1 Introduction

Let \({\mathscr {M}}\) be a von Neumann algebra, let \({\mathscr {N}}\) be its von Neumann subalgebra, and let \(\{\rho _{\theta }:\theta \in \Theta \}\) be a family of normal states on \({\mathscr {M}}\). In [4] the notion of strong sufficiency of the subalgebra \({\mathscr {N}}\) for the family \(\{\rho _{\theta }:\theta \in \Theta \}\) was introduced (and further investigated in [5]) as a generalisation of sufficiency in Umegaki’s sense considered earlier in [13, 14]. In this operator algebra setup, strong sufficiency of \({\mathscr {N}}\) means the existence of a normal two-positive map \(\alpha :{\mathscr {M}}\rightarrow {\mathscr {N}}\) such that

$$\begin{aligned} \rho _{\theta }\circ \alpha =\rho _{\theta },\qquad \theta \in \Theta . \end{aligned}$$

If the map \(\alpha \) is a normal conditional expectation onto \({\mathscr {N}}\), then we get sufficiency in Umegaki’s sense. An obvious and natural generalisation of the notion of strong sufficiency would consist in giving up the, rather technical, requirement of two-positivity and replacing it by positivity. Some work in this setup was done in [6], and in the present paper, we continue this line of investigation. As was pointed out in [6], this approach is additionally motivated by considerations from quantum hypothesis testing theory, where a different notion of sufficiency is employed, namely, the sufficiency of \({\mathscr {N}}\) means that we can find there an optimal measurement minimising the Bayes risk (cf. [7]).

In the first part of the paper, we show that for a \(\sigma \)-finite finite von Neumann algebra with a faithful family of normal states the minimal sufficient von Neumann subalgebra is sufficient in Umegaki’s sense, and obtain a proper version of the factorization theorem due to Jenčová and Petz. In the second part, we are dealing with the full algebra \({\mathbb {B}}({\mathcal {H}})\) and a family of pure states. There we give a description of the minimal sufficient subalgebra together with an explicit form of the conditional expectation defining this sufficiency.

2 Preliminaries and notation

Let \({\mathcal {H}}\) be a Hilbert space with a scalar product \(\langle \cdot |\cdot \rangle \) and norm \(\Vert \cdot \Vert \), and let \({\mathscr {M}}\) be an arbitrary von Neumann algebra of operators acting on \({\mathcal {H}}\), with identity \(\mathbb {1}\), and predual \({\mathscr {M}}_*\). \({\mathscr {M}}\) is said to be \(\sigma \)-finite (or countably decomposable) if any family of pairwise orthogonal projections in \({\mathscr {M}}\) is countable. In our considerations, we restrict attention to such algebras.

A state on \({\mathscr {M}}\) is a bounded positive linear functional \(\rho :{\mathscr {M}}\rightarrow {\mathbb {C}}\) of norm one. A state \(\rho \) is said to be normal if it is continuous in the \(\sigma \)-weak topology on \({\mathscr {M}}\), in other words, \(\rho \in {\mathscr {M}}_*\). For a normal state \(\rho \), its support, denoted by \({\text {s}}(\rho )\), is defined as the smallest projection e in \({\mathscr {M}}\) such that \(\rho (e)=\rho (\mathbb {1})\). In particular, we have

$$\begin{aligned} \rho ({\text {s}}(\rho )A)=\rho (A{\text {s}}(\rho ))=\rho (A),\qquad A\in {\mathscr {M}}, \end{aligned}$$

and if \(\rho ({\text {s}}(\rho )A{\text {s}}(\rho ))=0\) for \({\text {s}}(\rho )A{\text {s}}(\rho )\geqslant 0\) then \({\text {s}}(\rho )A{\text {s}}(\rho )=0\).

A normal state \(\rho \) is said to be faithful if for each positive element \(A\in {\mathscr {M}}\) from the equality \(\rho (A)=0\) it follows that \(A=0\). It is easily seen that the faithfulness of \(\rho \) is equivalent to the relation \({\text {s}}(\rho )=\mathbb {1}\). The existence of a normal faithful state on \({\mathscr {M}}\) is equivalent to \({\mathscr {M}}\) being \(\sigma \)-finite.

To each unit vector \(\psi \in {\mathcal {H}}\) there corresponds a normal state, denoted by the same symbol and called a vector state, defined as

$$\begin{aligned} \psi (A)=\langle \psi |A\psi \rangle ,\qquad A\in {\mathscr {M}}. \end{aligned}$$

By a slight abuse of language, we shall speak of the vector \(\psi \) itself as a pure state. By \(P_{[\psi ]}\) shall be denoted the projection onto the space spanned by the vector \(\psi \) (in Dirac notation, \(P_{[\psi ]}=|\psi \rangle \langle \psi |\)).

Let \(\{\rho _{\theta }:\theta \in \Theta \}\) be a family of normal states on a von Neumann algebra \({\mathscr {M}}\). This family is said to be faithful if for each positive element \(A\in {\mathscr {M}}\) from the equality \(\rho _{\theta }(A)=0\) for all \(\theta \in \Theta \) it follows that \(A=0\). Similarly to the case of one state, it is seen that the faithfulness of the family is equivalent to the relation

$$\begin{aligned} \bigvee _{\theta \in \Theta }{\text {s}}(\rho _{\theta })=\mathbb {1}. \end{aligned}$$

In particular, for a family of pure states \(\{\psi _{\theta }:\theta \in \Theta \}\), its faithfulness amounts to the equality \(\overline{{{\,\mathrm{Lin}\,}}\{\psi _{\theta }:\theta \in \Theta \}}={\mathcal {H}}\).

Let P be a projection in a von Neumann algebra \({\mathscr {M}}\). A reduced von Neumann algebra \({\mathscr {M}}_P\) is defined as an algebra of operators acting on the Hilbert space \(P({\mathcal {H}})\) by the formula

$$\begin{aligned} {\mathscr {M}}_P=\{PA|P({\mathcal {H}}):A\in {\mathscr {M}}\}. \end{aligned}$$

Let \(\{{\mathscr {M}}_i:i\in I\}\) be a family of von Neumann algebras acting on Hilbert spaces \({\mathcal {H}}_i\). The direct sum von Neumann algebra is defined as

$$\begin{aligned} \sum _{i\in I}{}^{\oplus }{\mathscr {M}}_i=\left\{ A=(A_i)_{i\in I}:A_i\in {\mathscr {M}}_i\text { for all } i\in I,\, \sup _{i\in I}\Vert A_i\Vert <\infty \right\} , \end{aligned}$$

where the operators \(A=(A_i)_{i\in I}\) act on the Hilbert space

$$\begin{aligned} {\mathcal {H}}=\bigoplus _{i\in I}{\mathcal {H}}_i \end{aligned}$$

as

$$\begin{aligned} A((\xi _i)_{i\in I})=(A_i\xi _i)_{i\in I}. \end{aligned}$$

Let \({\mathscr {M}}\) and \({\mathscr {N}}\) be von Neumann algebras. A linear map \(\alpha :{\mathscr {M}}\rightarrow {\mathscr {N}}\) is said to be normal if it is continuous in the \(\sigma \)-weak topologies on \({\mathscr {M}}\) and \({\mathscr {N}}\), respectively. It is called unital if \(\alpha (\mathbb {1})=\mathbb {1}\).

The algebra of all bounded linear operators on \({\mathcal {H}}\) will be denoted by \({\mathbb {B}}({\mathcal {H}})\). For arbitrary \({\mathscr {T}}\subset {\mathbb {B}}({\mathcal {H}})\) by \({\mathscr {T}}_h\) will be denoted the set of hermitian elements of \({\mathscr {T}}\).

The Jordan product of two operators \(A,B\in {\mathbb {B}}({\mathcal {H}})\) will be denoted by the symbol ‘\(\circ \)’:

$$\begin{aligned} A\circ B=\frac{1}{2}(AB+BA), \end{aligned}$$

and by a slight abuse of notation, the same symbol will be used for superpositions of a functional \(\varphi \) on \({\mathscr {M}}\) and a map \({\mathbb {E}}\) on \({\mathscr {M}}\): \(\varphi \circ {\mathbb {E}}\), as well as of two maps \({\mathbb {E}},{\mathbb {F}}\) on \({\mathscr {M}}\): \({\mathbb {E}}\circ {\mathbb {F}}\).

For arbitrary \({\mathscr {T}}\subset {\mathbb {B}}({\mathcal {H}})\), by \(W^*({\mathscr {T}})\), we shall denote the von Neumann algebra generated by \({\mathscr {T}}\), i.e., the smallest von Neumann algebra containing \({\mathscr {T}}\).

Let \({\mathscr {M}}\) be a von Neumann algebra, let \(\{\rho _{\theta }:\theta \in \Theta \}\) be a family of normal states on \({\mathscr {M}}\), and let \({\mathscr {N}}\) be a von Neumann subalgebra of \({\mathscr {M}}\). \({\mathscr {N}}\) is said to be sufficient for the family of states \(\{\rho _{\theta }:\theta \in \Theta \}\) if there exists a normal positive linear unital map \(\alpha :{\mathscr {M}}\rightarrow {\mathscr {N}}\) such that

$$\begin{aligned} \rho _{\theta }\circ \alpha =\rho _{\theta },\qquad \text {for all}\quad \theta \in \Theta . \end{aligned}$$
(1)

If the map \(\alpha \) above is two-positive, then \({\mathscr {N}}\) is said to be strongly sufficient, and if it is a conditional expectation onto \({\mathscr {N}}\), then \({\mathscr {N}}\) is said to be sufficient in Umegaki’s sense. It can be shown that for finite-dimensional von Neumann algebras and families of faithful states sufficiency and strong sufficiency are equivalent. The notion of sufficiency in Umegaki’s sense appeared in [13, 14], while strong sufficiency was introduced in [4]. If the algebra \({\mathscr {N}}\) is sufficient and contained in any other sufficient (sufficient in strong or Umegaki’s sense) algebra, then \({\mathscr {N}}\) is said to be minimal. It is clear that a minimal sufficient (in any sense) subalgebra is unique (if it exists). On account of [6, Theorem 1], for a faithful family of normal states on \({\mathscr {M}}\) there exists the minimal sufficient von Neumann subalgebra \({\mathscr {M}}_{\text {min}}\) of \({\mathscr {M}}\).

3 Sufficiency in finite von Neumann algebras

In this section, we show that for a \(\sigma \)-finite finite von Neumann algebra \({\mathscr {M}}\) and a faithful family of normal states on \({\mathscr {M}}\) the minimal sufficient von Neumann subalgebra of \({\mathscr {M}}\) is sufficient in Umegaki’s sense. Moreover, we also obtain a version of the factorization theorem.

In what follows, we shall need a few basic facts from modular theory of von Neumann algebras; however, this theory will be employed only to a limited extent. For its full account, the reader is referred to [10]. Let \(\varphi \) be a normal faithful state on \({\mathscr {M}}\). There exists a group of automorphisms of \({\mathscr {M}}\), \(\{\sigma _t^\varphi :t\in {\mathbb {R}}\}\), called the modular automorphism group, associated with \(\varphi \) in a canonical way. The algebra of fixed points of the modular automorphism group, denoted by \({\mathscr {M}}^\varphi \), is called the centralizer of \(\varphi \), and the following relation holds

$$\begin{aligned} {\mathscr {M}}^\varphi =\{A\in {\mathscr {M}}:\text { for each }B\in {\mathscr {M}}, \; \varphi (AB)=\varphi (BA)\}. \end{aligned}$$

Let \({\mathscr {N}}\) be a von Neumann subalgebra of \({\mathscr {M}}\). If \(\sigma _t^\varphi ({\mathscr {N}})={\mathscr {N}}\) for all \(t\in {\mathbb {R}}\), then there is a normal faithful conditional expectation \({\mathbb {F}}\) from \({\mathscr {M}}\) onto \({\mathscr {N}}\) such that

$$\begin{aligned} \varphi \circ {\mathbb {F}}=\varphi . \end{aligned}$$

Let \({\mathscr {A}}\) be a subspace of \({\mathscr {M}}\) containing the identity operator \(\mathbb {\mathbb {1}}\). \({\mathscr {A}}\) is said to be a JW*-subalgebra of \({\mathscr {M}}\) if it is \(\sigma \)-weakly closed and closed with respect to the operation of taking adjoints and the Jordan product. Then the self-adjoint part of \({\mathscr {A}}\) is a JW-algebra, in particular, it is generated by projections (cf. [3]). Let \({\mathbb {E}}\) be a normal positive faithful unital projection from the JW-algebra \({\mathscr {M}}_h\) onto a JW-algebra \({\mathscr {A}}\). On account of [3, Lemma 4.4.13], we have

$$\begin{aligned} {\mathbb {E}}(A\circ B)=A\circ {\mathbb {E}}(B) \end{aligned}$$
(2)

for all \(A\in {\mathscr {A}}_h\), \(B\in {\mathscr {M}}_h\). Let now \({\mathbb {E}}\) be a normal positive faithful unital projection from \({\mathscr {M}}\) onto a JW*-algebra \({\mathscr {A}}\). Then \({\mathbb {E}}|{\mathscr {M}}_h\) is as above , and hence it satisfies Eq. (2). Decomposing arbitrary \(A\in {\mathscr {A}}\) and \(B\in {\mathscr {M}}\) as \(A=A_1+iA_2\), \(B=B_1+iB_2\) where \(A_1,A_2\in {\mathscr {A}}_h\), \(B_1,B_2\in {\mathscr {M}}_h\), we obtain the equality

$$\begin{aligned} {\mathbb {E}}(A\circ B)=A\circ {\mathbb {E}}(B) \end{aligned}$$
(3)

for all \(A\in {\mathscr {A}}\), \(B\in {\mathscr {M}}\). Using the formula

$$\begin{aligned} ABA=2A\circ (A\circ B)-A^2\circ B, \quad A\in {\mathscr {A}},\,B\in {\mathscr {M}}, \end{aligned}$$

we get

$$\begin{aligned} {\mathbb {E}}(SBS)=S{\mathbb {E}}(B)S \end{aligned}$$
(4)

for each \(B\in {\mathscr {M}}\) and each symmetry \(S\in {\mathscr {A}}_h\) (symmetry means that \(S^2=\mathbb {1}\)).

The lemma below is a JW*-version of a known result on uniqueness of conditional expectation for von Neumann algebras.

Lemma 1

Let \(\varphi \) be a normal faithful state on \({\mathscr {M}}\), let \({\mathscr {A}}\) be a JW*-subalgebra of \({\mathscr {M}}\), and let \({\mathbb {E}}_i:{\mathscr {M}}\rightarrow {\mathscr {A}}\), \(i=1,2\), be normal positive unital projections onto \({\mathscr {A}}\) such that

$$\begin{aligned} \varphi \circ {\mathbb {E}}_1=\varphi \circ {\mathbb {E}}_2. \end{aligned}$$

Then \({\mathbb {E}}_1={\mathbb {E}}_2\).

Proof

Observe first that the family \(\{\varphi _H:H\in {\mathscr {A}}\}\) of normal linear functionals on \({\mathscr {M}}\) defined as

$$\begin{aligned} \varphi _H(B)=\varphi (H\circ B), \quad B\in {\mathscr {M}}, \end{aligned}$$

separates the points of \({\mathscr {A}}\). Indeed, if we have \(\varphi _H(A)=0\) for some \(A\in {\mathscr {A}}\) and all \(H\in {\mathscr {A}}\), then taking \(H=A^*\) we get

$$\begin{aligned} 0=\varphi _{A^*}(A)=\frac{1}{2}\varphi (A^*A+AA^*), \end{aligned}$$

and the faithfulness of \(\varphi \) yields \(A^*A=0\), i.e., \(A=0\). Now for arbitrary \(B\in {\mathscr {M}}\), we have by virtue of the equality (3)

$$\begin{aligned} \varphi _H({\mathbb {E}}_1(B))=\varphi (H\circ {\mathbb {E}}_1(B))=\varphi ({\mathbb {E}}_1(H\circ B))=\varphi (H\circ B)=\varphi _H(B), \end{aligned}$$

and analogously

$$\begin{aligned} \varphi _H({\mathbb {E}}_2(B))=\varphi _H(B), \end{aligned}$$

showing that \(\varphi _H({\mathbb {E}}_1(B))=\varphi _H({\mathbb {E}}_2(B))\), hence \({\mathbb {E}}_1(B)={\mathbb {E}}_2(B)\). \(\square \)

Now we are ready to prove that for a faithful family of normal states the minimal sufficient subalgebra is also sufficient in Umegaki’s sense.

Theorem 2

Let \(\{\rho _{\theta }:\theta \in \Theta \}\) be a faithful family of normal states on a \(\sigma \)-finite finite von Neumann algebra \({\mathscr {M}}\) with a normal faithful tracial state \(\tau \). The minimal sufficient von Neumann subalgebra \({\mathscr {M}}_{\text {min}}\) of \({\mathscr {M}}\) is sufficient in Umegaki’s sense.

Proof

As in the proof of Theorem 1 in [6], we let \({\mathfrak {S}}\) be the family of all normal positive linear unital maps on \({\mathscr {M}}\) such that the states \(\rho _{\theta }\) are invariant with respect to the maps from \({\mathfrak {S}}\). It is seen that \({\mathfrak {S}}\) is a non-empty (because it contains the identity map) semigroup. Let \({\mathscr {A}}\) be the set of fixed points of the maps from \({\mathfrak {S}}\), i.e.,

$$\begin{aligned} {\mathscr {A}}=\{A\in {\mathscr {M}}:\alpha (A)=A\quad \text {for all} \quad \alpha \in {\mathfrak {S}}\}. \end{aligned}$$
(5)

Then \({\mathscr {A}}\) is a JW*-algebra, and on account of [6, Theorem 1] we have

$$\begin{aligned} {\mathscr {M}}_{\text {min}}=W^*({\mathscr {A}})=W^*({\mathscr {A}}_h), \end{aligned}$$

moreover, there is a normal positive faithful unital projection \({\mathbb {E}}\) from \({\mathscr {M}}\) onto \({\mathscr {A}}\) such that for all \(\theta \in \Theta \)

$$\begin{aligned} \rho _\theta \circ {\mathbb {E}}=\rho _\theta . \end{aligned}$$

Put

$$\begin{aligned} \varphi =\tau \circ {\mathbb {E}}. \end{aligned}$$

Then \(\varphi \) is a normal state on \({\mathscr {M}}\) such that

$$\begin{aligned} \varphi \circ {\mathbb {E}}=\varphi . \end{aligned}$$

Let S be an arbitrary symmetry in \({\mathscr {A}}_h\). Taking BS instead of B in the formula (4), we get

$$\begin{aligned} {\mathbb {E}}(SB)=S{\mathbb {E}}(BS)S, \end{aligned}$$

hence

$$\begin{aligned} \varphi (SB)=\tau ({\mathbb {E}}(SB))=\tau (S{\mathbb {E}}(BS)S)=\tau ({\mathbb {E}}(BS))=\varphi (BS). \end{aligned}$$

An arbitrary projection P in \({\mathscr {A}}_h\) has the form \(P=\frac{1}{2}(S+\mathbb {1})\) where \(S=2P-\mathbb {1}\) is a symmetry, thus

$$\begin{aligned} \varphi (PB)=\varphi (BP) \end{aligned}$$

for every such projection. Since \({\mathscr {A}}_h\) is a JW-algebra, linear combinations of its projections are \(\sigma \)-weakly dense in \({\mathscr {A}}_h\) (cf. [3]) and the \(\sigma \)-weak continuity of \(\varphi \) yields

$$\begin{aligned} \varphi (AB)=\varphi (BA) \end{aligned}$$

for each \(A\in {\mathscr {A}}_h\). Since \(B\in {\mathscr {M}}\) was arbitrary, this means that \(A\in {\mathscr {M}}^\varphi \), thus

$$\begin{aligned} {\mathscr {A}}_h\subset {\mathscr {M}}^\varphi . \end{aligned}$$

Since \({\mathscr {M}}^\varphi \) is a von Neumann algebra, this yields

$$\begin{aligned} {\mathscr {M}}_{\text {min}}=W^*({\mathscr {A}}_h)\subset {\mathscr {M}}^\varphi . \end{aligned}$$

(The reasoning above follows the proof of Haagerup and Størmer [2, Lemma 2.2].) Since for the modular automorphism group we have

$$\begin{aligned} \sigma _t^\varphi (A)=A \quad \text {for all }A\in {\mathscr {M}}^\varphi , \end{aligned}$$

we get \(\sigma _t^\varphi ({\mathscr {M}}_{\text {min}})={\mathscr {M}}_{\text {min}}\) for all \(t\in {\mathbb {R}}\), thus there is a normal faithful conditional expectation \({\mathbb {F}}\) from \({\mathscr {M}}\) onto \({\mathscr {M}}_{\text {min}}\) such that

$$\begin{aligned} \varphi \circ {\mathbb {F}}=\varphi . \end{aligned}$$

We have

$$\begin{aligned} ({\mathbb {E}}\circ {\mathbb {F}})^2={\mathbb {E}}\circ ({\mathbb {F}}\circ {\mathbb {E}})\circ {\mathbb {F}}={\mathbb {E}}\circ {\mathbb {E}}\circ {\mathbb {F}}={\mathbb {E}}\circ {\mathbb {F}}\end{aligned}$$

showing that \({\mathbb {E}}\circ {\mathbb {F}}\) is a normal positive unital projection onto \({\mathscr {A}}\). Moreover,

$$\begin{aligned} \varphi \circ ({\mathbb {E}}\circ {\mathbb {F}})=(\tau \circ {\mathbb {E}})\circ ({\mathbb {E}}\circ {\mathbb {F}})=(\tau \circ {\mathbb {E}})\circ {\mathbb {F}}=\varphi \circ {\mathbb {F}}=\varphi =\varphi \circ {\mathbb {E}}, \end{aligned}$$

and Lemma 1 yields

$$\begin{aligned} {\mathbb {E}}\circ {\mathbb {F}}={\mathbb {E}}. \end{aligned}$$

Thus for each \(\theta \in \Theta \), we have

$$\begin{aligned} \rho _{\theta }\circ {\mathbb {F}}=(\rho _{\theta }\circ {\mathbb {E}})\circ {\mathbb {F}}=\rho _{\theta }\circ ({\mathbb {E}}\circ {\mathbb {F}})=\rho _{\theta }\circ {\mathbb {E}}=\rho _{\theta }, \end{aligned}$$

which means that \({\mathscr {M}}_{\text {min}}\) is sufficient in Umegaki’s sense. \(\square \)

An important factorization theorem was formulated in [4, Theorem 4] in the case when the algebra \({\mathscr {M}}\) is a countable direct sum of type I factors. However, as it was shown in [6, Examples 2 and 3], its setup needs substantial corrections, and we shall see that this theorem holds in full generality in finite von Neumann algebras. For this purpose, let us introduce some necessary notions. Since the proof of this theorem is in its main part a repetition of the proof in [4, Section 4] with some additional reasoning, we shall follow closely the notation employed there. However, we shall use some parts of the theory of measurable operators and noncommutative \(L^p\)-spaces (actually only \(L^1\)-space) for a brief account of which we refer the reader to the “Appendix”.

For a linear operator A on a Hilbert space \({\mathcal {H}}\), by \({\mathcal {D}}(A)\), we denote the domain of A, and by \({\bar{A}}\)—the closure of A.

Let \({\mathscr {M}}\) be a finite von Neumann algebra with a normal faithful finite trace \(\tau \), let \({\mathscr {M}}_0\) be a von Neumann subalgebra of \({\mathscr {M}}\), and let \({\mathscr {M}}_1={\mathscr {M}}'_0\cap {\mathscr {M}}\)be the relative commutant of \({\mathscr {M}}_0\). By \({\widetilde{{\mathscr {M}}}}\) we denote the algebra of measurable operators, and by \(L^1({\mathscr {M}},\tau ),\,L^1({\mathscr {M}}_0,\tau )\), \(L^1({\mathscr {M}}_1,\tau )\) the corresponding \(L^1\)-spaces. (Here we make use of the assumption that \({\mathscr {M}}\) is finite, otherwise there is no guarantee that \(\tau |{\mathscr {M}}_0\) or \(\tau |{\mathscr {M}}_1\) is semifinite.) Let \(\{\rho _{\theta }:\theta \in \Theta \}\) be a faithful family of normal states on \({\mathscr {M}}\), and let, as before, for some convex combination

$$\begin{aligned} \omega =\sum _{k=1}^{\infty }\lambda _k\rho _{\theta _k}, \end{aligned}$$

\(\omega \) be a faithful state on \({\mathscr {M}}\). Denote

$$\begin{aligned} \omega _0=\omega |{\mathscr {M}}_0, \quad \omega _1=\omega |{\mathscr {M}}_1,\quad \tau _0=\tau |{\mathscr {M}}_0, \quad \tau _1=\tau |{\mathscr {M}}_1, \end{aligned}$$

and let \(D_{\omega }\in L^1({\mathscr {M}},\tau ),\,D_{\omega _0}\in L^1({\mathscr {M}}_0,\tau _0),\, D_{\omega _1}\in L^1({\mathscr {M}}_1,\tau _1)\) be the densities of \(\omega ,\,\omega _0\) and \(\omega _1\), respectively. Now the reasoning of [4, p. 269] can be repeated word for word upon observing that the modular automorphism groups \(\{\sigma _t^{\omega }\}\), \(\{\sigma _t^{\omega _0}\}\) and \(\{\sigma _t^{\omega _1}\}\) have the same form as in [4], i.e., for instance

$$\begin{aligned} \sigma _t^{\omega }(A)=D_{\omega }^{it}AD_{\omega }^{-it},\quad A\in {\mathscr {M}}, \end{aligned}$$

and similarly for the remaining two (cf. [10, Proposition 4.7]). This leads to the equality

$$\begin{aligned} D_{\omega }^{it}=D_{\omega _0}^{it}D_{\omega _1}^{it}z^{it},\quad t\in {\mathbb {R}}, \end{aligned}$$
(6)

for some positive self-adjoint z with trivial null-space, affiliated with the centre of \({\mathscr {M}}_1\).

Denote by \(D_{\theta },\,D_{\theta ,0}\) the densities of the states \(\rho _{\theta }\) and \(\rho _{\theta }|{\mathscr {M}}_0\), respectively. Now the factorization theorem reads

Theorem 3

The subalgebra \({\mathscr {M}}_0\) is sufficient for a faithful family of normal states \(\{\rho _{\theta }:\theta \in \Theta \}\) if and only if

$$\begin{aligned} D_{\theta }=\overline{D_{\theta ,0}D_{\omega _1}z}, \end{aligned}$$
(7)

where z is a positive self-adjoint operator with trivial null-space, affiliated with the centre of \({\mathscr {M}}_1\).

Proof

Fix \(\theta \), and let

$$\begin{aligned} D_{\theta ,0}&=\int _0^{\infty }\lambda \,E_0({\text {d}}\lambda ),\\ D_{\omega _1}&=\int _0^{\infty }\lambda \,E_1({\text {d}}\lambda ),\\ z&=\int _0^{\infty }\lambda \,E_z({\text {d}}\lambda ), \end{aligned}$$

be the spectral decompositions of \(D_{\theta ,0},\,D_{\omega _1}\) and z. Since \(D_{\theta ,0}\) is affiliated with \({\mathscr {M}}_0\), \(D_{\omega _1}\) is affiliated with \({\mathscr {M}}_1\), and z is affiliated with the centre of \({\mathscr {M}}_1\), it follows that the spectral measures \(E_0,\,E_1\) and \(E_z\) take values in \({\mathscr {M}}_0,\,{\mathscr {M}}_1\) and the centre of \({\mathscr {M}}_1\), respectively, thus they mutually commute. Consequently, there exists a spectral measure E in \({\mathscr {M}}\), and nonnegative Borel functions \(g_0,\,g_1\) and h such that

$$\begin{aligned} D_{\theta ,0}&=\int _{-\infty }^{\infty }g_0(\lambda )\,E({\text {d}}\lambda ),\\ D_{\omega _1}&=\int _{-\infty }^{\infty }g_1(\lambda )\,E({\text {d}}\lambda ),\\ z&=\int _{-\infty }^{\infty }h(\lambda )\,E({\text {d}}\lambda ). \end{aligned}$$

From these equalities, we obtain

$$\begin{aligned} D_{\theta ,0}D_{\omega _1}z\subset \int _{-\infty }^{\infty }g_0(\lambda )g_1(\lambda ) h(\lambda )\,E({\text {d}}\lambda ):=B. \end{aligned}$$

Operator B on the right hand side of the above relation is a self-adjoint operator affiliated with \({\mathscr {M}}\), i.e., it belongs to the algebra \({\widetilde{{\mathscr {M}}}}\) of measurable operators, while on the left hand side we have a product of operators from \({\widetilde{{\mathscr {M}}}}\). From the properties of \({\widetilde{{\mathscr {M}}}}\), it follows that

$$\begin{aligned} \overline{D_{\theta ,0}D_{\omega _1}z}=B. \end{aligned}$$
(8)

Moreover, we have

$$\begin{aligned} B^{it}&=\bigg [\int _{-\infty }^{\infty }g_0(\lambda )g_1(\lambda ) h(\lambda )\,E({\text {d}}\lambda )\bigg ]^{it}\\&=\int _{-\infty }^{\infty }g_0(\lambda )^{it}g_1(\lambda )^{it} h(\lambda )^{it}\,E({\text {d}}\lambda )=D_{\theta ,0}^{it}D_{\omega _1}^{it}z^{it}. \end{aligned}$$

Now the proof follows the lines of the proof of Theorem 4 in [4]. The crucial observation is that also in the case of, in general unbounded, operators \(D_{\theta },\,D_{\omega },\,D_{\theta ,0}\) and \(D_{\omega _0}\) the following equalities for the Connes cocycles hold true

$$\begin{aligned} \begin{aligned} \left[ D\rho _{\theta }:D\omega \right] _t&=[D\rho _{\theta }:D\tau ]_t[D\tau :D\omega ]_t\\&=[D\rho _{\theta }:D\tau ]_t[D\omega :D\tau ]_t^{-1} =D_{\theta }^{it}D_{\omega }^{-it}, \end{aligned} \end{aligned}$$
(9)

(cf. Araki and Masuda [1, Theorem C.1] for the chain rule in the first equality above, Strătilă [10, Corollary 3.4] for the second equality, and Strătilă [10, Corollary 4.8] for the third). Similarly,

$$\begin{aligned}{}[D\rho _{\theta ,0}:D\omega _0]_t=D_{\theta ,0}^{it}D_{\omega _0}^{-it}. \end{aligned}$$
(10)

Assume that \({\mathscr {M}}_0\) is sufficient. Then on account of [4, Theorem 1], we have \([D\rho _{\theta }:D_{\omega }]_t=[D\rho _{\theta ,0}:D_{\omega _0}]_t\), thus from (9), (10) and the fact that \(D_{\omega }^{it}\) are unitary, we get

$$\begin{aligned} D_{\theta }^{it}=D_{\theta ,0}^{it}D_{\omega _0}^{-it}D_{\omega }^{it}. \end{aligned}$$

Using (6) together with the fact that \(D_{\omega _0}^{it}\) are also unitary, we obtain

$$\begin{aligned} D_{\theta }^{it}=D_{\theta ,0}^{it}D_{\omega _0}^{-it} D_{\omega _0}^{it}D_{\omega _1}^{it}z^{it}=D_{\theta ,0}^{it} D_{\omega _1}^{it}z^{it}=B^{it}. \end{aligned}$$
(11)

From this easily follows that \(D_{\theta }=B\). Indeed, denoting by \({\mathcal {N}}(T)\) the null-space of the operator T, we have from the spectral theorem

$$\begin{aligned} {\mathcal {N}}(D_{\theta }^{it})={\mathcal {N}}(D_{\theta })\qquad \text {and} \qquad {\mathcal {N}}(B^{it})={\mathcal {N}}(B), \end{aligned}$$

so

$$\begin{aligned} {\mathcal {N}}(D_{\theta })= {\mathcal {N}}(D_{\theta }^{it}) ={\mathcal {N}}(B^{it})={\mathcal {N}}(B). \end{aligned}$$

Denote by P the projection onto the common null-space above, and let

$$\begin{aligned} D_{\theta }=\int _0^{\infty }\lambda \,E'({\text {d}}\lambda ) \end{aligned}$$

be the spectral decomposition of \(D_{\theta }\). Then

$$\begin{aligned} P+D_{\theta }=\int _0^{\infty }[\chi _{\{0\}}(\lambda )+\lambda ]\,E'({\text {d}}\lambda ), \end{aligned}$$

where \(\chi _{\{0\}}\) denotes the indicator function of the set \(\{0\}\). Hence

$$\begin{aligned} (P+D_{\theta })^{it}&=\int _0^{\infty }[\chi _{\{0\}}(\lambda )+\lambda ]^{it}\,E'({\text {d}}\lambda ) =\int _0^{\infty }[\chi _{\{0\}}(\lambda )+\lambda ^{it}]\,E'({\text {d}}\lambda )\\ {}&= P+\int _0^{\infty }\lambda ^{it}\,E'({\text {d}}\lambda )=P+D_{\theta }^{it}, \end{aligned}$$

and similarly

$$\begin{aligned} (P+B)^{it}=P+B^{it}, \end{aligned}$$

which on account of the equality (11) yields

$$\begin{aligned} (P+D_{\theta })^{it}= (P+B)^{it}. \end{aligned}$$

But the operators \(P+D_{\theta }\) and \(P+B\) have trivial null-spaces, which means that the equality above is the identity of two unitary groups. It follows that their generators must be the same, so

$$\begin{aligned} P+D_\theta =P+B, \end{aligned}$$

i.e.,

$$\begin{aligned} D_{\theta }=B, \end{aligned}$$

and the equality (8) proves the claim.

Assume now that the equality (7) holds. In terms of the algebra \({\widetilde{{\mathscr {M}}}}\), it can be rewritten as

$$\begin{aligned} D_{\theta }=D_{\theta ,0}\cdot D_{\omega _1}\cdot z. \end{aligned}$$

(where the central dot stands for the strong multiplication—see the “Appendix”). Denote

$$\begin{aligned} \omega _n=\sum _{k=1}^n\lambda _k\rho _{\theta _k}. \end{aligned}$$

Then by virtue of the isomorphism between \({\mathscr {M}}_*\) and \(L^1({\mathscr {M}},\tau )\), we have

$$\begin{aligned} D_{\omega _n}=s -\sum _{k=1}^n\lambda _kD_{\theta _k}, \end{aligned}$$

where \(s -\sum \) denotes the strong sum. The same holds of course for \(\omega _{n,0}=\omega _n|{\mathscr {M}}_0\), i.e.,

$$\begin{aligned} D_{\omega _{n,0}}=s -\sum _{k=1}^n\lambda _kD_{\theta _{k,0}}, \end{aligned}$$

so from the formula (7) and the fact that \({\widetilde{{\mathscr {M}}}}\) is an algebra, we obtain

$$\begin{aligned} \begin{aligned} D_{\omega _n}&=s - \sum _{k=1}^n\lambda _kD_{\theta _k}= s - \sum _{k=1}^n\lambda _kD_{\theta _{k,0}}\cdot D_{\omega _1}\cdot z\\&=D_{\omega _{n,0}}\cdot D_{\omega _1}\cdot z. \end{aligned} \end{aligned}$$
(12)

Since \(\omega _n\rightarrow \omega \) in norm in \({\mathscr {M}}_*\), we infer that \(D_{\omega _n}\rightarrow D_{\omega }\) in the norm \(\Vert \cdot \Vert _1\), and by the same token \(D_{\omega _{n,0}}\rightarrow D_{\omega _0}\). According to Lemma 5 (see the Appendix), we have \(D_{\omega _n}\rightarrow D_{\omega }\) and \(D_{\omega _{n,0}}\rightarrow D_{\omega _0}\) in measure. Since \({\widetilde{{\mathscr {M}}}}\) is a topological \({}^*\)-algebra with respect to the measure topology, the formula (12) yields

$$\begin{aligned} D_{\omega }= D_{\omega _0}\cdot D_{\omega _1}\cdot z, \end{aligned}$$

in other words

$$\begin{aligned} D_{\omega }=\overline{D_{\omega _0}D_{\omega _1}z}. \end{aligned}$$
(13)

Now we proceed as in the first part of the proof. Since the operators \(D_{\omega _0},\,D_{\omega _1}\) and z commute, from the equalities (7) and (13), we obtain

$$\begin{aligned} D_{\theta }^{it}=D_{\theta ,0}^{it}D_{\omega _1}^{it}z^{it},\qquad D_{\omega }^{it}=D_{\omega _0}^{it}D_{\omega _1}^{it}z^{it}. \end{aligned}$$

Thus on account of commutation properties and the fact that \(\{D_{\omega _1}^{it}\}\) and \(\{z^{it}\}\) are unitary groups, we get

$$\begin{aligned}{}[D\rho _{\theta }:D\omega ]_t&=D_{\theta }^{it}D_{\omega }^{-it}= D_{\theta ,0}^{it}D_{\omega _1}^{it}z^{it}D_{\omega _0}^{-it}D_{\omega _1}^{-it}z^{-it}\\&=D_{\theta ,0}^{it}D_{\omega _0}^{-it}=[D\rho _{\theta ,0}:D\omega _0]_t, \end{aligned}$$

which by virtue of [4, Theorem 1] yields the sufficiency of \({\mathscr {M}}_0\). \(\square \)

4 Sufficiency for pure states on \({\mathbb {B}}({\mathcal {H}})\)

In this section, we shall be dealing with the full algebra \({\mathscr {M}}={\mathbb {B}}({\mathcal {H}})\) and pure (\(\equiv \) vector) states. Let \(\{\psi _{\theta }:\theta \in \Theta \}\) be a faithful family of pure states. According to [6, Theorem 1], there exists the minimal sufficient for this family subalgebra of \({\mathbb {B}}({\mathcal {H}})\). It turns out that this algebra admits a straightforward description.

Theorem 4

Let \(\{\psi _{\theta }:\theta \in \Theta \}\) be a faithful family of pure states on \({\mathbb {B}}({\mathcal {H}})\), and let \({\mathscr {M}}_{\text {min }}\) be the minimal sufficient for this family subalgebra of \({\mathbb {B}}({\mathcal {H}})\). Then

  1. (i)

    \({\mathscr {M}}_{\text {min }}\) is generated by \(P_{[\psi _{\theta }]}\), i.e., \({\mathscr {M}}_{\text {min }}=W^*(\{P_{[\psi _{\theta }]}:\theta \in \Theta \})\).

  2. (ii)

    There exists a decomposition

    $$\begin{aligned} {\mathcal {H}}=\bigoplus _i{\mathcal {H}}_i \end{aligned}$$

    such that

    $$\begin{aligned} {\mathscr {M}}_{\text {min }}=\sum _i{}^{\oplus }{\mathbb {B}}({\mathcal {H}}_i). \end{aligned}$$
    (14)

Proof

First observe that \({\text {s}}(\psi _{\theta })=P_{[\psi _{\theta }]}\). As in Theorem 2, denote by \({\mathfrak {S}}\) the family of all normal positive unital maps on \({\mathscr {M}}\) such that the states \(\psi _{\theta }\) are invariant with respect to the maps from \({\mathfrak {S}}\). On account of [12, Theorem 4], for every \(\theta \in \Theta \), we have \(\alpha (P_{[\psi _{\theta }]})=P_{[\psi _{\theta }]}\) for each \(\alpha \in {\mathfrak {S}}\), which by virtue of [6, Theorem 1] yields the relation \(P_{[\psi _{\theta }]}\in {\mathscr {M}}_{\text {min}}\).

For the sake of brevity, set \({\mathcal {R}}=\{\psi _{\theta }:\theta \in \Theta \}\), and define on \({\mathcal {R}}\) an equivalence relation as follows: \(\psi _{\theta '}\equiv \psi _{\theta ''}\) if there exists a finite string \(\psi _{\theta _1},\dots ,\psi _{\theta _m}\) such that \(\psi _{\theta _1}=\psi _{\theta '}\), \(\psi _{\theta _m}=\psi _{\theta ''}\) and

$$\begin{aligned} \langle \psi _{\theta _j}|\psi _{\theta _{j+1}}\rangle \ne 0 \qquad \text {for}\quad j=1,\dots ,m-1. \end{aligned}$$

Let

$$\begin{aligned} {\mathcal {R}}=\bigcup _i{\mathcal {R}}_i \end{aligned}$$

be the partition of \({\mathcal {R}}\) determined by the relation \(\equiv \). Denote \({\mathcal {R}}_{i}=\{\psi _{\theta }:{\theta } \in {\Theta _{i}} \}\). It is obvious that for all \(i',\,i''\) such that \(i'\ne i''\), we have \({\mathcal {R}}_{i'}\perp {\mathcal {R}}_{i''}\). Put

$$\begin{aligned} P_i=\bigvee _{\theta \in \Theta _i}P_{[\psi _{\theta }]}. \end{aligned}$$

It follows that \(P_i\) are pairwise orthogonal, and the faithfulness of \(\{\psi _{\theta }:\theta \in \Theta \}\) yields

$$\begin{aligned} \sum _iP_i=\mathbb {1}. \end{aligned}$$

Denote \({\mathcal {H}}_i=P_i({\mathcal {H}})\). Then

$$\begin{aligned} {\mathcal {H}}=\bigoplus _i{\mathcal {H}}_i. \end{aligned}$$

Fix arbitrary i and consider the family of pure states \({\mathcal {R}}_i=\{\psi _{\theta }:\theta \in \Theta _i\}\) on the Hilbert space \({\mathcal {H}}_i\). Take arbitrary \(\psi _{\theta '},\psi _{\theta ''}\in {\mathcal {R}}_i\), and let \(\psi _{\theta _1},\dots ,\psi _{\theta _m}\), with \(\psi _{\theta _1}=\psi _{\theta '}\) and \(\psi _{\theta _m}=\psi _{\theta ''}\) be the string as in the definition of the equivalence relation \(\equiv \). From this definition, it follows that \(\psi _{\theta _j}\equiv \psi _{\theta _k}\) for all \(j,k=1,\dots ,m\), thus \(\psi _{\theta _j}\in {\mathcal {R}}_i\) for all \(j=1,\dots ,m\). Consequently, \(P_{[\psi _{\theta _j}]}\leqslant P_i\), so we can consider the projections \(P_{[\psi _{\theta _j}]}\) on the Hilbert space \({\mathcal {H}}_i\). We have

$$\begin{aligned} P_{[\psi _{\theta _1}]}\ldots P_{[\psi _{\theta _m}]}= aE_{\psi _{\theta _1},\psi _{\theta _m}} =aE_{\psi _{\theta '},\psi _{\theta ''}}, \end{aligned}$$
(15)

where

$$\begin{aligned} a=\langle \psi _{\theta _1}|\psi _{\theta _2}\rangle \cdots \langle \psi _{\theta _{m-1}}|\psi _{\theta _m}\rangle \ne 0, \end{aligned}$$

and for \(\xi ,\eta \in {\mathcal {H}}\) the operator \(E_{\xi ,\eta }\) is defined as

$$\begin{aligned} E_{\xi ,\eta }\zeta =\langle \eta |\zeta \rangle \xi ,\qquad \zeta \in {\mathcal {H}}. \end{aligned}$$
(16)

(The formula (15) is obvious when using Dirac notation since then on the left hand side we have simply

$$\begin{aligned} |\psi _{\theta _1}\rangle \langle \psi _{\theta _1}|\psi _{\theta _2}\rangle \cdots \langle \psi _{\theta _{m-1}}|\psi _{\theta _m}\rangle \langle \psi _{\theta _m}| = \langle \psi _{\theta _1}|\psi _{\theta _2}\rangle \cdots \langle \psi _{\theta _{m-1}}|\psi _{\theta _m}\rangle |\psi _{\theta _1}\rangle \langle \psi _{\theta _m}|, \end{aligned}$$

which is just the right-hand side. In this notation, we have \(E_{\xi ,\eta }=|\xi \rangle \langle \eta |\).)

Denote \({\mathscr {N}}=W^*(\{P_{[\psi _{\theta }]}:\theta \in \Theta \})\). On account of the above considerations, we infer that \(E_{\psi _{\theta '},\psi _{\theta ''}}\in {\mathscr {N}}\) for all \(\psi _{\theta '},\psi _{\theta ''}\in {\mathcal {R}}_i\).

Let

$$\begin{aligned} 0\ne \xi =\sum _{j=1}^ma_j\psi _{\theta _j},\qquad \psi _{\theta _j}\in {\mathcal {R}}_i, \end{aligned}$$

be an arbitrary finite linear combination of vectors from \({\mathcal {R}}_i\). We have

$$\begin{aligned} P_{[\xi ]}=\frac{1}{\Vert \xi \Vert ^2}\sum _{j,k=1}^ma_j{\bar{a}}_kE_{\psi _{\theta _j},\psi _{\theta _k}}. \end{aligned}$$

(This is again most easily seen by using Dirac notation since then

$$\begin{aligned} |\xi \rangle \langle \xi |=\left| \sum _{j=1}^ma_j\psi _{\theta _j}\Bigg \rangle \Bigg \langle \sum _{k=1}^ma_k\psi _{\theta _k}\right| =\sum _{j,k=1}^ma_j{\bar{a}}_k|\psi _{\theta _j}\rangle \langle \psi _{\theta _k}|.) \end{aligned}$$

It follows that \(P_{[\xi ]}\in {\mathscr {N}}\). Vectors \(\xi \) as above lie densely in \({\mathcal {H}}_i\), so arbitrary \(\eta \in {\mathcal {H}}_i\) is a limit of some \(\xi _n\), consequently,

$$\begin{aligned} P_{[\xi _n]}\rightarrow P_{[\eta ]} \quad \textit{weakly}, \end{aligned}$$

hence \(P_{[\eta ]}\in {\mathscr {N}}\) for each \(\eta \in {\mathcal {H}}_i\). Linear combinations of these projections restricted to the subspace \({\mathcal {H}}_i\) lie densely in \({\mathbb {B}}({\mathcal {H}}_i)\), so for the reduced von Neumann algebra \({\mathscr {N}}_{P_i}\), we get the inclusion

$$\begin{aligned} {\mathbb {B}}({\mathcal {H}}_i)\subset {\mathscr {N}}_{P_i}, \end{aligned}$$

and thus

$$\begin{aligned} {\mathbb {B}}({\mathcal {H}}_i)={\mathscr {N}}_{P_i}. \end{aligned}$$
(17)

From the definition of \(P_i\), it follows that \(P_i\) commutes with all \(P_{[\psi _{\theta }]}\), \(\theta \in \Theta \), because

$$\begin{aligned} P_iP_{[\psi _{\theta }]}= {\left\{ \begin{array}{ll} P_{[\psi _{\theta }]}, &{}\quad \text {if }\theta \in \Theta _i\\ 0, &{}\quad \text {otherwise.} \end{array}\right. } \end{aligned}$$

Since the algebra \({\mathscr {N}}\) is generated by the projections \(P_{[\psi _{\theta }]}\), we obtain that \(P_i\in {\mathscr {N}}'\). For each \(A\in {\mathscr {N}}\), we have

$$\begin{aligned} A=\sum _iAP_i=\sum _iP_iA, \end{aligned}$$

showing that

$$\begin{aligned} {\mathscr {N}}=\sum _i{}^{\oplus }{\mathscr {N}}_{P_i}, \end{aligned}$$

and the formula (17) gives the representation

$$\begin{aligned} {\mathscr {N}}=\sum _i{}^{\oplus }{\mathbb {B}}({\mathcal {H}}_i). \end{aligned}$$
(18)

In the first part of the proof, we obtained the relation \(P_{[\psi _{\theta }]}\in {\mathscr {M}}_{\text {min}}\) for all \(\theta \in \Theta \) which yields the inclusion

$$\begin{aligned} {\mathscr {N}}\subset {\mathscr {M}}_{\text {min}}. \end{aligned}$$

Consider a map \({\mathbb {E}}:{\mathbb {B}}({\mathcal {H}})\rightarrow {\mathscr {N}}\) defined as

$$\begin{aligned} {\mathbb {E}}(A)=\sum _iP_iAP_i. \end{aligned}$$

It is easily seen that \({\mathbb {E}}\) is a normal conditional expectation such that the states \(\psi _{\theta }\) are \({\mathbb {E}}\)-invariant, thus \({\mathscr {N}}\) is sufficient in Umegaki’s sense. Since \({\mathscr {M}}_{\text {min}}\) is minimal sufficient, we obtain

$$\begin{aligned} {\mathscr {M}}_{\text {min}}\subset {\mathscr {N}}, \end{aligned}$$

showing that \({\mathscr {M}}_{\text {min}}={\mathscr {N}}\), and the formula (18) together with the definition of \({\mathscr {N}}\) as \(W^*(\{P_{[\psi _{\theta }]}:\theta \in \Theta \})\) finish the proof. \(\square \)