Return to Question

The group von Neumann algebra $L\Gamma$ is a factor if and only if the group $\Gamma$ is ICC (i.e. infinite conjugacy class property). Moreover if $\Gamma$ is nontrivial then $L\Gamma$ is a $\mathrm{II}_1$ factor.

Question 0.1 (Sakai, Problem 4.4.10 here, 1971): Is every $\mathrm{II}_1$ factor a group von Neumann algebra?
Answer (Connes here, 1975): No, because there is a $\mathrm{II}_1$ factor which is not anti-isomorphic to itself (whereas $L\Gamma$ is so).

Question 0.2: Is every $\mathrm{II}_1$ factor anti-isomorphic to itself a group von Neumann algebra?
Answer (Jones, here, 1980): No, because there is a $\mathrm{II}_1$ factor which is anti-isomorphic to itself but without involutory antiautomorphisms (whereas $L\Gamma$ has so).

The answers for the next two questions were pointed out by Jiang's comments.

Question 0.3 (Remark 5.7 in previous paper): Is a $\mathrm{II}_1$ factor with an involutory antiautomorphism a group von Neumann algebra?
Answer (Ioana here, 2010): No, because there are $\mathrm{II}_1$ factors $M$ whose amplifications $M^t$ (with $t \neq 1$) admit an involutory antiautomorphism but are not group von Neumann algebras (see Corollary F, Corollary 10.1 and Remark 10.3).

Question 0.4: Is a $\mathrm{II}_1$ factor with an involutory antiautomorphism stably isomormphic to a group von Neumann algebra (i.e. of the form $(L\Gamma)^t$)?
Answer (Boutonnet here, 2013): No, because there are crossed product von Neumann algebra $L^{\infty}(X,\mu) \rtimes_{\sigma} \Gamma$, with $σ$ a free ergodic pmp-action of an ICC group $Γ$ on a probability space $(X,μ)$, which are not stably isomorphic to a group von Neumann algebra.

Question: Is a $\mathrm{II}_1$ factor with an involutory antiautomorphism stably isomorphic to a crossed product $A \rtimes G$, with $A$ an abelian von Neumann algebra and $G$ a group?

Recall that if the abelian von Neumann algebra $A$ is just $\mathbb{C}$ then $A \rtimes G = LG$. If the action of $G$ on $A$ is free then $A \rtimes G$ is a factor iff the action is ergodic. Now, note that $A \rtimes G$ could also be a factor without the action being free.

Remark: The existence of a Kac algebra generating a $\mathrm{II}_1$ factor not stably isomorphic to any $A⋊G$ (as above) would be very interesting.

The initial question and then Connes' paper were pointed out to me by Keshab Chandra Bakshi.

The group von Neumann algebra $L\Gamma$ is a factor if and only if the group $\Gamma$ is ICC (i.e. infinite conjugacy class property). Moreover if $\Gamma$ is nontrivial then $L\Gamma$ is a $\mathrm{II}_1$ factor.

Question 0.1 (Sakai, Problem 4.4.10 here, 1971): Is every $\mathrm{II}_1$ factor a group von Neumann algebra?
Answer (Connes here, 1975): No, because there is a $\mathrm{II}_1$ factor which is not anti-isomorphic to itself (whereas $L\Gamma$ is so).

Question 0.2: Is every $\mathrm{II}_1$ factor anti-isomorphic to itself a group von Neumann algebra?
Answer (Jones, here, 1980): No, because there is a $\mathrm{II}_1$ factor which is anti-isomorphic to itself but without involutory antiautomorphisms (whereas $L\Gamma$ has so).

The answers for the next two questions were pointed out by Jiang's comments.

Question 0.3 (Remark 5.7 in previous paper): Is a $\mathrm{II}_1$ factor with an involutory antiautomorphism a group von Neumann algebra?
Answer (Ioana here, 2010): No, because there are $\mathrm{II}_1$ factors $M$ whose amplifications $M^t$ (with $t \neq 1$) admit an involutory antiautomorphism but are not group von Neumann algebras (see Corollary F, Corollary 10.1 and Remark 10.3).

Question 0.4: Is a $\mathrm{II}_1$ factor with an involutory antiautomorphism stably isomormphic to a group von Neumann algebra (i.e. of the form $(L\Gamma)^t$)?
Answer (Boutonnet here, 2013): No, because there are crossed product von Neumann algebra $L^{\infty}(X,\mu) \rtimes_{\sigma} \Gamma$, with $σ$ a free ergodic pmp-action of an ICC group $Γ$ on a probability space $(X,μ)$, which are not stably isomorphic to a group von Neumann algebra.

Question: Is a $\mathrm{II}_1$ factor with an involutory antiautomorphism stably isomorphic to a crossed product $A \rtimes G$, with $A$ an abelian von Neumann algebra and $G$ a group?

Recall that if the action of the group $G$ on the abelian von Neumann algebra $A$ is free then $A \rtimes G$ is a factor iff the action is ergodic. Now $A \rtimes G$ can be a factor without the action being free, for example when $A = \mathbb{C}$, because then $A \rtimes G = LG$.

Remark: The existence of a Kac algebra generating a $\mathrm{II}_1$ factor not stably isomorphic to any $A⋊G$ (as above) would be very interesting.

The initial question and then Connes' paper were pointed out to me by Keshab Chandra Bakshi.

The group von Neumann algebra $L\Gamma$ is a factor if and only if the group $\Gamma$ is ICC (i.e. infinite conjugacy class property). Moreover if $\Gamma$ is nontrivial then $L\Gamma$ is a $\mathrm{II}_1$ factor.

Question 0.1 (Sakai, Problem 4.4.10 here, 1971): Is every $\mathrm{II}_1$ factor a group von Neumann algebra?
Answer (Connes here, 1975): No, because there is a $\mathrm{II}_1$ factor which is not anti-isomorphic to itself (whereas $L\Gamma$ is so).

Question 0.2: Is every $\mathrm{II}_1$ factor anti-isomorphic to itself a group von Neumann algebra?
Answer (Jones, here, 1980): No, because there is a $\mathrm{II}_1$ factor which is anti-isomorphic to itself but without involutory antiautomorphisms (whereas $L\Gamma$ has so).

The answers for the next two questions were pointed out by Jiang's comments.

Question 0.3 (Remark 5.7 in previous paper): Is a $\mathrm{II}_1$ factor with an involutory antiautomorphism a group von Neumann algebra?
Answer (Ioana here, 2010): No, because there are $\mathrm{II}_1$ factors $M$ whose amplifications $M^t$ (with $t \neq 1$) admit an involutory antiautomorphism but are not group von Neumann algebras (see Corollary F, Corollary 10.1 and Remark 10.3).

Question 0.4: Is a $\mathrm{II}_1$ factor with an involutory antiautomorphism stably isomormphic to a group von Neumann algebra (i.e. of the form $(L\Gamma)^t$)?
Answer (Boutonnet here, 2013): No, because there are crossed product von Neumann algebra $L^{\infty}(X,\mu) \rtimes_{\sigma} \Gamma$, with $σ$ a pmp-action of an ICC group $Γ$ on a probability space $(X,μ)$, which are not stably isomorphic to a group von Neumann algebra.

Question: Is a $\mathrm{II}_1$ factor with an involutory antiautomorphism stably isomorphic to a group von Neumann algebra $L \Gamma$ or a crossed product $L^{\infty}(X,\mu) \rtimes_{\sigma} \Gamma$ (as above)?
[if not, more generally, stably isomorphic to $A⋊Γ$ with $A$ an abelian von Neumann algebra?]

Remark: The existence of a Kac algebra generating a $\mathrm{II}_1$ factor not stably isomorphic to any $A⋊Γ$ (as above) would be very interesting.

The initial question and then Connes' paper were pointed out to me by Keshab Chandra Bakshi.

The group von Neumann algebra $L\Gamma$ is a factor if and only if the group $\Gamma$ is ICC (i.e. infinite conjugacy class property). Moreover if $\Gamma$ is nontrivial then $L\Gamma$ is a $\mathrm{II}_1$ factor.

Question 0.1 (Sakai, Problem 4.4.10 here, 1971): Is every $\mathrm{II}_1$ factor a group von Neumann algebra?
Answer (Connes here, 1975): No, because there is a $\mathrm{II}_1$ factor which is not anti-isomorphic to itself (whereas $L\Gamma$ is so).

Question 0.2: Is every $\mathrm{II}_1$ factor anti-isomorphic to itself a group von Neumann algebra?
Answer (Jones, here, 1980): No, because there is a $\mathrm{II}_1$ factor which is anti-isomorphic to itself but without involutory antiautomorphisms (whereas $L\Gamma$ has so).

The answers for the next two questions were pointed out by Jiang's comments.

Question 0.3 (Remark 5.7 in previous paper): Is a $\mathrm{II}_1$ factor with an involutory antiautomorphism a group von Neumann algebra?
Answer (Ioana here, 2010): No, because there are $\mathrm{II}_1$ factors $M$ whose amplifications $M^t$ (with $t \neq 1$) admit an involutory antiautomorphism but are not group von Neumann algebras (see Corollary F, Corollary 10.1 and Remark 10.3).

Question 0.4: Is a $\mathrm{II}_1$ factor with an involutory antiautomorphism stably isomormphic to a group von Neumann algebra (i.e. of the form $(L\Gamma)^t$)?
Answer (Boutonnet here, 2013): No, because there are crossed product von Neumann algebra $L^{\infty}(X,\mu) \rtimes_{\sigma} \Gamma$, with $σ$ a free ergodic pmp-action of an ICC group $Γ$ on a probability space $(X,μ)$, which are not stably isomorphic to a group von Neumann algebra.

Question: Is a $\mathrm{II}_1$ factor with an involutory antiautomorphism stably isomorphic to a crossed product $A \rtimes G$, with $A$ an abelian von Neumann algebra and $G$ a group?

Recall that if the abelian von Neumann algebra $A$ is just $\mathbb{C}$ then $A \rtimes G = LG$. If the action of $G$ on $A$ is free then $A \rtimes G$ is a factor iff the action is ergodic. Now, note that $A \rtimes G$ could also be a factor without the action being free.

Remark: The existence of a Kac algebra generating a $\mathrm{II}_1$ factor not stably isomorphic to any $A⋊G$ (as above) would be very interesting.

The initial question and then Connes' paper were pointed out to me by Keshab Chandra Bakshi.

The group von Neumann algebra $LG$ is a factor if and only if the group $G$ is ICC (i.e. infinite conjugacy class property). Moreover if $G$ is nontrivial then $LG$ is a $\mathrm{II}_1$ factor.

The initial question is to ask whether every $\mathrm{II}_1$ factor is of this form (Problem 4.4.10 in this book of S. Sakai, 1971). A. Connes answered negatively in this paper (1975) by building a $\mathrm{II}_1$ factor which is not anti-isomorphic to itself (whereas $LG$ is so). This leads to ask whether every $\mathrm{II}_1$ factor anti-isomorphic to itself is of that form. V. Jones answered negatively in this paper (1980) by building a $\mathrm{II}_1$ factor which is anti-isomorphic to itself but without involutory antiautomorphisms (whereas $LG$ has so). Then:

Question: Is a $\mathrm{II}_1$ factor with an involutory antiautomorphism a group von Neumann algebra?
Answer (after Jiang's comment): No, A. Ioana built in this paper (2010) $\mathrm{II}_1$ factors $M$ whose amplifications $M^t$ (with $t \neq 1$) admit an involutory antiautomorphism but are not group von Neumann algebras, see Corollary F, Corollary 10.1 and Remark 10.3. Then:

Updated question: Is a $\mathrm{II}_1$ factor with an involutory antiautomorphism of the form $(LG)^t$?

In other words: Does a $\mathrm{II}_1$ factor with an involutory antiautomorphism admit an amplification which is a group von Neumann algebra? If not, it there still an abstract characterization of such $\mathrm{II}_1$ factors?

The initial question and then Connes' paper were pointed out to me by Keshab Chandra Bakshi.

The group von Neumann algebra $L\Gamma$ is a factor if and only if the group $\Gamma$ is ICC (i.e. infinite conjugacy class property). Moreover if $\Gamma$ is nontrivial then $L\Gamma$ is a $\mathrm{II}_1$ factor.

Question 0.1 (Sakai, Problem 4.4.10 here, 1971): Is every $\mathrm{II}_1$ factor a group von Neumann algebra?
Answer (Connes here, 1975): No, because there is a $\mathrm{II}_1$ factor which is not anti-isomorphic to itself (whereas $L\Gamma$ is so).

Question 0.2: Is every $\mathrm{II}_1$ factor anti-isomorphic to itself a group von Neumann algebra?
Answer (Jones, here, 1980): No, because there is a $\mathrm{II}_1$ factor which is anti-isomorphic to itself but without involutory antiautomorphisms (whereas $L\Gamma$ has so).

The answers for the next two questions were pointed out by Jiang's comments.

Question 0.3 (Remark 5.7 in previous paper): Is a $\mathrm{II}_1$ factor with an involutory antiautomorphism a group von Neumann algebra?
Answer (Ioana here, 2010): No, because there are $\mathrm{II}_1$ factors $M$ whose amplifications $M^t$ (with $t \neq 1$) admit an involutory antiautomorphism but are not group von Neumann algebras (see Corollary F, Corollary 10.1 and Remark 10.3).

Question 0.4: Is a $\mathrm{II}_1$ factor with an involutory antiautomorphism stably isomormphic to a group von Neumann algebra (i.e. of the form $(L\Gamma)^t$)?
Answer (Boutonnet here, 2013): No, because there are crossed product von Neumann algebra $L^{\infty}(X,\mu) \rtimes_{\sigma} \Gamma$, with $σ$ a pmp-action of an ICC group $Γ$ on a probability space $(X,μ)$, which are not stably isomorphic to a group von Neumann algebra.

Question: Is a $\mathrm{II}_1$ factor with an involutory antiautomorphism stably isomorphic to a group von Neumann algebra $L \Gamma$ or a crossed product $L^{\infty}(X,\mu) \rtimes_{\sigma} \Gamma$ (as above)? 
[if not, more generally, stably isomorphic to $A⋊Γ$ with $A$ an abelian von Neumann algebra?]

Remark: The existence of a Kac algebra generating a $\mathrm{II}_1$ factor not stably isomorphic to any $A⋊Γ$ (as above) would be very interesting.

The initial question and then Connes' paper were pointed out to me by Keshab Chandra Bakshi.