数理解析研究所講究録
1077 巻 1999 年 79-87

79

TYPE III FACTORS ARISING AS
AMALGAMATED FREE PRODUCTS

YOSHIMICHI UEDA (植田好道)

Graduate School of Mathematics, Kyushu UniversityFukuoka, 810-8560, Japan
Department of Mathematics, University of California
Berkeley, CA 94720, USA

1. Introduction.
Type III factors arising as free products were studied in detail by F.

$\mathrm{R}\dot{\mathrm{a}}\mathrm{d}\mathrm{u}\mathrm{l}\mathrm{e}\mathrm{s}\mathrm{C}\mathrm{u}$,L. Barnett, K. Dykema and D. Shlykhtenko mainly based on D. Voiculescu’s powerfulmachine. (However, the question of type classification is not yet completed. An attempt
to the question can be also found in [U3].) On the other hand, amalgamated freeproducts in the type III setting had never been investigated before our study. In thisnote, we will report our recent study on type III factors arising as amalgamated freeproducts.

2. Amalgamated free products of von Neumann algebras.
Let$A\supseteq D\subseteq B$be (a-finite) von Neumann algebras and$E_{D}^{A}$: $Aarrow D,$$E_{D}^{B}$

be faithful normal conditional expectations. The amalgamated free product ofover$D$with respect to$E_{D}^{A}$

and$E_{D}^{B}$: $Barrow D$$A$

and $B$is defined as a von Neumann algebra $M$ with unital

Research fellow of JSPS
Typeset by$A_{\Lambda 4}S_{-}\mathrm{z}\mathrm{E}\mathrm{x}$80(normal) embeddings ofconditional expectation(1)

and$A$$B$$A$

and $B$ into $M$ which coincide on $D$ and a faithful normal$E_{D}^{M}$: $Marrow D$ such thatgenerate $M$ as subalgebras of $M$ ;(2) the restrictions of$E_{D}^{M}$

to$A$

and $B$ (as subalgebras of $M$ ) are just$E_{D}^{A}$

and$E_{D}^{B}$

respectively;(3)

and$A$$B$(as subalgebras of $M$ ) are free with respect to$E_{D}^{M}$.

We denote the amalgamated free product by$(M, E_{D}^{M})=(A, E_{D}^{A})*_{D}(B, E_{D}^{B})$.

In analysis on type III factors, modular automorphisms have central importance.
Hence, we should at first compute the modular automorphisms

fixed faithful normal state$\varphi$$\sigma_{t}^{\varphi\circ E^{M}}D,$$t\in \mathrm{R}$, for aon $D$ . This can be done by using the freeness condition.

Theorem 1. (cf. [U2]) We have$\sigma_{t}^{\varphi\circ E_{D}^{M}\mathrm{o}E^{A}}|A=\sigma_{t}\varphi D$

for$t\in \mathrm{R}$,$\sigma_{t}^{\varphi\circ E^{M}}D|_{Bt}=\sigma^{\varphi\circ E^{B}}D$.

3. Amalgamated free products over Cartan subalgebras.

Amalgamated free products of factors over their common Cartan subalgebras were
investigated as a first step of our attempt towards investigation on amalgamated freeproducts in the type III setting.

Our main theorem is as follows:

81Theorem 2.

Let$([\mathrm{U}2])$

non-atomic ($i.e.,$$A$$A$

and $B$ be factors with separable preduals having a commonand $B$ are not of type ) Cartan subalgebra D. Let$I$$(M, E_{D}^{M})=(A, E_{D}^{A})*_{D}(B, E_{D}^{B})$be the amalgamated free product. Then there exists a faithful normal state$\varphi$

on $D$ suchthat$(A_{\varphi\circ E_{D}^{A}})/\mathrm{n}M\subseteq A$

Moreover, if $A$ is of type

$III_{\lambda}(\lambda\neq 0)$.

, the above state$\varphi$can be chosen in such a way

that$(A_{\varphi \mathrm{o}E_{D}^{A}})’\cap A=^{\mathrm{c}1}$.

The analogous result holds for $B$ .
The proof depends on several results related to ergodic theory and the averaging
technique, but the crucial idea is very simple. Indeed, the theorem is proved based on theessentially same simple spirit as in the proof of the classical fact that

where$a$is one of the free generators inRemark. (The type$I$case.) If

$A$

and

$L(\mathrm{F}_{2})$$B$$\{a\}’\cap L(\mathrm{F}_{2})=\{a\}^{\prime/}$,.

are type$I$factors$\mathrm{h}\mathrm{a}_{\vee}\mathrm{v}\mathrm{i}\mathrm{n}\mathrm{g}$a common Cartan

subalgebra $D$ , then $A=B$ , that is, $A\supseteq D\subseteq B$ is isomorphic to $A\supseteq D\subseteq A$ . In

this case, it can be shown that the amalgamated free product ofisomorphic to$M$$L(\mathrm{F}_{n-1})\otimes A$

when$A$(or $B$ ) is of type

$I_{n}$$A$

and $B$ over $D$ is(possibly $n=\infty$ ) (and hence

is not of type III). This computation is essentially contained in [D3], and also T.

Sakamoto checked the same formula in the$C^{*}$-case. Furthermore, the same formula also

holds even in the level of groupoid free products (see below). It should be remarked

that the relative commutant property in the theorem does not hold in the type

$I$

case.

82

Indeed, via the above isomorphism $M=A*_{D}B\cong L(\mathrm{F}_{n-1})\otimes A$ the subalgebra
corresponds to$\mathrm{C}1\otimes A$$A$, and hence $A’\cap M\cong L(\mathrm{F}_{n-1})$ . This shows that the non-typecase is different from the type

$I$

$I$

case.

After the work [U2] was completed, H. Kosaki began to study amalgamated freeproducts over Cartan subalgebras from the view-point of measured groupoids (see [K]).
He discussed, among other things, the construction (in a rigorous measurable fashion)of “free products” of discrete measured equivalence relations as well as some detailed
analyses in the type$I$

setting. His approach might be useful to clarify the above-

mentioned difference between the type$I$

case and the non-type

$I$

case.

Theorem 2, in particular, shows that the resulting amalgamated free product $M$ is afactor. Moreover, based on Theorem 2 we can obtain the following type classification
results:(1) If $M$ is of type $III_{0}$ , then both ofand $B$ must be of type $III_{0;}$$A$(2) If either$A$or$B$

is of type(3) If either$A$or$B$

is of type $III_{1}$ , then $M$ is of type $III_{1}$ ;(4) If$A$

is of type

$III_{\lambda}$$II_{1}$and $M$ is semi-finite, then $M$ is of type

and $B$ is of typeindependent, then $M$ is of type

$III_{\mu}$$III_{1}$such that$\log\lambda$$II_{1}$;and $\log\mu$ are rationally.

The assertions (3), (4) follow from a simple computation of the$\mathrm{T}$-set $T(M)$ based on

the relative commutant property of Theorem 2 together with the assertion (1).
It should be here mentioned that an example of AFD type

$II_{1}$(or$II_{\infty}$) factors

having a common Cartan subalgebra whose amalgamated free product is of type$0<\lambda\leq 1$, can be constructed.

$III_{\lambda}$,83

To get more detailed information on type classification we should compute the flowof weights of the resulting amalgamated free product $M$ in terms of those of$B$$A$and. This can be done based on the techniques used in the proof of Theorem 2 together

with a fact, suggested by H. Kosaki, on the continuous cores (or the Takesaki duals) ofamalgamated hee products. Indeed, we can prove the following:
Theorem 3.

of weights
flows ofHere,Let$([\mathrm{U}2])$$(X_{M}, F_{t}M)$$X_{D}=\mathrm{Y}\cross \mathrm{R}$$(X_{D}, F_{t}D)$and $M=A*_{D}B$ be as in Theorem 2. The flow

is determined as the (unique) maximal common factor flow of the$weight_{\mathit{8}}(X_{A}, F_{t}^{A}),$

Cartan subalgebra ofones of$A\supseteq D\subseteq B$$(X_{B}, F_{t}^{B})$and$(X_{D}, F_{t}^{D})$.

with $D=L^{\infty}(Y)$ and $F_{t}^{D}(y, s)=(y, s+t)$ . Since $D$ is a common$A$and$B$, the flows of weights $(X_{A}, F_{t}A)$ and$(X_{B}, F_{t}^{B})$are factor. Hence, the above (unique) maximal common factor flow exists.

The above formulation in terms of ergodic theory was suggested by T. Hamachi, while
the author’s original formulation is written in terms of the continuous cores$A$

and$B$

as follows: The centerof the centers$Z(\overline{A})$and$Z(\overline{B})$$\mathcal{Z}(\overline{M})$$\overline{A}$and$\overline{B}$ofof the continuous core of $M$ isjust the intersection.

Theorem 3, in particular, shows that in the case that $A=B$ , the resulting flow ofweights$(X_{M}, F_{t}^{M})$is just $(X_{A}, F_{t}A)$ (or$(X_{B},$ $F^{B})t$).

Remark. The above theorem (Theorem 3) can be proved under a weaker condition.
The cores$\overline{M}:=M\lambda\sigma^{\varphi \mathrm{o}E_{D}^{M}}\mathrm{R},\overline{A}:=A\rangle\triangleleft_{\sigma^{\varphi \mathrm{o}E_{D}^{A}}}\mathrm{R},\overline{B}:=B\rangle\triangleleft_{\sigma^{\varphi \mathrm{o}E_{D}^{B}}}\mathrm{R}$$D\rangle\triangleleft_{\sigma^{\varphi}}\mathrm{R}=D\otimes\lambda(\mathrm{R})^{\prime/}$have the natural inclusion relations:$\overline{M}=\overline{A}\vee\overline{B}\supseteq\overline{A},\overline{B}\supseteq\overline{D}$.

and$\overline{D}$

$:=$

84

As is well-known in the subfactor theory, the conditional expectations

can be lifted as$\overline{E_{D}^{M}}$:$\overline{M}arrow\tilde{D},\overline{E_{D}^{A}}$:$\overline{A}arrow\overline{D}$In the current setting, we have known thatand$\overline{E_{D}^{B}}$:$\overline{B}arrow\overline{D}$$E_{D}^{M},$$E_{D}^{A}$

and$E_{D}^{B}$by the quite natural way.

$Z(\overline{M})=Z(\overline{A})\cap Z(\overline{B})\subseteq\overline{D}$, and hence wehave$\overline{M}=\int_{X_{M}}^{\oplus}\overline{M}(\omega)d\mu(\omega)\supseteq$$\overline{A}=\int_{X_{M}}^{\oplus}\overline{A}(\omega)d\mu(\omega)$,$\overline{B}=\int_{X_{M}}^{\oplus}\overline{B}(\omega)d\mu(\omega)$$\supseteq\overline{D}=\int_{X_{M}}^{\oplus}\overline{D}(\omega)d\mu(\omega)$

under the identification $Z(\overline{M})=L^{\infty}(x_{M,\mu})$ . Based on Voiculescu’s striking result$([\mathrm{V}2])$

we can show the following:

Proposition 4. Let

is of type III. For a. .
$e$for $r>1$ .

and $M=A*_{D}B$ be as in Theorem 2. Suppose that $M$$A\supseteq D\subseteq B$$\omega$, the type

$II_{\infty}$

factor$\overline{M}(\omega)$

is not of the form

$L(\mathrm{F}_{r})\otimes B(\mathcal{H})$

Therefore, if $M$ is of type $III_{\lambda}(0<\lambda<1)$ , then the typeappearing in its discrete decomposition is also not isomorphic to$II_{\infty}$$L(\mathrm{F}_{r})\otimes B(\mathcal{H})$factorfor any$r>1$ .

This proposition suggests us that type III factors arising as our amalgamated freeproducts are different from those arising as free products. Indeed, K. Dykema showed,

in [D2], that the discrete cores of many type

$III_{\lambda}$

factors arising as free products of.

AFD von Neumann algebras are of the form

$L(\mathrm{F}_{\infty})\otimes B(\mathcal{H})$

Remark. (The case that $M$ is of type) The factor $M$ is not isomorphic to any$II_{1}.$

free group factor even in the case that $M$ is of type

Voiculescu’s striking result$([\mathrm{V}2])$.

$II_{1}$. This also follows from D.

85

In closing this section, it is mentioned that the (non-)amenability of our amalgamatedfree product $M$ is unknown. (Of course, if $M$ is of type

$II_{1}$, then $M$ is always not

amenable thanks to the presence of a copy of the free group factor
4. The crossed-product by a minimal action of $SU_{q}(2)$$L(\mathrm{F}_{2})$in $M.$ ).

In [U1], the author introduced (and established) the concept of free products of actionsof quantum groups, and by using it a minimal action of the quantum group $SU_{q}(n)$ wasconstructed. However, the question of type classification of the fixed-point algebra (orequivalently the crossed-product thanks to the Takesaki duality) of the minimal action
has been unsolved. The crossed product can be written as a certain amalgamated freeproduct, and hence this question can be regarded as that for an amalgamated freeproduct. The author recently solved it as follows:
Theorem 5.

$([\mathrm{U}4])$in [U1] is of type

The crossed-product by the minimal action of $SU_{q}(2)$ constructed$III_{q^{2}}$(and hence the fixed-point algebra is also of typeThe main part of the proof is as follows: If a real number$t$$III_{q^{2}}$is in the$\mathrm{T}$).

-set of thecrossed-product, then the following equations hold:$x\otimes(q^{2it_{X}})+u\otimes(q^{2it}v)=x\otimes(wXw^{*})+u\otimes(wvw^{*})$;$v\otimes x+y\otimes v=v\otimes(wXw^{*})+y\otimes(wvw*)$

for some unitary$w$

on the standard Hilbert space $L^{2}(SU_{q}(2))$ . Herestandard generators of $SU_{q}(2)$ (see$[\mathrm{M}^{2}\mathrm{N}^{2}\mathrm{U}]$generators with respect to the Haar state$SU_{q}(2))$$h$be thefor example). By the orthogonality of the(thanks to the Peter-Weyl type theorem for

we obtain$q^{2it}x=wXw^{*}=x$ ,$x,$ $u,$ $v,$ $y$$q^{2it*}v=wvw=v$ .

86Hence the number$t$is in$( \frac{-2\pi}{\log q^{2}})\mathbb{Z}$.

Furthermore, it can be shown that the ambient factor and the fixed-point algebra

under the above minimal action of $SU_{q}(2)$ have the identical flow of weights.

The details together with further investigations (related to subfactor theory) will bepresented elsewhere.

Acknowledgment. The author thanks H. Kosaki for discussions on free products ofmeasured equivalence relations and for informing of his recent attempt. He also thanksT. Hamachi for discussions on ergodic flows.

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