# Tagged Questions

Subtag of [tag:oa.operator-algebras] for questions about von Neumann algebras, that is, weak operator topology closed, unital, *-subalgebras of bounded operators on a Hilbert space.

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### Is there an operator algebraic reformulation of the invariant subspace problem?

Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators. Invariant subspace problem: Let $T \in B(H)$. Is there a non-trivial closed $T$-invariant ...
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### Is the fundamental group of $II_{1}$ factors invariant under a relation?

In order to define the equivalence relation, let's first recall the Tomita-Takesaki modular theory and conditional expectation for von Neumann algebras. Let $H$ be a separable Hilbert space and $B(H)$...
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### What's a noncommutative set?

This issue is for logicians and operator algebraists (but also for anyone who is interested). Let's start by short reminders on von Neumann algebra (for more details, see [J], [T], [W]): Let $H$ ...
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### What are the intermediate subfactors of the tensor product of two maximal subfactors?

Let $(N_1 \subset M_1)$ and $(N_2 \subset M_2)$ be two maximal subfactors. Their tensor product, the subfactor $(N_1 \otimes N_2 \subset M_1 \otimes M_2)$, admits four obvious intermediate ...
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### Is there a maximal finite depth infinite index irreducible subfactor?

A subfactor $N \subset M$ is irreducible if $N' \cap M = \mathbb{C}$. It's maximal if it admits no non-trivial intermediate subfactors $N \subset P \subset M$. It's cyclic if its lattice of ...
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### About the classification of infinite depth irreducible finite index maximal subfactors

The Temperley Lieb subfactors $A_{\infty}$ are the first examples of infinite depth irreducible finite index maximal subfactors. We can see these subfactors as coming from the simple Lie group $SU(2)$....
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### The cyclic subfactors theory: a quantum arithmetic?

Context: First recall some results: - Actions of finite groups on the hyperfinite type $II_{1}$ factor $R$ (Jones 1980). - A Galois correspondence for depth 2 irreducible subfactors (Izumi-Longo-...
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### Projections in a W*-algebra as a continuous lattice?

A continuous lattice is a complete lattice $L$ in which every element $y$ is equal to $\bigvee${$x \in L \mid x \ll y$} where $x \ll y$ ("x approximates y" or "x is way below y") if for any directed ...
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### Is there an infinite depth irreducible finite index maximal subfactor (other than Temperley Lieb) ?

A subfactor $N \subset M$ is maximal if it admits no non-trivial intermediate subfactors $N \subset P \subset M$. Is there an infinite depth irreducible finite index maximal subfactor (other than ...
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### Are there only finitely many maximal irreducible amenable subfactors at fixed finite index?

A subfactor $N \subset M$ is maximal if it admits no non-trivial intermediate subfactors $N \subset P \subset M$. Question: Are there only finitely many maximal irreducible amenable subfactors at ...
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### When does a $W^*$-algebra have a standard Borel spectrum?

EDIT: André Henriques has commented below that the correct separability condition is not weak-* separability as I have written below, but separability of the predual. This post came out a bit long, ...
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### Weights on Von Neuman factors

Let A be a type $I$ factor on a Hilbert space H. Let $\varphi$ be a semi-finite normal weight on $A^{+}$ is it possible to say that then there exist Hilbert spaces $H_2 \subset H_1$ and an isomorphism ...
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### Find a lower bound for a pre-invariant $Fol(L(F_m), X_m)$

In the paper of Bannon and Ravichandran, A Folner invariant for type $\rm{II}_1$ factors, they defined an invariant $Fol(M)$ for a separable type $\rm{II}_1$ factor $M$, especially for the free group ...
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### Commutant of a von Neumann algebra as the linear span of unitaries.

I'm reading chapter 4 of Gerard Murphy's C*-algebras book and am confused by a statement in his proof of theorem 4.1.10. In his proof, he says, "$A'$ is the linear span of its unitaries" (where $A'$ ...
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### Limits of von Neumann algebras

Consider (abstract) von Neumann algebras topologised by their weak*-topology arising from the unique predual. In the theory of topological vector spaces, there is a natural notion of an inductive ...
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### Finite projection in Von Neumann algebra

I had the following question when I am learning von Neumann algebras: Let p be a finite projection in a finite von Neumann algebra $M$, let $p>p_1>p_2>\cdots$ be a decreasing sequence of ...
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### Von Neumann algebra associated to the infinite Cuntz algebra

The Cuntz algebra $\mathcal{O}_{\infty}$ is the universal $C^*$-algebra generated by countably infinitely many isometries $s_i$ satisfying the relations $s_i^*s_j = \delta_{ij}$ (there is no condition ...
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### Which W*-algebras are the duals of C*-coalgebras?

A Banach algebra (assumed associative and unital) is precisely a monoid object in the monoidal category of Banach spaces, short linear maps, and the projective tensor product. A Banach coalgebra is ...
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### Extensions of completely positive mappings

I would like to ask the following two questions. Let $1_{\mathcal{H}}\in \mathcal{A}\subset\mathcal{B}\subset\overline{\mathcal{A}}^{SOT}\subset\mathbb{B}(\mathcal{H})$ be a sequence of $C^{\ast}$-...
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### Is it true that there are exactly two conjugacy classes of order two elements in Out(R)?

In the title, $R$ stands for the hyperfinite III1 factor. An order two element $\alpha\in Out(M)$ ($M$ any factor) has an invariant $c(\alpha)\in H^3(\mathbb Z/2,S^1)=\mathbb Z/2$. Q: Is $c$ the ...
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### How well do we know relative commutants in $L(\mathbb{F}_\infty)$?

Let $H=K_1\oplus K_2$ be infinite dimensional Hilbert spaces. Voiculescu's free Gaussian functor gives us free group factors $L(H)$, $L(K_1)$, $L(K_2)$ acting on the full Fock space $\Gamma(H)$ and, ...
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### The monotone closure of a $C^*$-algebra

Related to Jon's question, I have two questions. Let $\mathcal{A}$ be a concrete $C^*$-algebra on a Hilbert space $\mathcal{H}$. For any selfadjoint subset $S$ of $\mathbb{B}(\mathcal{H})$, let $S^m$ ...
Let $M$ be a separable von Neumann algebra and let $A$ be a (von Neumann-)dense *-subalgebra. Suppose that $\alpha,\alpha_1,\alpha_2,\dots$ are automorphisms of $M$, such that for every $a \in A$, $$\... 1answer 390 views ### Denseness of inner automorphisms inside automorphisms of hyperfinite type III_1 factor Let R be the hyperfinite type III_1 factor, and let Aut(R) be its group of automorphisms, equipped with the u-topology (topology of pointwise convergence on the predual). An automorphism \... 1answer 357 views ### Injective von Neumann algebra Let G be a non-amenable countable discrete group. How can I show that the group von Neumann algebra L(G) has no injective direct summand? 0answers 125 views ### Number of projections in a von Neumann algebra Suppose that a von Neumann algebra A acts on \ell_2(I) and I has the minimal cardinality for which it holds. Can we caluclate (or at least give a reasonable lower bound) for the cardinality of ... 1answer 229 views ### Intersections of maximal abelian von Neumann algebras Let H be separable Hilbert space. Let A be a maximal abelian von Neumann subalgebra of B(H), and B an abelian von Neumann algebra with A\cap B={\mathbb C}I, where I is the indentity ... 1answer 294 views ### Idempotent homomorphisms of von Neumann algebras Is there any description of unital idempotent (F^2(x)=F(x)) morphisms of a von Neumann algebra into itself? Or, equivalently, of weakly closed subalgebras which are retracts as von Neumann algebras? 1answer 589 views ### Do Baumslag-Solitar Group von Neumann algebras have Property \Gamma? A type II_{1} factor M with trace \tau has Property \Gamma if for every finite subset \{ x_{1}, x_{2},..., x_{n} \} \subseteq M and each \epsilon >0, there is a unitary element u in ... 1answer 343 views ### When an AW*-algebra is a W*-algebra In a very old book of Kaplansky "Rings of operators", on p. 123 one can find the following sentence: It is a standing conjecture that an AW{}^\ast-algebra is W{}^\ast if its center is W{}^\ast. ... 2answers 569 views ### Polar decomposition in C*-algebras A very nice feature of W*-algebras is the following: once you have an element a of a W*-algebra M, and a=u|a| (the polar decomposition), then u\in M. It seems that it carries over to AW*-... 0answers 314 views ### Does this inequality of negative relative entropy and quantum relative entropy hold? Hello, everyone! Question I have a question about the relationship between general relative entropy and general quantum relative entropy: Given a unit vector |i\rangle and two Hermitian matrices ... 1answer 316 views ### Topology of the “normal spectrum” of a commutative von Neumann algebra Kadison and Ringrose define normal states \omega of a von Neumann algebra A as such that \omega(H_\alpha)\to \omega(H) for each monotone increasing net of operators H_\alpha with least upper ... 1answer 254 views ### Abelian sub-W*-algebras Let M be a von Neumann algebra which acts faithfully on a Hilbert space of density character \kappa but does not on a Hilbert space of density character \lambda<\kappa (that is, the density ... 1answer 715 views ### Does the hyperfinite II_1 factor admit two irreducible representations that are not unitarily equivalent? Regarding the hyperfinite II_{1} factor R as C^{*}-algebra, is it known whether any two irreducible representations of R are unitarily equivalent? If it is known that there exists a pair of ... 1answer 130 views ### Endomorphism of a type III_1 factor Is a unital, injective, ultraweakly continuous *-endomorphism f:M\rightarrow M of a III_1 factor M inner? I.e. is there a unitary U\in M: f(-)=U(-)U^*? 2answers 440 views ### Normalizer of a von Neumann algebra Let (A,H) be a von Neumann algebra in standard form (which means that H=L^2A), and recall that the automorphism group Aut(A) acts on both A and H. Let$$N:=\{u\in U(H): uAu^*=A\} be the ...
Let $M$ be a von Neumann algebra and $\varphi$ be a normal weight on it, and let $A$ be its definition subalgebra. We still denote $\varphi$ the extension to $A$ as a linear positive functional. It ...