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6
votes
1answer
287 views

von Neumann automorphisms: does convergence on a dense algebra imply $u$-convergence?

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 $a \in A$, $$ ...
15
votes
1answer
370 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 ...
3
votes
1answer
324 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?
1
vote
0answers
121 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 ...
4
votes
1answer
224 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 ...
2
votes
1answer
278 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? ...
11
votes
1answer
555 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 ...
3
votes
1answer
313 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${}^\ast$. ...
3
votes
2answers
423 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 ...
2
votes
0answers
299 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 ...
3
votes
1answer
286 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 ...
4
votes
1answer
238 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 ...
10
votes
1answer
666 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 ...
2
votes
1answer
125 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 $U\in M:$ $f(-)=U(-)U^*$?
6
votes
2answers
416 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 ...
1
vote
1answer
245 views

Continuity of a weight on its definition domain in a von Neumann algebra

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 ...
15
votes
2answers
1k views

About the category of von neumann algebras

I am looking for one (or more) reference about properties of the category of von Neumann algebra. More precisely, in an answer of a previous question, Dmitri Pavlov mentions that the $W^*$ category ...
5
votes
1answer
311 views

Reference for embedding an infinite direct product of matrix algebras into the hyperfinite $II_1$ factor

In some calculations I am writing up, $\newcommand{\cR}{{\mathcal R}}$ I want to add - as a fairly throwaway remark - that any countable product (= $\ell^\infty$-direct sum) of matrix algebras can be ...
6
votes
2answers
424 views

Is there an inclusion of $L_\infty(G)$ into $C_0(G)^{**}$?

This is probably a silly question with which I am stuck however. Let $G$ be a locally compact group. It seems to me that there is a canonical map of $L_\infty(G)$ into $M=C_0(G)^{**}$ (the latter is ...
2
votes
2answers
311 views

ED compact $K$ such that $C(K)$ is not a dual Banach space

Almost every mathematician working in von Neumann algebras says that there are certainly extremely disconnected compact spaces $K$ such that $C(K)$ is not a dual space (they are not hyperstonean). ...
16
votes
0answers
639 views

Relative commutants of abelian von Neumann algebras

This question arose from the discussion over at the question Centralizers in $C^*$-algebras. Which von Neumann algebras $N$ satisfy the property that $A' \cap N = B' \cap N \implies A = B$, for all ...
2
votes
1answer
654 views

Centralizers in C*-algebra

Let $a,b\in A$ be self-adjoint elements in $C^*$-algebra $A$ with equal centralizers, $\{x\in A; [a,x]=0\}=\{x\in A; [b,x]=0\}$. Can we say anything about the correspondence between $a$ and $b$? For ...
11
votes
1answer
495 views

A left inverse for the comultiplication on a Hopf von Neumann algebra

Edit: incorrect claim at end of earlier version; thanks to Matthew Daws for pointing this out in comments. $\newcommand{\cM}{{\mathcal M}}\newcommand{\stp}{{\overline{\otimes}}}$The following ...
1
vote
1answer
228 views

Disintegration of von Neumann algebra

Let $B=\int_{Y}^{\oplus}B_yd_y$ be the direct integral decomposition of vNa $B$ into factors and if $P=\int_{Y}^{\oplus} p_ydy$ is a projection in $B$ and $p_y$ is equivalent to a projection $q_y$ in ...
3
votes
0answers
374 views

Morphism of von Neumann Algebras

Hello, Is there a counterexample to the following statement: let $A,B$ two von Neumann algebras, every morphism $A \rightarrow B$ of $C^* $-algebras is a $W^*$-homomorphism ? ( a $W^* ...
11
votes
1answer
704 views

W*-completion of a C*-algebra?

tl;dr: Is there such a thing as a W*-completion of a C*-algebra, and if so, where can I read about it? I'm wondering about the relationship between (abstract) C*-algebras and W*-algebras. On the one ...
17
votes
0answers
511 views

The Mackey Topology on a Von Neumann Algebra

Every von Neumann algebra $\mathcal M$ is the dual of a unique Banach space $\mathcal M_* $. The Mackey topology on $\mathcal M$ is the topology of uniform convergence on weakly compact subsets of ...
14
votes
0answers
554 views

Which complete Boolean algebras arise as the algebras of projections of commutative von Neumann algebras?

Projections in an arbitrary commutative von Neumann algebra form a complete Boolean algebra. Moreover, a morphism of commutative von Neumann algebras induces a continuous morphism of the corresponding ...
7
votes
1answer
437 views

How does the right regular of GL(n, R) and GL(n,Qp) decompose?

The question is contained in the title. I would guess that this question is already answered in the literature. Given the reductive group $GL(n)$ over a complete local field, how does the right ...
8
votes
1answer
441 views

Lifting of a ucp map with values in a von Neumann algebra ultraproduct of matrix algebras

Let $u:A \to \prod_{\mathcal U} M_n$ be a unital completely positive map (ucp) from a unital separable $C^*$algebra into the von Neumann algebra ultraprodut $\prod_{\mathcal U} M_n$. Here $\mathcal ...
13
votes
4answers
904 views

Are almost commuting hermitian matrices close to commuting matrices (in the 2-norm)?

I consider on $M_n(\mathbb C)$ the normalized $2$-norm, i.e. the norm given by $\|A\|_2 = \sqrt{Tr(A^* A)/n}$. My question is whether a $k$-uple of hermitian matrices that are almost commuting (with ...
24
votes
1answer
1k views

Tomita-Takesaki versus Frobenius: where is the similarity?

I've often heard Alain Connes say that the modular flow of Tomita-Takesaki theory should be thought of as a characteristic zero analog of the Frobenius endomorphism. ... can anyone justify this ...
5
votes
1answer
289 views

representing tensor-C*-categories in BIM

Given a factor M (=von Neumann alg. with center ℂ), let us write BIM for the ⊗-C*-category of M-M-bimodules. Which ⊗-C*-categories can one faithfully embed into BIM? ⓵ Are ...
18
votes
0answers
697 views

local equivalence of loop group representations

Let $G$ be a compact, simple, connected, simply connected (cscsc) Lie group, and let its smooth loop group $LG:=C^\infty(S^1,G)$. Given an interval $I\subset S^1$, we have the local loop group $$ L_IG ...
4
votes
1answer
465 views

topologies on U(H)

There are many topologies on the algebra $B(H)$ of bounded operators on Hilbert space: the weak, strong, ultraweak (also called σ-weak), ultrastrong (also called σ-strong), and some ...
0
votes
0answers
138 views

For a separable v-N algebra M, how to see $M \rtimes \mathbb{R}$ as a subalgebra of $M \otimes B(L^2)$?

For a v-N algebra $M$ acting as bounded operators on a separable Hilbert space $H$, how to see $M \rtimes \mathbb{R}$ as a subalgebra of $M \otimes B(L^2(\mathbb{R})$? Why I am confused is because ...
2
votes
2answers
396 views

Spatial isomorphisms of tensor product of factors

Suppose $N \subset M$ are two factors, neither of them Type I, acting on a separable Hilbert space $H$. Let $\pi_1$ be a faithful normal representation of $N$ and $\pi_2$ a faithful normal ...
5
votes
3answers
905 views

What is the difference between a primary representation and a irreducible representation?

I am currently reading some of Mackey's work on unitary representation. Given a locally compact group $G$ and a unitary representation $\pi : G\rightarrow U(H)$. As far as I understood it, the ...
1
vote
1answer
207 views

when is an element of $M_n(M)$ $\ast$-free from $M_n(\mathbb{C})$ for a $\ast$-non-commutative probability space $M$.

From "Lectures on the combinatorics of free probability" by Nica and Speicher we have a necessary sufficient criteria for an element of $M_n(M)$ being free from $M_n(\mathbb{C})$ for a non-commutative ...
4
votes
2answers
285 views

v-Na generated by

Given two free semicirculars X_1 and X_2 and a projection h in the von-Neumann algebra generated by X_1, how does one show that the von-Neumann algebra generated by {X_1, hX_2(1-h)} is a factor? It is ...
5
votes
2answers
509 views

Presenting the Hyperfinite II_1 Factor

It's well known that all hyperfinite $\mathrm{II}_1$ factors are isomorphic. I risk the wrath of MathOverflow elders to ask if a particular isomorph is easier than others to handle. In particular, is ...
5
votes
1answer
701 views

Subspaces of a Subfactor

Is the following true? Let $\mathcal N \subset \mathcal M$ be a subfactor. There is a bijective correspondence between the ultraweakly closed subspaces of $\mathcal M$ that are bimodules over ...
9
votes
4answers
938 views

What kind of completion is this?

Let $X$ be a compact Hausdorff space, and $C(X)$ the unital commutative C*-algebra of continuous functions on it. The double Banach dual $C(X)^{**}$ is a commutative von Neumann algebra and hence has ...
5
votes
1answer
386 views

Is there a trivial construction of the trace on the Jones basic construction?

Let $N$ be a type $II_{1}$-factor with trace $\tau$, and $B$ a von Neumann subalgebra. The existence of the semifinite trace on the Jones basic construction $\langle N, e_{B} \rangle$ is reasonably ...
9
votes
1answer
770 views

Connes's unpublished manuscript on correspondences, anyone?

There exist unpublished notes on correspondences of von Neumann algebras due to Connes. This is often cited, but I've never seen a copy. It would be nice to have this, say, to maybe look further into ...
2
votes
0answers
330 views

Connes fusion and the composition of completely positive maps

Let $N$ be a type $II_{1}$-factor with trace $\tau$. An $N-N$ correspondence is a Hilbert $N$-bimodule $H$ where the left and right actions are both ultraweakly continuous. Equivalently, a ...
7
votes
0answers
279 views

Canonical Time Evolution for Type $II_{1}$-Factors?

This question was spurred by the answer of Steve Huntsman to the MO question here. The Tomita-Takesaki modular automorphism group gives rise to a canonical time evolution on a type $III$ factor ...
0
votes
1answer
199 views

How coarse is the coarse correspondence?

Let $M$ denote a finite von Neumann algebra with trace $\tau$, and $L^{2}(M)$ denote the standard (trivial) M-M correspondence (binormal bimodule). The coarse correspondence is $L^{2}(M) ...
11
votes
4answers
723 views

What is the right definition of “real von Neumann algebra”?

Recall that a real C*-algebra is a Banach $\ast$-algebra $A$ over $\mathbb{R}$ which satisfies the standard C* identity and which also has the property that $1 + a^{\ast}a$ is invertible in the ...
2
votes
0answers
220 views

If the flip automorphism of a finite factor can be connected to the identity is it approximately inner?

This intriguing question is due to Sorin Popa, so I'm marking it community wiki. I'd like to know the status of the question, and if it is still open, perhaps gather some new ideas for attacking it. ...