Srinivasan have introduced other invariants of E0-semigroups on II1 factors in [12], and they have investigated non-cocycle conjugate E0-semigroups on factors in [13] by using the modular conjugation of Tomita-Takesaki theory.

A CP0-semigroups is a $\sigma$-weakly continuous semigroup of normal completely positive maps on a von Neumann algebra. The theory of Arveson’s product systems influenced the constructions of minimal dilations of CP0-semigroups. Roughly speaking, a dilation of a CP0-semigroup is an extension of it to an E0-semigroup in a suitable sense. Stinespring’s dilation theorem can not be applied to CP0-semigroups, and some researchers have shown an existence of the minimal dilation of a given CP0-semigroup gradually. In [6] and [7], Bhat has shown it in the cases when $M$ is $\mathcal{B}(\mathcal{H})$ and a C∗-algebra, respectively, in which we do not assume the $\sigma$-weakly continuity for semigroups. In [8], Bhat-Skeide constructed minimal dilations by a method which is valid for both of the von Neumann algebra case and the C∗-algebra case. Also, we know Muhly-Solel’s ([14]) and Arveson’s ([4]) constructions, which differ from each other, of the minimal dilation of a CP0-semigroup on a von Neumann algebra. It is more difficult to construct an example of E0-semigroups than CP0-semigroups in general, however the existence of (minimal) dilations gives rise to E0-semigroups from CP0-semigroups. This is one of benefits of the dilation theory. Also, in [17], we have clarified a direct relationships between Bhat-Skeide’s and Muhly-Solel’s constructions of the minimal dilation of a discrete CP0-semigroup, which is different from one described by Skeide’s commutant duality in [20] and [21].

There have been no approaches to the classification theory of E0-semigroups on a von Neumann algebra and the dilation theory of CP0-semigroups by the W∗-bimodule (which is not von Neumann bimodule) theory. In this paper, we attempt to give a W∗-bimodule approach to their field by a way reflected by Bhat-Skeide’s works in [8].

We give an outline of this paper. We will recall the notions of W∗-bimodule, relative tensor product, CP0-semigroup and E0-semigroup in Section 2.

In Section 3, we will provide a concept of product system of W∗-bimodules, where adopted tensor products are relative tensor product introduced by Connes[9]. This is a direct extension of Arveson’s product system. A unit $\Xi$ of a product system $H$ of W∗-$M$-bimodules induces an E0-semigroup on ${\rm End}(\mathfrak{H}_{M})$, where $\mathfrak{H}$ is the inductive limit of $H$ with respect to parameters. The E0-semigroup is called the dilation of the pair $(H,\Xi)$. We prove that cocycles of the dilation of the pair $(H,\Xi)$ and units of $H$ are the essentially same.

In Section 4, we will find a one-to-one correspondence between CP0-semigroups on a von Neumann algebra $M$ and pairs of product systems of W∗-$M$-bimodules and units up to unit preserving isomorphism. The correspondence enables as to translate the $\sigma$-weak continuity of CP0-semigroups into a continuity of units. Also, the dilation of the pair associated with a given CP0-semigroup $T$, gives a dilation of $T$. The product system of W∗-bimodules associated with a CP0-semigroup $T$ describes a relation between Bhat-Skeide’s and Muhly-Solel’s constructions of the minimal dilation of $T$. This is an extension to the continuous case of the relation in the discrete case in [17]. Some relationships among the two constructions and Arveson’s construction have not been clarified yet.

In Section 5, we consider the product system associated the heat semigroup $\{e^{t\Delta}\}_{t\geq 0}$ given by the Laplacian $\Delta$ on a compact Riemannian manifold, and its dilation by the method in Section $4$, as an example. We will show that the W∗-bimodules appearing in the construction of the product system associated with $\{e^{t\Delta}\}_{t\geq 0}$ are realized as $L^{2}$-spaces with respect to measures given by the heat kernel. We will reconstruct the dilation in more detail under this identification.

We can get the product system $H^{\alpha}$ of W∗-bimodules (and the unit) from an E0-semigroup $\alpha$ on a von Neumann algebra $M$ as CP0-semigroups by the above correspondence. We will classify E0-semigroups on $M$ by product systems of W∗-bimodules: two E0-semigroups $\alpha$ and $\beta$ on $M$ are cocycle equivalent if and only if $H^{\alpha}\cong H^{\beta}$ in Section 6. Hence, this enables as to classify E0-semigroups up to cocycle conjugacy by product systems of W∗-bimodules. Also, we will get a unit of a given E0-semigroup $\theta$ on II1 factor from a unit of the product system $H^{\theta}$ associated with $\theta$.

2. Preliminaries

In this section, we recall the notions of W∗-bimodule, relative tensor product, CP0-semigroup, E0-semigroup, tensor product related to CP0-semigroup and partition, which will be used in the later sections.

W∗-bimodules are Hilbert spaces on which von Neumann algebras act from the left and the right. More precisely, for von Neumann algebras $N$ and $M$, a Hilbert space $\mathcal{H}$ with normal $*$-representations of $N$ and the opposite von Neumann algebra $M^{\circ}$ of $M$ is a W∗-$N$-$M$-bimodule if their representations commute. When $N=\mathbb{C}$ or $M=\mathbb{C}$, we call $\mathcal{H}$ a right W∗-$M$-module or a left W∗-$N$-module, respectively. We write a W∗-$N$-$M$-bimodule, a right W∗-$M$-module and a left W∗-$N$-module by ${}_{N}\mathcal{H}_{M},\ \mathcal{H}_{M}$ and ${}_{N}\mathcal{H}$, respectively.

Let $N$ be a von Neumann algebra, $\mathcal{H}_{N}$ and $\mathcal{K}_{N}$ be right W∗-$N$-modules, and ${}_{N}\mathcal{H}^{\prime}$ and ${}_{N}\mathcal{K}^{\prime}$ be left W∗-$N$-modules. $\mbox{\rm{Hom}}(\mathcal{H}_{N},\mathcal{K}_{N})$ and $\mbox{\rm{Hom}}({}_{N}\mathcal{H}^{\prime},{}_{N}\mathcal{K}^{\prime})$ are the sets of all right and left $N$-linear bounded maps, respectively. If $\mathcal{H}=\mathcal{K}$ and $\mathcal{H}^{\prime}=\mathcal{K}^{\prime}$, they are denoted by $\mbox{\rm{End}}(\mathcal{H}_{N})$ and $\mbox{\rm{End}}({}_{N}\mathcal{H}^{\prime})$, respectively.

We denote the standard space of a von Neumann algebra $M$ by $L^{2}(M)$. The standard space $L^{2}(M)$ contains all left and right GNS-spaces and we have $[\phi]\overline{M\psi^{\frac{1}{2}}}=\overline{\phi^{\frac{1}{2}}M}[\psi]$ in $L^{2}(M)$ and $\overline{\phi^{\frac{1}{2}}M}=[\phi]L^{2}(M)$ for all $\phi,\psi\in M_{*}^{+}$. In particular, we have $\overline{\phi^{\frac{1}{2}}M}=L^{2}(M)=\overline{M\phi^{\frac{1}{2}}}$ for each faithful $\phi\in M_{*}$. This observation will be helpful under the assumption which a von Neumann algebra has a faithful normal state in the later sections. We refer the reader to [25, Chapter IX], [28], [26] and [27] for details of the definition and properties of standard spaces included in the modular theory.

Now, we shall recall (left) relative tensor products. For more details, see [9, Chapter 5, Appendix B], [16] or [25, Chapter IX, Section 3]. Suppose $\mathcal{H}$ is a W∗-$M$-$N$-bimodule and $\mathcal{K}$ is a W∗-$N$-$P$-bimodule. Let $\phi$ be a faithful normal state on $N$. A vector $\xi\in\mathcal{H}$ is called a (left) $\phi$-bounded vector if there is $c>0$ such that $\|\xi x\|\leq c\|\phi^{\frac{1}{2}}x\|$ for all $x\in M$. We denote the set of all $\phi$-bounded vectors in $\mathcal{H}$ by $\mathcal{D}(\mathcal{H};\phi)$. The (left) relative tensor product $\mathcal{H}\otimes_{\phi}^{N}\mathcal{K}$ is the completion $\overline{\mathcal{D}(\mathcal{H};\phi)\otimes_{{\rm alg}}\mathcal{K}}$ with respect to an inner product defined by

[TABLE]

for each $\xi_{1},\xi_{2}\in\mathcal{D}(\mathcal{H};\phi)$ and $\eta_{1},\eta_{2}\in\mathcal{K}$, where $\pi_{\phi}(\xi):L^{2}(N)\ni\phi^{\frac{1}{2}}x\to\xi x\in\mathcal{H}$ and we usually use a notation $\xi\phi^{-\frac{1}{2}}\eta$ rather than $\xi\otimes\eta$. Also, we can define the right relative tensor product by right $\phi$-bounded vectors.

Remark 2.1**.**

Left and right relative tensor products can be defined by the way which is independent on a choice of $\phi$ (see [5]). If we denote the left and the right relative tensor product by $\mathcal{H}\otimes_{l}^{N}\mathcal{K}$ and $\mathcal{H}\otimes_{r}^{N}\mathcal{K}$, respectively for W∗-bimodules ${}_{M}\mathcal{H}_{N}$ and ${}_{N}\mathcal{K}_{P}$, we already know the W∗-bimodule isomorphism $\mathcal{H}\otimes_{l}^{N}\mathcal{K}\cong\mathcal{H}\otimes_{r}^{N}\mathcal{K}$. In [18], we have constructed the isomorphism $\mathcal{H}\otimes_{l}^{N}\mathcal{K}\cong\mathcal{H}\otimes_{r}^{N}\mathcal{K}$ by the canonical way, and shown that the two W∗-bicategories of W∗-bimmodules with left and right tensor products as tensor functors are monoidally equivalent.

Now, we provide the basic notions related with CP0-semigroups and E0-semigroups. A family $T=\{T_{t}\}_{t\geq 0}$ of normal UCP-maps $T_{t}$ on a von Neumann algebra $M$ is called a CP0-semigroup if $T_{0}={\rm id}_{M},\ T_{s}T_{t}=T_{s+t}$ for all $s,t\geq 0$, and for every $x\in M$ and $\phi\in M_{*}$, the function $\phi(T_{t}(x))$ on $[0,\infty)$ is continuous. If each $T_{t}$ is a $*$-homomorphism, $T$ is called an E0-semigroup. A CP0-semigroup (E0-semigroup) without the continuity is called an algebraic CP0-semigroup (algebraic E0-semigroup, respectively).

Example 2.2**.**

Let $\{v_{t}\}_{t\geq 0}$ be a family of isometries $v_{t}$ in a von Neumann algebra $M$ such that $v_{s+t}=v_{s}v_{t}$ for all $s,t\geq 0$ and $v_{0}=1_{M}$. Suppose $\{v_{t}\}_{t\geq 0}$ is strongly continuous with respect to the parameter. If we define $T=\{T_{t}\}_{t\geq 0}$ by $T_{t}(x)=v_{t}^{*}xv_{t}$ for each $x\in M$ and $t\geq 0$, then $T$ is a CP0-semigroup. If each $v_{t}$ is unitary, $T$ is an E0-semigroup.

Example 2.3**.**

The CCR heat flow is a CP0-semigroup $T$ which has the noncommutative Laplacian $\Delta$ as generators. This will be immediately and concretely defined by the Weyl system. For more details, see [4, Section 7].

Let $\mathcal{H}=L^{2}(\mathbb{R})$ and $M=\mathcal{B}(\mathcal{H})$. For ${\bf x}=(x,y)\in\mathbb{R}^{2}$, the concrete Weyl operator is $W_{\bf x}=\exp(\frac{xy}{2}i)U_{x}V_{y}$, where $\{U_{x}\}_{x\in\mathbb{R}}$ and $\{V_{x}\}_{x\in\mathbb{R}}$ are the unitary groups which have the position operator $Q$ and the momentum operator $P$ as generators, respectively, i.e.

[TABLE]

for $f\in L^{2}(\mathbb{R})$ and $t,x\in\mathbb{R}$. Then, the family $\{W_{\bf x}\}_{{\bf x}\in\mathbb{R}^{2}}$ of the unitaries satisfies the Weyl relations

[TABLE]

for ${\bf x}_{1}=(x_{1},y_{1}),{\bf x}_{2}=(x_{2},y_{2})\in\mathbb{R}^{2}$. The CCR heat flow is defined as the unique CP0-semigroup $T=\{T_{t}\}_{t\geq 0}$ on $M$ satisfying $T_{t}(W_{\bf x})=\exp(-t\|x\|^{2})W_{\bf x}$ for all ${\bf x}\in\mathbb{R}^{2}$ and $t\geq 0$. More precisely, we define $T_{t}$ for $t\geq 0$ by a weak integral $T_{t}(x)=\int_{\mathbb{R}^{2}}W_{\frac{{\bf x}}{\sqrt{2}}}xW_{\frac{{\bf x}}{\sqrt{2}}}^{*}d\mu_{t}({\bf x})$ for each $x\in M$, where $\mu_{t}$ is the probability measure whose Fourier transformation is $u_{t}({\bf x})=\exp(-t\|{\bf x}\|^{2})$.

According to Stinespring’s dilation theorem, for a UCP-map $T$ from a C∗-algebra $A$ into $\mathcal{B}(\mathcal{H})$, there exist a Hilbert space $\mathcal{K}$, a unital representation of $A$ on $\mathcal{K}$ and an isometry $v:\mathcal{H}\to\mathcal{K}$ such that $T(a)=v^{*}\pi(a)v$ for all $a\in A$. However, Stinespring’s theorem does not apply to CP0-semigroup. The notion of dilation of CP0-semigroups are introduced as follows:

Definition 2.4**.**

Let $T=\{T_{t}\}_{t\geq 0}$ be a CP0-semigroup on a von Neumann algebra $M$. A dilation of $T$ consists of a von Neumann algebra $N$, a projection $p\in N$ and an E0-semigroup $\{\theta_{t}\}_{t\geq 0}$ on $N$ such that $M=pNp$ and $T_{t}(x)=p\theta_{t}(x)p$ for all $x\in M$ and $t\geq 0$. In addition, if $N$ is generated by $\theta_{[0,\infty)}(M)$ and the central support of $p$ in $N$ is $1_{N}$, the dilation is said to be minimal.

Note that a minimal dilation of a CP0-semigroup is unique (if it exists). The existence of minimal dilations is proved by Bhat-Skeide and Muhly-solel. In Section 4, a relation between the two constructions will be clarified. Arveson also constructed the minimal dilation by other approach in [4] (or [3]).

The notion of cocycle for E0-semigroups which is useful for the classification of E0-semigroups, is introduced as the following definition.

Definition 2.5**.**

Let $\theta$ be an E0-semigroup on a von Neumann algebra $M$. A family $w=\{w_{t}\}_{t\geq 0}\subset M$ is called a right cocycle for $\theta$ if $w_{s+t}=\theta_{t}(w_{s})w_{t}$ for all $s,t\geq 0$. If each $w_{t}$ is unitary (contractive), then $w$ is called a right unitary (contractive, respectively) cocycle.

Definition 2.6**.**

Two E0-semigroups $\alpha$ and $\beta$ on a von Neumann algebra $M$ are said to be cocycle equivalent if there exists a strongly continuous right unitary cocycle $w$ such that $\beta_{t}(x)=w_{t}^{*}\alpha_{t}(x)w_{t}$ for all $t\geq 0$ and $x\in M$. Then, the E0-semigroup $\beta$ is called the cocycle perturbation of $\alpha$ with respect to $w$.

Let $\alpha$ and $\beta$ be E0-semigroups on von Neumann algebras $M$ and $N$, respectively, and $\Phi:M\to N$ is a $*$-isomorphism. The conjugation $\beta^{\Phi}$ of $\beta$ with respect to $\Phi$ is an E0-semigroup on $M$ defined by $\beta_{t}^{\Phi}=\Phi^{-1}\circ\beta_{t}\circ\Phi$ for each $t\geq 0$. If $\beta^{\Phi}$ is a cocycle perturbation of $\alpha$, we say $\alpha$ and $\beta$ are cocycle conjugate.

We will establish a product system of W∗-bimodule from a given CP0-semigroup in Section 4. For this, we prepare a W∗-bimodule equipped with a information of a given normal UCP-map as follows. Let $T$ be a normal UCP-map on a von Neumann algebra $M$ and $\mathcal{H}$ a W∗-$M$-$N$-bimodule. We define $M\otimes_{T}\mathcal{H}$ as the completion of the algebraic tensor product $M\otimes_{{\rm alg}}\mathcal{H}$ with respect to an inner product defined by

[TABLE]

for each $x,y\in M$ and $\xi,\eta\in\mathcal{H}$. The Hilbert space $M\otimes_{T}\mathcal{H}$ has the canonical W∗-$M$-$N$-bimodule structure: $a(x\otimes\xi)b=(ax)\otimes(\xi b)$ for each $a,x\in M,\ b\in N$ and $\xi\in\mathcal{H}$. When $\mathcal{H}=L^{2}(M)$, we can provide the following formula related to the relative tensor products and normal UCP-maps, which will be used for computing inner products in later arguments.

Proposition 2.7**.**

Let $T$ be a normal UCP-map on a von Neumann algebra $M$. For $x,y\in M$, we have $x\otimes y\phi^{\frac{1}{2}}\in\mathcal{D}(M\otimes_{T}L^{2}(M);\phi)$. Moreover, we have

[TABLE]

for $x_{1},x_{2},y_{1},y_{2}\in M$.

Proof.

For $x^{\prime},y^{\prime},z\in M$, we can compute as

[TABLE]

and hence we have $\pi_{\phi}(x_{1}\otimes y_{1}\phi^{\frac{1}{2}})^{*}(x^{\prime}\otimes y^{\prime}\phi^{\frac{1}{2}}z^{\prime})=y_{1}^{*}T(x_{1}^{*}x^{\prime})y^{\prime}\phi^{\frac{1}{2}}z^{\prime}$. Thus, we conclude that $\pi_{\phi}(x_{1}\otimes y_{1}\phi^{\frac{1}{2}})^{*}\pi_{\phi}(x_{2}\otimes y_{2}\phi^{\frac{1}{2}})\phi^{\frac{1}{2}}z=y_{1}^{*}T(x_{1}^{*}x_{2})y_{2}\phi^{\frac{1}{2}}z.$ $\Box$

Finally, we prepare notations related with partitions. We fix $t>0$. Let $\mathfrak{P}_{t}$ be the set of all finite tuples $\mathfrak{p}=(t_{1},\cdots,t_{n})$ with $t_{i}>0$ such that $\sum_{i=1}^{n}t_{i}=t$. For $\mathfrak{p}=(t_{1},\cdots,t_{n})\in\mathfrak{P}_{t}$, we define $\#\mathfrak{p}=n$. Let $\mathfrak{p}=(t_{1},\cdots,t_{n}),\mathfrak{q}=(s_{1},\cdots,s_{m})\in\mathfrak{P}_{t}$. We define the joint tuple by $\mathfrak{p}\lor\mathfrak{q}=(t_{1},\cdots,t_{n},s_{1},\cdots,s_{m})$ for $\mathfrak{p}=(t_{1},\cdots,t_{n}),\mathfrak{q}=(s_{1},\cdots,s_{m})\in\mathfrak{P}_{t}$. Also, we write $\mathfrak{p}\succ\mathfrak{q}$ if there exist partitions $\mathfrak{q}_{i}\in\mathfrak{P}_{s_{i}}$ for $i=1,\cdots,m$ such that $\mathfrak{p}=\mathfrak{q}_{1}\lor\cdots\lor\mathfrak{q}_{m}$. Let $\mathfrak{P}_{0}$ be the singleton of the empty tuple $()$ satisfying $\mathfrak{p}\lor()=()\lor\mathfrak{p}=\mathfrak{p}$. Note that when we consider partitions of an interval $[0,t]$, treating $\mathfrak{P}_{t}$ or the set $\mathfrak{P}_{t}^{\prime}$ of all finite tuples $(t_{1},\cdots,t_{n})$ such that $t=t_{n}>t_{n-1}>\cdots>t_{1}>0$ is equivalent because $\mathfrak{P}_{t}$ and $\mathfrak{P}_{t}^{\prime}$ are order isomorphic via a map $\mathfrak{o}:\mathfrak{P}_{t}\to\mathfrak{P}_{t}^{\prime}$ defined by $\mathfrak{o}(t_{1},t_{2},\cdots,t_{n})=\left(\sum_{i=1}^{1}t_{i},\sum_{i=1}^{2}t_{i},\cdots,\sum_{i=1}^{n}t_{i}\right)$ for each $\mathfrak{p}=(t_{1},\cdots,t_{n})\in\mathfrak{P}_{t}$.

3. Product systems of W∗-bimodules and maximal dilations

In this section, we will introduce a new concepts of product system of W∗-bimodules and their units which are inspired by the definitions of Arveson’s and Bhat-Skeide’s product systems and units. We will construct an algebraic E0-semigroup from a given unit of a product system of W∗-bimodules by taking the inductive limit.

Definition 3.1**.**

Let $M$ be a von Neumann algebra and $H=\{\mathcal{H}_{t}\}_{t\geq 0}$ a family of W∗-$M$-bimodules with $\mathcal{H}_{0}=L^{2}(M)$. If there exist bimodule unitaries $U_{s,t}:\mathcal{H}_{s}\otimes^{M}\mathcal{H}_{t}\to\mathcal{H}_{s+t}$ for each $s,t\geq 0$ such that

[TABLE]

for each $r,s,t\geq 0$, and $U_{0,t}$ and $U_{s,0}$ are the canonical identifications, then the pair $(H,\{U_{s,t}\}_{s,t\geq 0})$ is called a product system of W∗-$M$-bimodules, where the notation $\otimes^{M}$ denotes the relative tensor product over $M$.

Remark 3.2**.**

Precisely speaking, associativity (3.1) means that the following diagram commutes.

[TABLE]

Here, the morphism $a$ is the associativity isomorphism which is discussed in detail in [18]. By [18, Theorem 3.2], we can choose either the left or the right relative tensor product. We will construct a product system of W∗-bimodules from a given CP0-semigroup by left relative tensor products.

Remark 3.3**.**

The notion of product system of W∗-bimodules is a direct extension of Arveson’s product system. Indeed, if $M=\mathbb{C}$, then a product system of W∗-$\mathbb{C}$-bimodules is just an Arveson’s product system without a measurable structure. For the purpose of this paper, measurable structures for product systems of W∗-bimodules are not necessary.

Definition 3.4**.**

Let $M$ be a von Neumann algebra with a faithful normal state $\phi$ and $(H,\{U_{s,t}\}_{s,t\geq 0})$ a product system of W∗-$M$-bimodules. A family $\Xi=\{\xi(t)\}_{t\geq 0}$ of $\xi(t)\in\mathcal{D}(\mathcal{H}_{t};\phi)$ is called a unit of $H$ with respect to $\phi$ if $\xi(0)=\phi^{\frac{1}{2}}$ and

[TABLE]

for all $s,t\geq 0$.

If a unit $\Xi=\{\xi(t)\}_{t\geq 0}$ satisfies $\pi_{\phi}(\xi(t))^{*}\pi_{\phi}(\xi(t))=1_{M}\ (\|\pi_{\phi}(\xi(t))^{*}\pi_{\phi}(\xi(t))\|\leq 1)$, it is said to be unital (contractive, respectively).

We introduce a natural notion of isomorphism between product systems of W∗-$M$-bimodules and generating property for units as follows:

Definition 3.5**.**

Let $(H,\{U_{s,t}\}_{s,t\geq 0})$ and $(K,\{V_{s,t}\}_{s,t\geq 0})$ be product systems of W∗-$M$-bimodules. An isomorphism is a family $u^{\otimes}=\{u_{t}\}_{t\geq 0}$ of $M$-bilinear unitaries $u_{t}:\mathcal{H}_{t}\to\mathcal{K}_{t}$ satisfying

[TABLE]

for all $s,t\geq 0$. Then, the product system $H$ is said to be isomorphic to $K$ and we denote as $H\cong K$.

Definition 3.6**.**

Let $H=\{\mathcal{H}_{t}\}_{t\geq 0}$ be a product system of W∗-$M$-bimodules with $M$-bilinear unitaries $\{U_{s,t}\}_{s,t\geq 0}$. For $t\geq 0$ and $\mathfrak{p}=(t_{1},\cdots,t_{n})\in\mathfrak{P}_{t}$, we denote the $M$-bilinear unitary

[TABLE]

from $\mathcal{H}_{t_{1}}\otimes^{M}\cdots\otimes^{M}\mathcal{H}_{t_{n}}$ onto $\mathcal{H}_{t}$, where $t_{i}^{\prime}=t_{i+1}+\cdots+t_{n}$ and ${\rm id}_{t_{1},\cdots,t_{i}}={\rm id}_{t_{1}}\otimes\cdots\otimes{\rm id}_{t_{i}}$. A unital unit $\Xi=\{\xi(t)\}_{t\geq 0}$ with respect to $\phi$ is said to be generating when the set

[TABLE]

is dense in $\mathcal{H}_{t}$ for all $t\geq 0$.

Remark 3.7**.**

Like Arveson’s and Bhat-Skeide’s product system, every product system of W∗-$M$-bimodules does not always have a unit with respect to a faithful normal state $\phi$ on $M$. We say that a product system of W∗-$M$-bimodules equipped with a unit with respect to $\phi$ is $\phi$-spatial. From now, we fix a faithful normal state $\phi$ on a von Neumann algebra $M$ through this paper . When we say a unit merely, suppose that it is a unit with respect to $\phi$ of a $\phi$-spatial product system.

Let $\Xi$ be a unital unit of a $\phi$-spatial product system $H$ of W∗-$M$-bimodules. We shall define the inductive limit $\mathfrak{H}$ of $H$ which depends on $\Xi$ and an algebraic E0-semigroup on ${\rm End}(\mathfrak{H}_{M})$ as follows: for $0\leq s\leq t$, we define a right $M$-linear isometry $b_{t,s}:\mathcal{H}_{s}\to\mathcal{H}_{t}$ by

[TABLE]

for each $\xi\in\mathcal{H}_{s}$. Note that $b_{t,s}\circ b_{s,r}=b_{t,r}$ for $0\leq r\leq s\leq t$. Let $\mathfrak{H}$ be the inductive limit of the inductive system $(H,\{b_{t,s}\}_{s\leq t})$ and $\kappa_{t}:\mathcal{H}_{t}\to\mathfrak{H}$ the canonical embedding for each $t\geq 0$. The right W∗-$M$-module $\mathfrak{H}$ is called the inductive limit of the pair $(H,\Xi)$.

Theorem 3.8**.**

Fix $t\geq 0$. There exists a right $M$-linear unitary $U_{t}:\mathfrak{H}\otimes^{M}\mathcal{H}_{t}\to\mathfrak{H}$

Proof.

For $s\geq 0$, $\xi_{s}\in\mathcal{D}(\mathcal{H}_{t};\phi)$ and $\eta_{t}\in\mathcal{H}_{t}$, we define

[TABLE]

We shall show that $U_{t}$ is an isometry. For $s\geq 0$, $\xi_{s},\xi_{s}^{\prime}\in\mathcal{H}_{s}$ and $\eta_{t},\eta_{t}^{\prime}\in\mathcal{H}_{t}$, we have

[TABLE]

This implies that for $s\geq 0$, $\xi_{s}\in\mathcal{H}_{s},\ \zeta_{r}\in\mathcal{H}_{r}$ and $\eta_{t},\eta_{t}^{\prime}\in\mathcal{H}_{t}$, in the general case, we have

[TABLE]

We shall check that $U_{t}$ is surjective. In the case when $s\leq t$, for $\eta=\kappa_{s}\eta_{s}\in\mathfrak{H}$, we can conclude that the image of $(\kappa_{0}\phi^{\frac{1}{2}})\phi^{-\frac{1}{2}}\kappa_{t}^{*}\kappa_{s}\eta_{s}$ by $U_{t}$ is $\eta$. In the case when $s>t$, let $\mathcal{D}$ be a subspace of $\mathcal{H}_{s-t}\otimes^{M}\mathcal{H}_{t}$ spanned by vectors $\eta_{s-t}\phi^{-\frac{1}{2}}\eta_{t}$ for all $\phi$-bounded vectors $\eta_{s-t}\in\mathcal{H}_{s-t}$ and $\eta_{t}\in\mathcal{H}_{t}$. For $\eta=\kappa_{s}\eta_{s}\in\mathfrak{H}$, $\eta_{s}$ can be approximated by vectors $U_{s-t,t}\zeta$ for some $\zeta\in\mathcal{D}$ and we have $U_{t}(\kappa_{s-t}\otimes{\rm id}_{\mathcal{H}_{t}})=\kappa_{s}U_{s-t,t}$ on $\mathcal{D}$. $\Box$

The von Neumann algebra $M$ can be represented faithfully on $\mathfrak{H}$ by

[TABLE]

for each $x\in M$ and $\xi\in\mathfrak{H}$. Note that $\pi(M)\subset{\rm End}(\mathfrak{H}_{M})$. Then, we can define an algebraic E0-semigroup $\theta$ on the von Neumann algebra ${\rm End}(\mathfrak{H}_{M})$ by

[TABLE]

for each $a\in{\rm End}(\mathfrak{H}_{M})$. The algebraic E0-semigroup $\theta$ is called the dilation of the pair $(H,\Xi)$. If we assume a continuity to the unit $\Xi$, then it becomes an E0-semigroup as follows:

Proposition 3.9**.**

If the unital unit $\Xi=\{\xi(t)\}_{t\geq 0}$ of the $\phi$-spatial product system $H$ of W∗-$M$-bimodules satisfies that

[TABLE]

for all $\xi\in\mathcal{D}(\mathfrak{H};\phi)$, then the dilation $\theta$ of the pair $(H,\Xi)$ is an E0-semigroup.

Proof.

For each $a\in{\rm End}(\mathfrak{H}_{M})$ and each $\phi$-bounded vector $\xi\in\mathfrak{H}$, we have

[TABLE]

when $t\to 0$. Thus, the map $t\mapsto\theta_{t}(a)$ is $\sigma$-weakly continuous for each $a\in{\rm End}(\mathfrak{H}_{M})$. $\Box$

Definition 3.10**.**

We say that a unit $\Xi$ with (3.6) is continuous.

A right cocycle $w=\{w_{t}\}_{t\geq 0}$ for $\theta$ is said to be adapted if $\kappa_{t}\kappa_{t}^{*}w_{t}\kappa_{t}\kappa_{t}^{*}=w_{t}$ for all $t\geq 0$, where $\kappa_{t}$ is the canonical embedding from $\mathcal{H}_{t}$ into $\mathcal{H}$. The following theorem describing a correspondence between cocycles and units, will enable as to classify E0-semigroups by product systems of W∗-bimodules in Section 6.

Theorem 3.11**.**

Let $\theta=\{\theta_{t}\}_{t\geq 0}$ be the dilation of a pair $(H,\Xi)$ of a $\phi$-spatial product system $H=\{\mathcal{H}_{t}\}_{t\geq 0}$ of W∗-$M$-bimodules and a continuous unital unit $\Xi=\{\xi(t)\}_{t\geq 0}$. There exists a one-to-one correspondence between contractive adapted right cocycles $w=\{w_{t}\}_{t\geq 0}$ on ${\rm End}(\mathfrak{H}_{M})$ and contractive units $\Lambda=\{\lambda(t)\}_{t\geq 0}$ of $H$ by relations $\lambda(t)=\kappa_{t}^{*}w_{t}\kappa_{0}\phi^{\frac{1}{2}}$ and $w_{t}=\pi_{\phi}(\kappa_{t}\lambda(t))\pi_{\phi}(\kappa_{0}\phi^{\frac{1}{2}})^{*}$ for all $t\geq 0$.

Proof.

Let $\{U_{s,t}\}_{s,t\geq 0}$ be a family giving the relative product system structure of $H$ and $\mathfrak{H}$ the inductive limit of $(H,\Xi)$.

Let $w=\{w_{t}\}_{t\geq 0}$ be a contractive adapted right cocycle for $\theta$. Note that each $\lambda(t)=\kappa_{t}^{*}w_{t}\kappa_{0}\phi^{\frac{1}{2}}$ is $\phi$-bounded. Moreover, for each $t\geq 0$, we have

[TABLE]

for each $x\in M$, and hence the contractivity of $w_{t}$ implies that $\|\pi_{\phi}(\lambda(t))^{*}\pi_{\phi}(\lambda(t))\|\leq 1$. We shall show that $\Lambda$ is a unit. For $s,t\geq 0$, $\kappa_{s+t}=U_{t}(\kappa_{s}\otimes{\rm id}_{t})U_{s,t}^{*}$ implies the following calculations.

[TABLE]

Conversely, let $\Lambda=\{\lambda(t)\}_{t\geq 0}$ be a contractive unit of $H$, and for each $t\geq 0$, $w_{t}=\pi_{\phi}(\kappa_{t}\lambda(t))\pi_{\phi}(\kappa_{0}\phi^{\frac{1}{2}})^{*}\in{\rm End}(\mathcal{H}_{M})$. For all $\xi\in\mathcal{H}$, the equation

[TABLE]

is implied from the approximation of $\kappa_{0}^{*}\xi$ by vectors as the form of $\phi^{\frac{1}{2}}x$. In particular,

[TABLE]

for all $t\geq 0,\ x\in M$, and hence $w_{t}=0$ on the orthogonal complement of the closed subspace $\kappa_{0}\kappa_{0}^{*}\mathcal{H}$. Thus, computations

[TABLE]

for every $x\in M$, implies that $w$ is a right cocycle. We shall show that $w$ is adapted. For all $t\geq 0$ and all $\xi\in\mathcal{H}$, by (3.8), we have

[TABLE]

By (3.8) again and the fact that the family $\{\kappa_{t}\kappa_{t}^{*}\}_{t\geq 0}$ is increasing, we have also

[TABLE]

We conclude that $\kappa_{t}\kappa_{t}^{*}w_{t}\kappa_{t}\kappa_{t}^{*}=w_{t}$, that is, the adaptedness.

We can check that the correspondence between contractive adapted right cocycles and contractive units is one-to-one by (3.8). $\Box$

Note that by (3.7) and (3.8), the unit associated with an adapted unitary right cocycle is unital, and the contractive adapted right cocycle associated with a unital unit preserves inner products on $\kappa_{0}\kappa_{0}^{*}\mathcal{H}$.

4. CP0-semigroups and units of product systems of W∗-bimodules

In this section, we will obtain a one-to-one correspondence between algebraic CP0-semigroups on $M$ and pairs of ($\phi$-spatial) product systems of W∗-$M$-bimodules and generating unital units up to unit preserving isomorphism. It will be shown that the dilation of the pair associated with a given CP0-semigroup $T$ is a dilation of $T$. Also, we will discuss a relation between the continuity of CP0-semigroups and one of units as follows: the unit associated with a CP0-semigroup is continuous, and conversely, a continuous unit gives rise to a CP0-semigroup.

First, we shall establish an algebraic CP0-semigroup from a unit. Let $\Xi=\{\xi(t)\}_{t\geq 0}$ be a unital unit of a $\phi$-spatial product system $H=\{\mathcal{H}_{t}\}_{t\geq 0}$ of W∗-$M$-bimodules. We define a unital linear map $T_{t}^{\Xi}$ on $M$ by

[TABLE]

for each $t\geq 0$ and $x\in M$.

Lemma 4.1**.**

The family $T^{\Xi}=\{T_{t}^{\Xi}\}_{t\geq 0}$ is an algebraic CP0-semigroup.

Proof.

By the definition, it is clear that each $T_{t}^{\Xi}$ is normal completely positive map.

For $s,t\geq 0$ and $x,y,z\in M$, we can compute as

[TABLE]

and hence $T_{s}^{\Xi}\circ T_{t}^{\Xi}=T_{s+t}^{\Xi}$. $\Box$

We can describe the continuity for the algebraic CP0-semigroup $T^{\Xi}$ as the one for the unit $\Xi$ as the following theorem.

Theorem 4.2**.**

Let $\mathfrak{H}$ be the inductive limit of the pair $(H,\Xi)$ and $U_{t}:\mathfrak{H}\otimes^{M}\mathcal{H}_{t}\to\mathfrak{H}$ the unitary in Theorem 3.8. The semigroup $T^{\Xi}$ associated with $\Xi$ is a CP0-semigroup if and only if

[TABLE]

holds for each $x\in M$.

Proof.

Suppose (4.2) for all $x\in M$. For $t\geq 0$ and $x,y,z\in M$, we have

[TABLE]

Thus, when $t\to+0$, the inner product $\langle T_{t}(x)\phi^{\frac{1}{2}}y,\phi^{\frac{1}{2}}z\rangle$ tends to $\langle\kappa_{0}(x\phi^{\frac{1}{2}})y,\kappa_{0}(\phi^{\frac{1}{2}})z\rangle=\langle x\phi^{\frac{1}{2}}y,\phi^{\frac{1}{2}}z\rangle.$ We conclude that for every $x\in M$, $T_{t}^{\Xi}(x)\to x$ weakly when $t\to+0$, and hence $T^{\Xi}$ is a CP0-semigroup by the boundedness of $\{\|T_{t}^{\Xi}(x)\|\}_{t\geq 0}$.

Conversely, we assume that $T^{\Xi}$ is a CP0-semigroup. We can compute as

[TABLE]

Thus, when $t\to+0$, we have $\|\kappa_{t}(x\xi(t))-\kappa_{0}(x\phi^{\frac{1}{2}})\|^{2}\to 0$. $\Box$

Definition 4.3**.**

We say that a unit $\Xi$ with (4.2) is weakly continuous.

Next, we shall construct a product system and a unit from a given algebraic CP0-semigroup $T$ on $M$. For $t>0$ and $\mathfrak{p}=(t_{1},\cdots,t_{n})\in\mathfrak{P}_{t}$, we define a W∗-$M$-bimodule

[TABLE]

where $M\otimes_{s}L^{2}(M)=M\otimes_{T_{s}}L^{2}(M)$ for each $s\geq 0$. Suppose $\mathfrak{p}=(t_{1},\cdots,t_{n})\succ\mathfrak{q}=(s_{1},\cdots,s_{m})$ in $\mathfrak{P}_{t}$, $\mathfrak{p}=\mathfrak{q}(s_{1})\lor\cdots\lor\mathfrak{q}(s_{m})$ and $\mathfrak{q}(s_{i})=(s_{i,1},\cdots,s_{i,k(i)})\in\mathfrak{P}_{s_{i}}$. We define a map $a_{\mathfrak{q}(s_{i})}:M\otimes_{s_{i}}L^{2}(M)\to\mathcal{H}^{T}(\mathfrak{q}(s_{i}),s_{i})$ by

[TABLE]

for each $x,y\in M$. We can check that $a_{\mathfrak{p}(s_{i})}$ is an $M$-bilinear isometry by Proposition 2.7. We define an isometry

[TABLE]

Then, the pair $(\{\mathcal{H}^{T}(\mathfrak{p},t)\}_{\mathfrak{p}\in\mathfrak{P}_{t}},\{a_{\mathfrak{p},\mathfrak{q}}\}_{\mathfrak{p}\succ\mathfrak{q}})$ is an inductive system of W∗-$M$-bimodules. Let $\mathcal{H}_{t}^{T}$ be the inductive limit and $\kappa_{\mathfrak{p},t}:\mathcal{H}^{T}(\mathfrak{p},t)\to\mathcal{H}_{t}^{T}$ the canonical embedding. Put $\mathcal{H}_{0}^{T}=L^{2}(M)$. Also, we define a family $\Xi^{T}=\{\xi^{T}(t)\}_{t\geq 0}$ by $\xi^{T}(t)=\kappa_{(t),t}(1_{M}\otimes\phi^{\frac{1}{2}})$ for each $t>0$ and $\xi^{T}(0)=\phi^{\frac{1}{2}}$.

Theorem 4.4**.**

The family $H^{T}=\{\mathcal{H}_{t}^{T}\}_{t\geq 0}$ is a $\phi$-spatial product system of W∗-$M$-bimodules and $\Xi^{T}$ is a generating unital unit of $H^{T}$.

Proof.

For $s,t>0$, we define a map $U_{s,t}^{T}:\mathcal{H}_{s}^{T}\otimes^{M}\mathcal{H}_{t}^{T}\to\mathcal{H}_{s+t}^{T}$ by

[TABLE]

for each $\mathfrak{q}=(s_{1},\cdots,s_{m})\in\mathfrak{P}_{s},\mathfrak{p}=(t_{1},\cdots,t_{n})\in\mathfrak{P}_{t},\ \xi_{\mathfrak{q}}\in\mathcal{D}(\mathcal{H}^{T}(\mathfrak{q},s);\phi)$ and $\eta_{\mathfrak{p}}\in\mathcal{H}^{T}(\mathfrak{p},t)$ Here, note that $\kappa_{\mathfrak{q},s}\xi_{\mathfrak{q}}$ is $\phi$-bounded. We shall show that $U_{s,t}^{T}$ is an isometry, i.e. the equation

[TABLE]

holds for all $\mathfrak{q},\mathfrak{q}^{\prime}\in\mathfrak{P}_{s},\ \mathfrak{p},\mathfrak{p}^{\prime}\in\mathfrak{P}_{t},\ \xi_{\mathfrak{q}}\in\mathcal{D}(\mathcal{H}^{T}(\mathfrak{q},s);\phi),\xi_{\mathfrak{q^{\prime}}}^{\prime}\in\mathcal{D}(\mathcal{H}^{T}(\mathfrak{q}^{\prime},s);\phi),\eta_{\mathfrak{p}}\in\mathcal{H}^{T}(\mathfrak{p},t)$ and $\eta_{\mathfrak{p}^{\prime}}^{\prime}\in\mathcal{H}^{T}(\mathfrak{p}^{\prime},t)$. If $\mathfrak{q}=\mathfrak{q}^{\prime}$ and $\mathfrak{p}=\mathfrak{p}^{\prime}$, we have

[TABLE]

In general case, since $\kappa_{\mathfrak{s},s}=\kappa_{\mathfrak{s}^{\prime},s}a_{\mathfrak{s}^{\prime},\mathfrak{s}}$ for all $\mathfrak{s},\mathfrak{s}^{\prime}\in\mathfrak{P}_{s}$ with $\mathfrak{s}^{\prime}\succ\mathfrak{s}$, if we take $\hat{\mathfrak{q}}\in\mathfrak{P}_{s},\hat{\mathfrak{p}}\in\mathfrak{P}_{t},$ such that $\hat{\mathfrak{q}}\succ\mathfrak{q},\mathfrak{q}^{\prime}$ and $\hat{\mathfrak{p}}\succ\mathfrak{p},\mathfrak{p}^{\prime}$, then

[TABLE]

In particular, we conclude that $U_{s,t}^{T}$ is well-defined and can be extended to an isometry from $\mathcal{H}_{s}^{T}\otimes^{M}\mathcal{H}_{t}^{T}$ to $\mathcal{H}_{s+t}^{T}$, and also denote the isometry by $U_{s,t}^{T}$ again. The surjectivity and the two-sides linearity of $U_{s,t}^{T}$ are obvious.

To show (3.1), it is enough to check it for $\phi$-bounded vectors with the form $\kappa_{\mathfrak{p},t}((x_{1}\otimes_{t_{1}}y_{1}\phi^{\frac{1}{2}})\phi^{-\frac{1}{2}}\cdots\phi^{-\frac{1}{2}}(x_{m}\otimes_{t_{m}}y_{m}\phi^{\frac{1}{2}}))$ for some $x_{i},y_{i}\in M$.

We conclude that $H^{T}$ is a product system of W∗-bimodules, and it is clear that $\Xi^{T}$ is a generating unital unit of $H^{T}$ by the definition of $\{U_{s,t}^{T}\}_{s,t\geq 0}$. $\Box$

Example 4.5**.**

Let $T=\{T_{t}\}_{t\geq 0}$ be an algebraic CP0-semigroup obtained by a semigroup $v$ of isometries in a von Neumann algebra $M$ in Example 2.2. For each $t\geq 0$, we can identify $M\otimes_{t}L^{2}(M)$ with $L^{2}(M)$ by a bilinear unitary $U^{t}(x\otimes_{t}y\phi^{\frac{1}{2}})=xv_{t}y\phi^{\frac{1}{2}}$ for $x,y\in M$. For $t\geq 0$ and $\mathfrak{p}\in\mathfrak{P}_{t}$, the unitaries induce a bilinear unitary $U^{\mathfrak{p}}:\mathcal{H}^{T}(\mathfrak{p},t)\to L^{2}(M)$ such that $U^{\mathfrak{p}}a_{\mathfrak{p},\mathfrak{q}}=U^{\mathfrak{q}}$ for all $\mathfrak{p}\succ\mathfrak{q}$.

Example 4.6**.**

We consider the CP0-semigroup generated by a family of stochastic matrices. Let $M=\mathbb{C}\oplus\mathbb{C}$ be a von Neumann algebra regarded as a von Neumann subalgebra of $M_{2}(\mathbb{C})$. Then, we have $L^{2}(M)=\mathbb{C}\oplus\mathbb{C}$. Let $T=\{T_{t}\}_{t\geq 0}$ be the CP0-semigroup on $M$ associated with stochastic matrices $\left\{\begin{pmatrix}e^{-t}&1-e^{-t}\\ 0&1\end{pmatrix}\right\}_{t\geq 0}$, that is, each $T_{t}$ is defined by $T_{t}(a\oplus b)=(e^{-t}a+(1-e^{-t})b)\oplus b$ for each $a,b\in\mathbb{C}$. By using the normalized canonical trace on $M_{2}(\mathbb{C})$, for $t\geq 0$ and $\mathfrak{p}=(t_{1},\cdots,t_{n})\in\mathfrak{P}_{t}$, it turns out that $\mathcal{H}^{T}(\mathfrak{p},t)$ coincides with $\mathbb{C}^{n}\oplus\mathbb{C}^{2}$ on which $M$ acts as

[TABLE]

for $a,b,x_{1},\cdots,x_{n},y,z\in\mathbb{C}$. The W∗-bimodule $\mathcal{H}^{T}(\mathfrak{p},t)$ depends on only $T,\ t$ and the number $n$ of the partition $\mathfrak{p}$, and $\mathcal{H}_{t}^{T}$ is infinite dimensional for all $t>0$.

Example 4.7**.**

We consider the W∗-$\mathcal{B}(\mathcal{H})$-bimodule $\mathcal{H}^{T}(\mathfrak{p},t)$ associated with the CCR heat flow in Example 2.3 for $t\geq 0$ and $\mathfrak{p}\in\mathfrak{P}_{t}$.

Let $\mathcal{H}=L^{2}(\mathbb{R})$ and $M=\mathcal{B}(\mathcal{H})$. Then, the standard space $L^{2}(\mathcal{B}(\mathcal{H}))$ of $M$ is isomorphic to $\mathcal{H}\otimes\mathcal{H}^{*}\cong\mathcal{C}_{2}(\mathcal{H})$. For $t\geq 0,\ x,x^{\prime}\in M$ and $\xi\otimes\eta^{*},\xi^{\prime}\otimes{\eta^{\prime}}^{*}\in\mathcal{H}\otimes\mathcal{H}^{*}$, the inner product on $M\otimes_{t}L^{2}(M)$ is given by $\langle x\otimes(\xi\otimes\eta^{*}),x^{\prime}\otimes(\xi^{\prime}\otimes{\eta^{\prime}}^{*})\rangle=\langle\eta^{\prime},\eta\rangle\int_{\mathbb{R}^{2}}\langle xW_{\frac{{\bf x}}{\sqrt{2}}}^{*}\xi,x^{\prime}W_{\frac{{\bf x}}{\sqrt{2}}}^{*}\xi^{\prime}\rangle d\mu_{t}({\bf x}).$ Let $\mathfrak{p}=(t_{1},\cdots,t_{n})\in\mathfrak{P}_{t}$. Fix a faithful normal state $\phi$ on $M$ and suppose $\rho\in\mathcal{C}_{1}(\mathcal{H})$ is associated with $\phi$ by $\phi(x)={\rm tr}(\rho x)$ for all $x\in M$. In terms of $\mathcal{C}_{2}(\mathcal{H})$, the inner product on $\mathcal{H}^{T}(\mathfrak{p},t)$ is

[TABLE]

for each $x_{1},\cdots,x_{n},y_{1},\cdots,y_{n},a_{1},\cdots,a_{n},b_{1},\dots,b_{n}\in M$. The properties (2.1) and $\mu_{s}*\mu_{t}=\mu_{s+t}$ ensure the fact that $a_{\mathfrak{p},\mathfrak{q}}$ defined by (4.4) is isometry for $\mathfrak{p},\mathfrak{q}\in\mathfrak{P}_{t}$.

In Section 5, we also will discuss the case of classical heat semigroups, that is, the construction of product system associated with the heat semigroup on a compact Riemannian manifold and its dilation in more detail.

We denote the inductive limit of the pair $(H^{T},\Xi^{T})$ associated with an algebraic CP0-semigroup $T$ by $\mathfrak{H}^{T}$, and the unitary: $\mathfrak{H}^{T}\otimes^{M}\mathcal{H}_{t}^{T}\to\mathfrak{H}^{T}$ in Theorem 3.8 by $U_{t}^{T}$. Also, $\{b_{t,s}^{T}\}_{s\leq t}$ is the family of isometries giving the inductive system for $H^{T}$ as (3.3) and $\pi^{T}$ is the faithful representation of $M$ on $\mathfrak{H}^{T}$ defined as (3.4).

Proposition 4.8**.**

If $T$ is a CP0-semigroup, then the unit $\Xi^{T}$ is continuous.

Proof.

Suppose that $t<\min\{s_{1},\cdots,s_{m}\}$ and $\xi=\kappa_{s}\kappa_{\mathfrak{q},s}((x_{1}\otimes_{s_{1}}y_{1}\phi^{\frac{1}{2}})\phi^{-\frac{1}{2}}\cdots\phi^{-\frac{1}{2}}(x_{m}\otimes_{s_{m}}y_{m}\phi^{\frac{1}{2}}))$ for some $s\geq 0,\ \mathfrak{q}=(s_{1},\cdots,s_{m})\in\mathfrak{P}_{s},\ x_{1},\cdots,x_{m},y_{1},\cdots,y_{m}\in M$. If we put $\mathfrak{p}^{\prime}=(t,s_{1}-t,t,s_{2}-t,t,\cdots,s_{m}-t,t)$, then we have $\mathfrak{p}^{\prime}\succ(t)\lor\mathfrak{p},\mathfrak{p}\lor(t)$. Now, we have

[TABLE]

On the other hand, we have

[TABLE]

By calculations of inner products and [23, Lemma A.2], when $t$ tend to 0, we conclude that $U_{t}(\xi\phi^{-\frac{1}{2}}\xi^{T}(t))$ converges to $\xi$. $\Box$

The dilation of the pair $(H^{T},\Xi^{T})$ associated with a CP0-semigroup gives a dilation of $T$ as follows:

Theorem 4.9**.**

Let $T$ be a CP0-semigroup on a von Neumann algebra $M$ and $\theta$ the dilation of the pair $(H^{T},\Xi^{T})$. The triple $({\rm End}(\mathfrak{H}_{M}^{T}),\pi^{T}(1_{M}),\theta)$ is a dilation of $T$. Moreover, if we denote by $N$ the von Neumann algebra generated by $\bigcup_{t\geq 0}\theta_{t}(\pi^{T}(M))$, the triple $(N,\pi^{T}(1_{M}),\theta|_{N})$ is the minimal dilation of $T$.

Proof.

We shall show that $\pi^{T}(T_{t}(x))=p\theta_{t}(\pi^{T}(x))p$ for all $t\geq 0$ and $x\in M$. For all $y,z\in M$, we have

[TABLE]

Thus we have $T_{t}(x)=\kappa_{0}^{*}\theta_{t}(\pi(x))\kappa_{0}$.

For $t\geq 0,\ \mathfrak{p}=(t_{1},\cdots,t_{n})\in\mathfrak{P}_{t},\ x_{1},\cdots,x_{n},y\in M$, we can check that

[TABLE]

Hence, we have $\overline{{\rm span}}(N\pi(1_{M})\mathfrak{H}^{T})\supset\overline{{\rm span}}(N\kappa_{0}L^{2}(M))=\mathfrak{H}^{T}$. Since the central support $c(\pi^{T}(1_{M}))$ of $\pi^{T}(1_{M})$ in $N$ is the projection onto $\overline{{\rm span}}(N\pi(1_{M})\mathfrak{H}^{T})$, we have $c(\pi^{T}(1_{M}))=1_{N}$. We conclude that the triplet $(N,\pi(1_{M}),\theta)$ is the minimal dilation of $T$. $\Box$

Remark 4.10**.**

Let $\theta$ be the dilation of a CP0-semigroup $T$ defined by (3.5). When $s\leq t$ and $\xi_{s}\in\mathcal{H}_{s}^{T}$, we have

[TABLE]

for all $a\in{\rm End}(\mathfrak{H}_{M}^{T})$, and hence the operator $\theta_{t}(a)\kappa_{s}\xi_{s}$ is depend on only the image $a(\kappa_{0}\phi^{\frac{1}{2}})$.

The following theorem asserts that the correspondence between algebraic CP0-semigroups and generating unital units is one-to-one up to unit preserving isomorphism.

Theorem 4.11**.**

Let $T$ be an algebraic CP0-semigroup on $M$ and $\Xi$ a generating unital unit of a $\phi$-spatial product system $H$ of W∗-$M$-bimodule. Then, we have $T^{\Xi^{T}}=T$ and there exists an isomorphism from $H^{T^{\Xi}}$ onto $H$, which preserves the units $\Xi^{T^{\Xi}}$ and $\Xi$.

Proof.

It is clear that $T^{\Xi^{T}}=T$. For $t\geq 0$ and a partition $\mathfrak{p}=(t_{1},\cdots,t_{n})\in\mathfrak{P}_{t}$, we define a map $u_{t}:\mathcal{H}_{t}^{\Xi^{T}}\to\mathcal{H}_{t}$ by

[TABLE]

for each $x_{1},\cdots,x_{n},y\in M$, where $\kappa_{\mathfrak{p},t}:\mathcal{H}^{T^{\Xi}}(\mathfrak{p},t)\to\mathcal{H}_{t}^{T^{\Xi}}$ is the canonical embedding. We can check that $u_{t}$ is an isometry. Since $\Xi$ is generating, $u_{t}$ can be extended as unitary from $\mathcal{H}_{t}^{T^{\Xi}}$ onto $\mathcal{H}_{t}$.

We must show that $U_{s,t}((u_{s}\xi_{s})\phi^{-\frac{1}{2}}(u_{t}\eta_{t}))=u_{s+t}U_{s,t}^{T^{\Xi}}(\xi_{s}\phi^{-\frac{1}{2}}\eta_{t})$ for all $\xi_{s}\in\mathcal{D}(\mathcal{H}_{s}^{T^{\Xi}};\phi)$ and all $\eta_{t}\in\mathcal{H}_{t}^{T^{\Xi}}$. It enough to show it for

[TABLE]

where $\mathfrak{q}=(s_{1},\cdots,s_{m})\in\mathfrak{P}_{s},\ \mathfrak{p}=(t_{1},\cdots,t_{n})\in\mathfrak{P}_{t}$ and $x_{1},\cdots,x_{m},z_{1},\cdots,z_{n},$ $w\in M$. We put $\zeta_{1}=x_{1}\xi(s_{1})\phi^{-\frac{1}{2}}\cdots\phi^{-\frac{1}{2}}x_{m}\xi(s_{m}),\ \zeta_{2}=z_{1}\xi(t_{1})\phi^{-\frac{1}{2}}\cdots\phi^{-\frac{1}{2}}z_{n-1}\xi(t_{n-1})\phi^{-\frac{1}{2}}z_{n}\xi(t_{n})w.$ By the associativity of $\{U_{s,t}\}_{s,t\geq 0}$, we have

[TABLE]

We conclude that $\{u_{t}\}_{t\geq 0}$ gives an isomorphism. $\Box$

As a corollary of Theorem 4.11, it will turn out that every weakly continuous generating unit is continuous. For this, we shall show that an isomorphism between product systems of W∗-bimodules induces a unitary between their inductive limits as follows: let $H=\{\mathcal{H}_{t}\}_{t\geq 0}$ and $K=\{\mathcal{K}_{t}\}_{t\geq 0}$ be product systems of W∗-$M$-bimodules with unital units $\Xi$ and $\Lambda$, respectively. Suppose a family $=\{u_{t}\}_{t\geq 0}$ is an isomorphism from $H$ onto $K$. Then, we can define the canonical right $M$-linear unitary $u$ from the inductive limit $\mathfrak{H}$ of $(H,\Xi)$ onto the one $\mathfrak{K}$ of $(K,\Lambda)$ by $u(\kappa_{t}^{H}(\xi_{t}))=\kappa_{t}^{K}u_{t}(\xi_{t})$ for each $t\geq 0$ and $\xi_{t}\in\mathcal{H}_{t}$, where $\kappa_{t}^{H}:\mathcal{H}_{t}\to\mathfrak{H}$ and $\kappa_{t}^{K}:\mathcal{K}_{t}\to\mathfrak{K}$ are the canonical embedding.

Corollary 4.12**.**

If $\Xi$ is a weakly continuous generating unital unit of a product system $H$ of W∗-bimodules, then $\Xi$ is continuous.

Proof.

Let $U_{t}^{T^{\Xi}}$ be the unitary giving the right W∗-$M$-module isomorphism $\mathfrak{H}^{T^{\Xi}}\otimes^{M}\mathcal{H}_{t}^{T^{\Xi}}\to\mathfrak{H}^{T^{\Xi}}$ for each $t\geq 0$. By Proposition 4.8, the unit $\Xi^{T^{\Xi}}=\{\xi^{T^{\Xi}}(t)\}_{t\geq 0}$ satisfies

[TABLE]

for all $\xi\in\mathcal{D}(\mathcal{H}^{T^{\Xi}};\phi)$. Suppose $\mathfrak{H}$ and $\mathfrak{H}^{T^{\Xi}}$ are the inductive limit of $(H,\Xi)$ and $(H^{T^{\Xi}},\Xi^{T^{\Xi}})$, respectively, and $\kappa_{t}:\mathcal{H}_{t}\to\mathfrak{H}$ and $\kappa_{t}^{T^{\Xi}}:\mathcal{H}_{t}^{T^{\Xi}}\to\mathfrak{H}^{T^{\Xi}}$ are the canonical embedding for each $t\geq 0$. Let $u$ be the unitary from $\mathfrak{H}^{T^{\Xi}}$ onto $\mathfrak{H}$ induced from the isomorphism $\{u_{t}\}_{t\geq 0}$ which is obtained by Theorem 4.11. For each $\xi_{s}\in\mathcal{H}_{s}^{T^{\Xi}}$, we have $uU_{t}^{T^{\Xi}}(\kappa_{s}^{T^{\Xi}}\xi_{s}\phi^{-\frac{1}{2}}\xi^{T^{\Xi}}(t))=U_{t}((u\kappa_{s}^{T^{\Xi}}\xi_{s})\phi^{-\frac{1}{2}}u_{t}\xi^{T^{\Xi}}(t))$ by (3.2) and (3.2). Thus, by (4.5) and the fact that $\{u_{t}\}_{t\geq 0}$ is unit preserving, we have $\|U_{t}(\xi\phi^{-\frac{1}{2}}\xi(t))-\xi\|=\|U_{t}((uu^{*}\xi)\phi^{-\frac{1}{2}}(u_{t}u_{t}^{*}\xi(t)))-\xi\|=\|U_{t}^{T^{\Xi}}((u^{*}\xi)\phi^{-\frac{1}{2}}\xi^{T^{\Xi}}(t))-u^{*}\xi\|$, and hence $\Xi$ is continuous. $\Box$

Remark 4.13**.**

Let $T$ be a CP0-semigroup on a von Neumann algebra acting on a separable Hilbert space $\mathcal{H}$. The product system $H^{T}=\{\mathcal{H}_{t}^{T}\}_{t\geq 0}$ of W∗-bimodules associated with $T$ gives a relation between Bhat-Skeide’s ([8]) and Muhly-Solel’s ([14]) constructions of the minimal dilation of $T$ as follows: a common point of the two methods is to establish a product system of von Neumann bimodules which has an information of $T$ by taking the inductive limits with respect to refinements of partitions. Let $\{E_{t}^{T}\}_{t\geq 0}$ and $\{E^{T}(t)\}_{t\geq 0}$ be the product systems of von Neumann bimodules associated with $T$ appearing in Bhat-Skeide’s and Muhly-Solel’s constructions, respectively. Note that each $E_{t}^{T}$ is a von Neumann $M$-bimodule and each $E^{T}(t)$ is a von Neumann $M^{\prime}$-bimodule. By the correspondence between W∗-bimodules and von Neumann bimodules (see [20, Section 2]), we have a correspondence.

[TABLE]

for each $t\geq 0$. This is an extension to the continuous case of the relation in the discrete case given by [17] and the proof of the correspondence (4.6) is essentially the same.

5. Heat semigroups on manifolds and product systems

In this section, we will consider the product system associated a heat semigroup $T$ and a dilation of $T$.

We shall recall the concept of heat semigroup on a compact Riemannian manifold. We refer the reader to [10] for their general theory. Let $\mathcal{M}$ be a compact Riemannian manifold with the normalized Riemannian measure $\mu$ associated with $\mathcal{M}$. We can define the self-adjoint positive (unbounded) operator $\Delta$ on $L^{2}(\mathcal{M})$ like the Laplacian on the Euclid space. The operator $\Delta$ is called the Laplacian (or Dirichlet Laplacian) on $\mathcal{M}$. The semigroup $T=\{T_{t}\}_{t\geq 0}$ of bounded operators on $L^{2}(\mathcal{M})$ defined by $T_{t}=e^{-t\Delta}$ for each $t\geq 0$, is called the heat semigroup on $\mathcal{M}$.

For $t>0$, it is known that there exists a measurable function $p_{t}$ on $\mathcal{M}\times\mathcal{M}$ such that

[TABLE]

for each $f\in L^{2}(\mathcal{M})$. The family $\{p_{t}\}_{t>0}$ called the heat kernel on $\mathcal{M}$ has the following properties:

1. (1)

$p_{t}(x,y)=p_{t}(y,z)\geq 0\ (t>0,\ x,y\in\mathcal{M}).$ 2. (2)

$\int_{\mathcal{M}}p_{t}(x,y)d\mu(y)=1\ (t>0,\ x\in\mathcal{M}).$ 3. (3)

$p_{s+t}(x,z)=\int_{\mathcal{M}}p_{t}(x,y)p_{t}(y,z)d\mu(z)\ (s,t>0,\ x,z\in\mathcal{M}).$

The equation (5.1) enables as to extend the heat semigroup $T$ to a semigroup on $L^{p}(\mathcal{M})$ for each $1\leq p\leq\infty$. We have the following continuity with respect to parameters:

[TABLE]

for each $f\in L^{p}(\mathcal{M})$ and $p=1,2$.

The heat semigroup $T$ on $\mathcal{M}$ is a CP0-semigroup on the commutative von Neumann algebra $M=L^{\infty}(\mathcal{M})$. A Noncommutative Laplacian and the associated CP0-semigroup on a type I factor are discussed in [4, Chapter 7]. They are noncommutative analogies of the Laplacian on and heat semigroup on a manifold.

Now, we shall compute the product system $H^{T}$ of W∗-bimodules associated with the heat semigroup $T$ on $M=L^{\infty}(\mathcal{M})$. For this, we introduce the following notations.

Definition 5.1**.**

For $t>0$ and $\mathfrak{p}=(t_{1},\cdots,t_{n})\in\mathfrak{P}_{t}$, we define a probability measure $\mu_{\mathfrak{p}}$ on $\mathcal{M}_{\mathfrak{p}}=\mathcal{M}^{n+1}$ by

[TABLE]

For convenience, we define as $\mathcal{M}_{()}=\mathcal{M}$ and $\mu_{()}=\mu$ for the empty partition $()$.

Definition 5.2**.**

Let $s,t>0,\ \mathfrak{p}\in\mathfrak{P}_{s},\ \mathfrak{q}\in\mathfrak{P}_{t}$ with $\#\mathfrak{p}=m,\ \#\mathfrak{q}=n$, and $f_{\mathfrak{p}},g_{\mathfrak{q}}$ and $h$ be functions on $\mathcal{M}_{\mathfrak{p}},\mathcal{M}_{\mathfrak{q}}$ and $\mathcal{M}$, respectively. We define functions $f_{\mathfrak{p}}\Box g_{\mathfrak{q}},f_{\mathfrak{p}}\Box h$ and $h\Box g_{\mathfrak{q}}$ on $\mathcal{M}_{\mathfrak{p}\lor\mathfrak{q}},\mathcal{M}_{\mathfrak{p}}$ and $\mathcal{M}_{\mathfrak{q}}$, respectively, by

[TABLE]

for $x_{i},y_{j},z\in\mathcal{M}$.

For $t>0$ and $\mathfrak{p}\in\mathfrak{P}_{t}$, the Hilbert space $L^{2}(\mathcal{M}_{\mathfrak{p}},\mu_{\mathfrak{p}})$ has a canonical W∗-$M$-bimodule structure given by $gf=g\Box f$ and $f\Box g$ for each $f\in L^{2}(\mathcal{M}_{\mathfrak{p}},\mu_{\mathfrak{p}})$ and $g\in M$. Then, we can obtain the following identification as W∗-$M$-bimodules.

Proposition 5.3**.**

For $t>0$ and $\mathfrak{p}\in\mathfrak{P}_{t}$, we have an isomorphism $H^{T}(\mathfrak{p},t)\cong L^{2}(\mathcal{M}_{\mathfrak{p}},\mu_{\mathfrak{p}})$ as W∗-$M$-bimodules.

Proof.

Let $\tau$ be the canonical faithful normal trace on $M=L^{\infty}(\mathcal{M})$ given by integrals on $\mathcal{M}$. Suppose $\mathfrak{p}=(t_{1},\cdots,t_{n})$. We define a $M$-bilinear map $u_{\mathfrak{p},t}:H^{T}(\mathfrak{p},t)\to L^{2}(\mathcal{M}_{\mathfrak{p}},\mu_{\mathfrak{p}})$ by

[TABLE]

for each $f_{1},g_{1},\cdots,f_{n},g_{n}\in M$, where $f_{i}(x_{i})$ denotes the function $f_{i}$ on $\mathcal{M}$ with variables $x_{i}$ and $g_{i}(x_{i+1})$ is similar. By the formula in Proposition 2.7, we have

[TABLE]

and hence $u_{\mathfrak{p},t}$ is an isometry.

We shall check that $u_{\mathfrak{p},t}$ is surjective. For an arbitrary $\varepsilon>0$ and $f\in L^{2}(\mathcal{M}_{\mathfrak{p}},\mu_{\mathfrak{p}})$, there exists $g\in C(\mathcal{M}^{n+1})$ such that $\|f-g\|_{L^{2}(\mathcal{M}_{\mathfrak{p}},\mu_{\mathfrak{p}})}<\varepsilon$. Since the space

[TABLE]

is dense in $C(\mathcal{M}^{n+1})$ with respect to the uniform convergence topology, there exist $N\in\mathbb{N}$ and functions $f_{i,j}\in C(\mathcal{M})$ for each $i=1,\cdots,n+1$ and $j=1,\cdots,N$ such that $\|g-\sum_{j=1}^{N}f_{1,j}(x_{1})\cdots f_{n+1,j}(x_{n+1})\|_{\infty}<\varepsilon$. Now, equations

[TABLE]

imply that the image $u_{\mathfrak{p},t}(D_{\mathfrak{p},t})$ of

[TABLE]

by $u_{\mathfrak{p},t}$ is dense in $L^{2}(\mathcal{M}_{\mathfrak{p}},\mu_{\mathfrak{p}})$, and hence $u_{\mathfrak{p},t}$ is unitary. $\Box$

Remark 5.4**.**

There is a connection between heat kernels and Brownian motions on Riemannian manifolds. Let $T$ be a heat semigroup on a compact Riemannian manifold $\mathcal{M}$ and $\{p_{t}\}_{t>0}$ the heat kernel associated with $T$. For $x\in\mathcal{M}$, there exist a probability space $(\Omega_{x},\mathbb{P}_{x})$ and an $\mathcal{M}$-valued stochastic process $\{X_{t}\}_{t\geq 0}$ such that

[TABLE]

for all Borel set $A\subset\mathcal{M}$, where $1_{A}$ is the characteristic function on $A$. Also, the family $\{\mu_{\mathfrak{p}}\mid\mathfrak{p}\in\bigcup_{t>0}\mathfrak{P}_{t}\}$ of probability measures describes joint distributions for $\{X_{t}\}_{t\geq 0}$ as follows: for a partition $\mathfrak{p}=(t_{1},\cdots,t_{n})\in\mathfrak{P}_{t}$ and Borel sets $A_{1},\cdots,A_{n}$, if we denote $\check{\mathfrak{p}}=(t_{2},t_{3},\cdots,t_{n})$, then we have

[TABLE]

Now, we shall reconstruct the product system associated with the heat semigroup $T$ and a dilation of $T$ under the identification in Proposition 5.3.

Let $\mathfrak{q}\succ\mathfrak{p}$ with $\mathfrak{p}=(t_{1},\cdots,t_{n})\in\mathfrak{P}_{t}$ and $\mathfrak{q}=\mathfrak{p}(t_{1})\lor\cdots\lor\mathfrak{p}(t_{n})$ with $\mathfrak{p}(t_{i})=(t_{i,1},\cdots,t_{i,k(i)})\in\mathfrak{P}_{t_{i}}$. The isometry $a_{\mathfrak{q},\mathfrak{p}}:L^{2}(\mathcal{M}_{\mathfrak{p}},\mu_{\mathfrak{p}})\to L^{2}(\mathcal{M}_{\mathfrak{q}},\mu_{\mathfrak{q}})$ is given by

[TABLE]

for each $f_{\mathfrak{p}}\in L^{2}(\mathcal{M}_{\mathfrak{p}},\mu_{\mathfrak{p}})$ and $x_{i,j},y\in\mathcal{M}$. By $\mathcal{H}_{t}^{T}$, we denote the inductive limit of the inductive system $(\{L^{2}(\mathcal{M}_{\mathfrak{p}},\mu_{\mathfrak{p}})\}_{\mathfrak{p}\in\mathfrak{P}_{t}},\{a_{\mathfrak{q},\mathfrak{p}}\}_{\mathfrak{p}\succ\mathfrak{q}})$. Let $\kappa_{\mathfrak{p},t}:L^{2}(\mathcal{M}_{\mathfrak{p}},\mu_{\mathfrak{p}})\to\mathcal{H}_{t}^{T}$ be the canonical embedding.

The family $\{U_{s,t}:\mathcal{H}_{s}^{T}\otimes^{M}\mathcal{H}_{t}^{T}\to\mathcal{H}_{s+t}^{T}\}_{s,t\geq 0}$ of $M$-bilinear unitaries giving the structure of the product system $H^{T}=\{\mathcal{H}_{t}^{T}\}_{t\geq 0}$ associated with the heat semigroup $T$ satisfies the follows: for $s,t>0,\ \mathfrak{p}\in\mathfrak{P}_{s},\ \mathfrak{q}\in\mathfrak{P}_{t}$ with $\#\mathfrak{p}=m,\ \#\mathfrak{q}=n$ and $f_{\mathfrak{p}}\in L^{\infty}(\mathcal{M}_{\mathfrak{p}},\mu_{\mathfrak{p}}),\ g_{\mathfrak{q}}\in L^{2}(\mathcal{M}_{\mathfrak{q}},\mu_{\mathfrak{q}}),\ h\in L^{2}(\mathcal{M})=\mathcal{H}_{0}^{T},\ h^{\prime}\in M=L^{\infty}(\mathcal{M})$, we have

[TABLE]

Also, the unit $\Xi^{T}=\{\xi^{T}(t)\}_{t\geq 0}$ associated with $T$ is given by $\xi^{T}(t)=\kappa_{(t),t}1_{\mathcal{M}^{2}}$ for each $t>0$ and $\xi^{T}(0)=1_{\mathcal{M}}$.

For $0<s\leq t,\ \mathfrak{p}=(s_{1},\cdots,s_{m})\in\mathfrak{P}_{s}$ and $f_{\mathfrak{p}}\in L^{2}(\mathcal{M}_{\mathfrak{p}},\mu_{\mathfrak{p}})$, the image of $\kappa_{\mathfrak{p},s}f_{\mathfrak{p}}$ by the isometry $b_{t,s}:\mathcal{H}_{s}^{T}\to\mathcal{H}_{t}^{T}$ is given by $b_{t,s}(\kappa_{\mathfrak{p},s}f_{\mathfrak{p}})=\kappa_{(t-s)\lor\mathfrak{p},t}\tilde{f}_{\mathfrak{p}}$, where $\tilde{f}_{\mathfrak{p}}\in L^{2}(\mathcal{M}_{(t-s)\lor\mathfrak{p}},\mu_{(t-s)\lor\mathfrak{p}})$ in the right side is a function defined by $\tilde{f}_{\mathfrak{p}}(x,x_{1},\cdots,x_{m},y)=f_{\mathfrak{p}}(x_{1},\cdots,x_{m},y)$ for each $x,x_{i},y\in\mathcal{M}$. If we also define $\tilde{f}\in L^{2}(\mathcal{M}_{(t)},\mu_{(t)})$ for $f\in L^{2}(\mathcal{M})$ by $\tilde{f}(x,y)=f(y)$ for each $x,y\in\mathcal{M}$, then $b_{t,0}f=\kappa_{(t),t}\tilde{f}$. We denote the inductive limit of $(\{\mathcal{H}_{t}^{T}\}_{t\geq 0},\{b_{t,s}\}_{s\leq t})$ by $\mathfrak{H}^{T}$ and let $\kappa_{t}:\mathcal{H}_{t}^{T}\to\mathfrak{H}^{T}$ be the canonical embedding. We can describe the isometry $b_{t,0}^{*}$ by heat kernels as follows:

Proposition 5.5**.**

For $t>0,\ \mathfrak{p}=(t_{1},\cdots,t_{n})\in\mathfrak{P}_{t}$ and $f_{\mathfrak{p}}\in L^{2}(\mathcal{M}_{\mathfrak{p}},\mu_{\mathfrak{p}})$, we have a formula

[TABLE]

for each $y\in\mathcal{M}$, where $\mathfrak{p}^{\prime}=(t_{1},\cdots,t_{n-1})$.

Proof.

Note that the function defined by the right hand belongs to $L^{2}(\mathcal{M})$ by Jensen’s inequality with respect to the convex function $h$ defined by $h(z)=z^{2}$ for each $z\in\mathbb{R}$.

For each $g\in L^{2}(\mathcal{M})$, we can compute as

[TABLE]

Thus, we have shown the desired equation. $\Box$

The right action of $M$ on the right W∗-$M$-module $\mathfrak{H}^{T}$ is given by $(\kappa_{t}\kappa_{\mathfrak{p},t}f_{\mathfrak{p}})g=\kappa_{t}\kappa_{\mathfrak{p},t}(f_{\mathfrak{p}}\Box g)$ for each $t\geq 0,\ \mathfrak{p}\in\mathfrak{P}_{t},\ f\in L^{2}(\mathcal{M}_{\mathfrak{p}},\mu_{\mathfrak{p}})$ and $g\in M=L^{\infty}(\mathcal{M})$. Clearly, for $t>0$, the identification $\mathfrak{H}^{T}\otimes^{M}\mathcal{H}_{t}^{T}\cong\mathfrak{H}^{T}$ as right W∗-modules is obtained by the right $M$-linear unitary

[TABLE]

where $f_{\mathfrak{p}}\in L^{\infty}(\mathcal{M}_{\mathfrak{p}},\mu_{\mathfrak{p}})$ and $g_{\mathfrak{q}}\in L^{2}(\mathcal{M}_{\mathfrak{q}},\mu_{\mathfrak{q}})$. Also, the unitary $U_{0}:\mathfrak{H}^{T}\otimes^{M}L^{2}(\mathcal{M})\to\mathfrak{H}^{T}$ is the canonical isomorphism.

By the embedding (3.4), we regard as $M\subset{\rm End}(\mathfrak{H}_{M}^{T})$. Then, the following direct computations imply that the triple of ${\rm End}(\mathfrak{H}_{M}^{T})$, the E0-semigroup $\theta$ on ${\rm End}(\mathfrak{H}_{M}^{T})$ defined by (3.5) and the projection $p=\kappa_{0}\kappa_{0}^{*}$ becomes a dilation of $T$. For $t>0,\ x\in\mathcal{M}$ and $f\in M=L^{\infty}(\mathcal{M}),\ g\in L^{2}(\mathcal{M})$, we have

[TABLE]

Here, the forth equality is implied from the formula in Proposition 5.5 in the case when $\mathfrak{p}=(t)$.

6. Classification of E0-semigroups

In this section, we shall classify E0-semigroups on a von Neumann algebra up to cocycle equivalence by the product systems of W∗-bimodules associated with their E0-semigroups as CP0-semigroups.

Let $\theta$ be an E0-semigroup on $M$. For each $t\geq 0$, let $\tilde{\mathcal{H}}^{\theta}_{t}=L^{2}(M)$ as sets with a left and a right actions of $M$ defined by $x\xi y=\theta_{t}(x)\xi y$ for each $x,y\in M$ and $\xi\in\tilde{\mathcal{H}}^{\theta}_{t}$, and $\tilde{\xi}^{\theta}(t)=\phi^{\frac{1}{2}}$. Then, the family $\tilde{H}^{\theta}=\{\tilde{\mathcal{H}}^{\theta}_{t}\}_{t\geq 0}$ is a product system of W∗-bimodules and $\tilde{\Xi}^{\theta}=\{\tilde{\xi}^{\theta}(t)\}_{t\geq 0}$ is a continuous generating unital unit.

Proposition 6.1**.**

Let $(H^{\theta},\Xi^{\theta})$ be the pair associated with $\theta$ as CP0-semigroups. There is an isomorphism $u^{\theta}=\{u^{\theta}_{t}\}_{t\geq 0}$ from $H^{\theta}$ onto $\tilde{H}^{\theta}$ preserving the units $\Xi^{\theta}$ and $\tilde{\Xi}^{\theta}$.

Proof.

For $t>0$, we define $u_{t}^{\theta}:\mathcal{H}_{t}^{\theta}\to\tilde{\mathcal{H}}_{t}^{\theta}$ by

[TABLE]

for each $\mathfrak{p}=(t_{1},\cdots,t_{n})\in\mathfrak{P}_{t}$ and $x_{1},\cdots,x_{n},y_{1},\cdots,y_{n}\in M$, where $\kappa_{\mathfrak{p},t}:\mathcal{H}^{\theta}(\mathfrak{p},t)\to\mathcal{H}_{t}^{\theta}$ is the canonical embedding. Put $u_{0}^{\theta}={\rm id}$. The family $u^{\theta}=\{u_{t}^{\theta}\}_{t\geq 0}$ is the desired isomorphism. $\Box$

Example 6.2**.**

For an E0-semigroup $\theta$ on $\mathcal{B}(\mathcal{H})$, the product system $H^{\theta}$ associated with $\theta$ is isomorphic to the product system $\tilde{H}^{\theta}=\{\tilde{\mathcal{H}}_{t}^{\theta}\}_{t\geq 0}$ of W∗-bimodules and ${\mathcal{H}}_{t}^{\theta}=\mathcal{H}\otimes\mathcal{H}^{*}$ with left and right actions of $\mathcal{B}(\mathcal{H})$ defined by $x(\xi\otimes\eta^{*})y=(\alpha_{t}(x)\xi)\otimes(y^{*}\eta)^{*}$ for each $t\geq 0,\ \xi,\eta\in\mathcal{H}$ and $x,y\in\mathcal{B}(\mathcal{H})$.

For an E0-semigroup $\theta$ on $M$, a family $\{f_{t}^{\theta}\}_{t\geq 0}$ of the trivial right $M$-linear unitaries $f_{t}^{\theta}:\tilde{\mathcal{H}}_{t}^{\theta}\ni\xi\mapsto\xi\in L^{2}(M)$ induces the right $M$-linear unitary $f^{\theta}:\tilde{\mathfrak{H}}^{\theta}\to L^{2}(M)$, where $\tilde{\mathfrak{H}}^{\theta}$ is the inductive limit of $(\tilde{H}^{\theta},\tilde{\Xi}^{\theta})$. Note that the all canonical embeddings $\kappa^{\theta}_{t}:\tilde{\mathcal{H}}^{\theta}_{t}\to\tilde{\mathfrak{H}}^{\theta}$ are unitaries and equal to each other. We have $\theta=T^{\tilde{\Xi}^{\theta}}$, and the E0-semigroup $\{(f^{\theta})^{*}\theta_{t}(f^{\theta}\cdot(f^{\theta})^{*})f^{\theta}\}_{t\geq 0}$ coincides with the dilation $\tilde{\theta}$ of the pair $(\tilde{H}^{\theta},\tilde{\Xi}^{\theta})$. We have the following classification of E0-semigroups.

Theorem 6.3**.**

Two E0-semigroups $\alpha$ and $\beta$ on a von Neumann algebra $M$ are cocycle equivalent if and only if $H^{\alpha}\cong H^{\beta}$.

Proof.

We will use the above notations for $\alpha$ and $\beta$ in this proof. If $w=\{w_{t}\}_{t\geq 0}\subset M$ is a unitary right cocycle for $\alpha$ and $\beta_{t}(\cdot)=w_{t}^{*}\alpha_{t}(\cdot)w_{t}$ for each $t\geq 0$, then $u_{t}:\tilde{\mathcal{H}}_{t}^{\beta}\ni x\phi^{\frac{1}{2}}\mapsto w_{t}x\phi^{\frac{1}{2}}\in\tilde{\mathcal{H}}_{t}^{\alpha}$ gives an isomorphism $\tilde{H}^{\beta}\cong\tilde{H}^{\alpha}$. Thus, we have $H^{\beta}\cong\tilde{H}^{\beta}\cong\tilde{H}^{\alpha}\cong H^{\alpha}$.

Conversely, suppose a family $\{u_{t}\}_{t\geq 0}$ of $M$-bilinear unitaries gives an isomorphism from $H^{\beta}$ onto $H^{\alpha}$. Let $u^{\alpha}$ be the isomorphism from $H^{\alpha}$ onto $\tilde{H}^{\alpha}$ in Proposition 6.1. Put $\lambda(t)=u_{t}^{\alpha}u_{t}\xi^{\beta}(t)\in\tilde{\mathcal{H}}_{t}^{\alpha}$ for each $t\geq 0$, and then $\Lambda=\{\lambda(t)\}_{t\geq 0}$ is a unital unit of $\tilde{H}^{\alpha}$. We have

[TABLE]

for all $x\in M$, that is, $\beta=T^{\Lambda}$. We can check that $\Lambda$ is weakly continuous and generating. Hence the unit $\Lambda$ is continuous by Corollary 4.12. We denote the right cocycle for $\tilde{\alpha}$ associated with $\Lambda$ by $w^{0}=\{w_{t}^{0}=\pi_{\phi}(\kappa_{t}^{\alpha}\lambda(t))\pi_{\phi}(\kappa_{0}^{\alpha}\phi^{\frac{1}{2}})^{*}\}_{t\geq 0}$ as Theorem 6.6.

By (3.8), each $w_{t}^{0}$ is isometry. Since the map $:\phi^{\frac{1}{2}}x\mapsto\lambda(t)x$ is isometry, we have $\overline{{\rm span}}\{\lambda(t)x\mid x\in M\}=L^{2}(M)$. Thus, by (3.9), each $w_{t}^{0}$ is surjective. Now, we shall show that $w^{0}$ is strongly continuous. For $s\geq 0$, by the continuity of $\Lambda$, we can check that $\kappa^{\alpha}_{t}\lambda(t)\to\kappa^{\alpha}_{s}\lambda(s)$ when $t\to s$. Let $\xi\in\tilde{\mathfrak{H}}^{\alpha}$ and $t\geq s$, by (3.8), we have

[TABLE]

Since $\pi_{\phi}(\kappa_{t}^{\alpha}\lambda(t))^{*}\pi_{\phi}(\kappa_{s}^{\alpha}\lambda(s))\to 1_{M}$ weakly when $t\to s$ or $s\to t$, (6) tends to $\langle\xi,\xi\rangle$ when $t\to s+0$, and by the symmetry, $\langle w_{t}^{0}\xi,w_{s}^{0}\xi\rangle$ also tends to $\langle\xi,\xi\rangle$ when $t\to s-0$. We conclude that $w_{t}^{0}\xi\to w_{s}^{0}\xi$ when $t\to s$.

Put $w_{t}=f^{\alpha}w_{t}^{0}(f^{\alpha})^{*}\in M$. Then, the family $w=\{w_{t}\}_{t\geq 0}$ is a strongly continuous right cocycle for $\alpha$. For all $t\geq 0$ and $x,y,z\in M$, since $w_{t}\phi^{\frac{1}{2}}x=\lambda(t)x$ and $\beta$ is given as (6.1), we have $\langle w_{t}^{*}\alpha_{t}(x)w_{t}\phi^{\frac{1}{2}}y,\phi^{\frac{1}{2}}z\rangle=\langle\pi_{\phi}(\lambda(t))^{*}\pi_{\phi}(x\cdot\lambda(t))\phi^{\frac{1}{2}}y,\phi^{\frac{1}{2}}z\rangle=\langle\beta_{t}(x)\phi^{\frac{1}{2}}y,\phi^{\frac{1}{2}}z\rangle$, and hence $\beta_{t}(x)=w_{t}^{*}\alpha_{t}(x)w_{t}$. $\Box$

Corollary 6.4**.**

Two E0-semigroups $\alpha$ and $\beta$ on von Neumann algebras $M$ and $N$, respectively, are cocycle conjugate if and only if there exists a $*$-isomorphism $\Phi:M\to N$ such that $H^{\alpha}$ and $H^{\beta^{\Phi}}$ are isomorphic.

Example 6.5**.**

Let $u=\{u_{t}\}_{t\geq 0}$ be a strongly continuous semigroup of unitaries $u_{t}$ in a von Neumann algebra $M$. We define $\theta_{t}(x)=u_{t}^{*}xu_{t}$ for each $t\geq 0$ and $x\in M$. The product system of W∗-bimodules associated with $\theta$ is isomorphic to the trivial product system $\{L^{2}(M)\}_{t\geq 0}$.

Proposition 6.6**.**

. Let $\theta$ be an E0-semigroup on a von Neumann algebra $M$. For a unit $\Xi=\{\xi(t)\}_{t\geq 0}$ of $H^{\theta}$ and $t\geq 0$, there exists a unique $a_{t}\in M$ such that $\xi(t)=a_{t}\phi^{\frac{1}{2}}$. The family $\{a_{t}\}_{t\geq 0}$ is a right cocycle for $\theta$.

Proof.

Fix $s,t\geq 0$. Since we have

[TABLE]

and $\phi$ is faithful, the equation $\theta_{t}(a_{s})a_{t}=a_{s+t}$ holds. $\Box$

Example 6.7**.**

Let $\theta$ be an E0-semigroup on a II1 factor $M$ and $\Xi$ be a unit of the product system $H^{\theta}$ of W∗-bimodules associated with $\theta$. For each $t\geq 0$, we define an operator $X_{t}^{\Xi}\in\mathcal{B}(L^{2}(M))$ by $X_{t}^{\Xi}(x\phi^{\frac{1}{2}})=\theta_{t}(x)a_{t}\phi^{\frac{1}{2}}$ for each $x\in M$, where $\{a_{t}\}_{t\geq 0}$ is the right cocycle for $\theta$ associated with $\Xi$ in Proposition 6.6. Then, the family $X^{\Xi}=\{X_{t}^{\Xi}\}_{t\geq 0}$ is a unit of $\theta$, that is, $X^{\Xi}$ is a semigroup satisfying $X_{t}^{\Xi}x=\theta_{t}(x)X_{t}^{\Xi}$ for all $t\geq 0$.

Acknowledgments

The author would like to express deeply gratitude to Shigeru Yamagami for his helpful comments. He also would like to thank Raman Srinivasan for a discussion. He is grateful to Yoshimichi Ueda for giving valuable informations and a chance of the discussion. He also is grateful to Yuhei Suzuki.