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Questions tagged [von-neumann-algebras]

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.

5
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2answers
169 views

Extension of a von Neumann algebra by a von Neumann algebra

I asked this question at MSE now I repeat it at MO: Let $A,B,C$ be $3$ unital $C^*$ algebras. Assume that we have the following short exact sequence of $C^*$-algebras: $$0\to A\to C\to B\...
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0answers
97 views

Topology of normal states of a von Neumann algebra

Is the topological space defined on the set of normal states of a von Neumann algebra equipped with a weak*-topology, first countable?
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0answers
71 views

Are these kinds of “crossed product” studied?

Let $M$ be a von Neumann algebra acting in a Hilbert space $H$, and let $\rho$ be a representation of a group $G$ on a Hilbert space $K$. Define $M\rtimes_\rho G$ to be a von Neumann algebra acting in ...
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0answers
57 views

About crossed product of the group von Neumann algebra

Let $G$ be a group with a normal subgroup $N$. If $K$ is a field, then it is well known in ring theory that $K[G]\cong K[N]*(G/N)$ ($*$ stands for the crossed product). Do we have such an isomorphism ...
3
votes
1answer
196 views

Fundamental group and group measure space construction

Let $N$ be a type ${\rm II}$ factor, with trace $\tau$. Consider its fundamental group$$ \mathcal{F}(N)= \{ \tau(p)/\tau(q) \ | \ p,q \text{ non-zero finite projections in } N \text{ and } pNp \simeq ...
6
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0answers
125 views

Group $C^*$ vs group von-Neumann algebras

Let $\Gamma$ be a countable (discrete) group (in what follows, make additional assumptions as you wish). Let $C^*_r(\Gamma)$ and $W^*_r(\Gamma)$ be the reduced $C^*$-algebra respectively the reduced ...
5
votes
1answer
207 views

Cartan subalgebra and group measure space construction

Let $N$ be a ${\rm II}_1$ factor. A maximal abelian self-adjoint subalgebra (MASA) is a $*$-subalgebra $A \subset N$ such that $A' \cap N = A$. It is called a Cartan subalgebra if moreover $\mathcal{N}...
0
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1answer
59 views

Examples of isomorphic W* algebra with non-homeomorphic weak topology

Due to the uniqueness of the predual, a W* algebra, when realized as a von Neumann algebra in any way, always has a unique, well-defined ultraweak (or $\sigma$-weak) topology. The same can be said ...
3
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2answers
152 views

Commutative direct summands of C*-algebras

I have a question about commutative direct summands of $C$*-algebras. Let $A$ be a $C$*-algebra (with unit) and suppose that its bidual $A^{**}$ has a commutative direct summand, that is, $A^{**}=B\...
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1answer
136 views

When is $\inf_{n\geq0}x^n\neq0$?

Let $M$ be a von Neumann algebra acting on a Hilbert space $H$. Let $x$ be a positive element of $M$ with $\|x\|=1$. So, $(x^n)_{n\in\mathbb N}$ is a decreasing sequence of positive elements and $y:=\...
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0answers
97 views

Unitary element of the group algebra

Let $G$ be a torsion free group. Are unitary elements of $\mathbb CG$ studied? By unitary element I mean an element $\alpha$ in $\mathbb CG$, such that $\alpha^*\alpha=1$? Do the triviality of unitary ...
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0answers
84 views

Does this element belong to $\mathbb CG$?

Let $G$ be a torsion-free group. Let $\alpha$ be a symmetric element of $\mathbb CG$, i.e. $\alpha^*=\alpha$, with $\|\alpha\|_1=\sum|\alpha(g)|<1$, so $\beta:=\sum_{n\ge 0}(-1)^n\alpha^n$ is an ...
24
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3answers
647 views

What does it mean for a category to admit direct integrals?

Given an infinite countable group $G$, the category of unitary representations of $G$ admits direct integrals. Namely, given a measure space $(X,\mu)$ and a measurable family of unitary $G$-reps $(H_x)...
3
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0answers
61 views

To what extend can a von Neumann algebra be determined by its projection lattice structure?

Let $ M, N $ be von Neumann algebras, $ P $ (resp. $Q$) the projection lattice of $M$ (resp. $N$). Any isomorphism $ \varphi : M \to N $ on the level of involutive algebras induces an isomorphism $ \...
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0answers
49 views

why can index be larger either the trace goes smaller or larger in index $\ge 4$ case?

I am studying the index of subfactor and basic construction recently. Suppose $M=\mathcal{R}$ to be the infinite hyperfinite II1 factor. For projection $p\in \mathcal{R}$, $p\mathcal{R}p$ is also a ...
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1answer
289 views

Topology of state space in von Neumann algebras

What are the sufficient conditions for a von Neumann algebra to have a first countable set of states with respect to the weak * operator topology?
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61 views

Support size of a zero divisor

Let $G$ and $\mathbb C[G]$ be a torsion free group and its group algebra. Is there a function $f:\mathbb N\rightarrow\mathbb R$, with $\lim_nf(n)=\infty$ such that if $0\neq\alpha,\beta\in\mathbb C[G]$...
2
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0answers
103 views

An example of non trivial projections in a group von Neumann algebra

Let $G$ and $\text{vN}(G)$ be a torsion free group and its group von Neumann algebra. Is there a characterization of non trivial projections in $\text{vN}(G)$? If not, is a certain class of them ...
3
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1answer
143 views

An analytical zero divisor

Let $G$, $\mathbb C[G]$ and $\text{vN}(G)$ be a torsion free group, it's group ring and group von Neumann algebra, resp.. Let $0\neq\alpha\in\mathbb C[G]$ and $0\neq p\neq1$ is a projection in the ...
2
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1answer
94 views

Direct proof a property of hyperstonean spaces

First, let me state some basic facts and definitions for my question. I believe these are well-known among experts working on von Neumann algebras, but let me state them anyway since my question is ...
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0answers
77 views

Conditional Expectation for von Neumann algebra

Let $M$ be a von Neumann algebra and $T: M\rightarrow \mathbb{C}$ be a finite normal faithful tracial map, s.t., if $\phi : M \rightarrow A \cap A^{*}$ is a conditional expectation, (A being a weak ...
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0answers
90 views

Is there an adaptation of the theory of standard forms and Tomita-Takesaki theory to the $\mathbb{Z}_{2}$-graded case?

Let $A$ be a von Neumann algebra acting on a Hilbert space $H$, and suppose that $\Omega \in H$ is a cyclic and separating vector for $A$. Then in Tomita-Takesaki theory one defines an unbounded ...
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0answers
75 views

An example of a sequence of finite projections

Let $A$ be a vn-algebra. Suppose that $x$ is an isometry with $\inf_{n\geq1} x^nx^{*n}=0$ (Note that $x^nx^{*n}$ are all projections). Let $e$ be a (non-zero) finite projection and put $q_n$ to be ...
3
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0answers
81 views

An example of a particular vn-algebra

Let $A$ be a vn-algebra. Let us suppose $e$ is a finite projection in $A$ and $x$ is an isometry (meaning $x^*x=1$) in $A$ such that $e$ does not commute with $x$. Then $\{q_n=x^nex^{*n}\}$ forms a ...
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1answer
55 views

The range projection of product of projections

Let $A$ be a von Neumann algebra. Let $p$ be a projection in $A$. Suppose that $e$ is a finite projection. Can we determine all types of vn-algebras in which $p-p\wedge(1-e)$ is a finite projection?...
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0answers
46 views

On direct integral of states of von Neumann algebras

Suppose we consider a direct integral of GNS states of a measure space in von Neumann algebra, get the new state by direct integral. Does the GNS represenation of the state breaks down into direct ...
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1answer
43 views

Subprojections of the sum of mutually orthogonal Abelian projections

Let $f_i$, $i=1,\dotsc,n$, be mutually orthogonal Abelian projections in a von Numann algebra, and let $e\leq\sum f_i$. Is it true that there exist mutually orthogonal Abelian projections $e_j$, $j=1,\...
3
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0answers
53 views

Characterizing fullness of a von Neumann algebra by the topology of its bimodules

Let $\mathcal{M}$ be a $\mathrm{II}_1$ factor. Among other characterizations, it is said to be full iff the adjoint map: $$ \mathrm{Ad}: U(\mathcal{M})/\mathbb{T} \longrightarrow \mathrm{Aut}(\...
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1answer
86 views

Does Borel functional calculus commute with *-isomorphism?

I am confused with the underlined equation in the following picture. I know that a *-isomorphism commutes with continuous functional calculus since every continuous functions on the compact subset of ...
6
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1answer
153 views

Trivializing unitary cocycles in abelian von Neumann algebras that are uniformly close to the trivial one

Suppose $M$ is an abelian von Neumann algebra, carrying a (point-ultraweakly) continuous action $G\curvearrowright M$ of a locally compact, second-countable group. Let $y: G\to\cal U(M)$ be an ...
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0answers
106 views

The Hahn-Hellinger Theorem [closed]

People can tell the question is not up to the mark, or research level question, but I felt without understanding the Hahn-Hellinger Theorem properly there is no point talking von Neumann algebras for ...
0
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1answer
80 views

Isomorphism of preduals implies isomorphism of the $W^*$-algebras or not?

Let $M$ and $N$ are two von Neumann algebras such that their preduals $M_{∗}$ and $N_{∗}$ are isomorphic in the sense of Banach spaces, does it imply M and N are $∗$-isomorphic or not??
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1answer
59 views

Clarification on predual on existence of separating vector

We know predual of a von Neumann algebra $M$ as a Banach space is independent of Hilbert space where the $M$ is represented. Now the question is if we represent $M$ in $B(\mathcal{H})$, where $M$ has ...
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0answers
66 views

Transmission of finite projections

Let $A$ be a Baer*-ring. Let us denote $L(x)$ by the left projection of $x$ (the smallest projection with $L(x)x=x$). Let $p$ be a finite projection in $A$. Is $L(xp)$ a finite projection for every $...
3
votes
1answer
118 views

How rich the group of unitary elements in a von Neumann algebra to get “Murray-von Neumann” equivalence?

Denote by $\sim$ the "Murray-von Neumann" equivalence in the projection lattice of a von Neumann algebra. It is well known that in a finite von Neumann algebra the equivalence of projections can be ...
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1answer
151 views

About separability of von Neumann algebras [closed]

Is a von Neumann algebra always separable in the $\sigma$-weak topology? If not, give a counterexample. Under what conditions will it be separable?
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72 views

References for hyperfinite factors

Can I have references of hyperfine $II_1$ factors where I can get structural properties to be studied and more characterizations.
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200 views

Do these limits exist?

Let $G$ be a (discrete) torsion free group with identity $e$. Recall that for an element $\alpha\in\mathbb C[G]$, the set of complex functions on $G$ with finite support, $\alpha^*\in\mathbb C[G]$ is ...
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2answers
147 views

On topology in von Neumann algebras

Suppose $M$ is von Neumann algebra, $A$ is $*$-algebra in $M$, further if $(A)_1$, the unit ball of $A$ is strong operator closed, does it implies $A$ is von Neumann algebra? I started proving this ...
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0answers
117 views

Schröder–Bernstein for representations of operator algebras

This is claimed in a Wikipedia Article: If two representations (of a $C^*$-algebra $A$) $\rho$ and $\sigma$, on Hilbert spaces $H$ and $G$ respectively, are each unitarily equivalent to a ...
3
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1answer
170 views

About some positive elements in a group von Neumann algebra

Let $G$ be a (discrete) torsion free group with identity $e$. Recall that for an element $\alpha=\sum a_gg$ in $\mathbb C[G]$ (complex functions on $G$ with compact support), $\alpha^*$ is defined to ...
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1answer
106 views

On predual of von Neumann algebra

Suppose $T_{n}$ converging $T$ in vN algebra $M$ in weak operator topology, can we conclude $||T_{n}||$ is uniformly bounded? Another question if a linear functional $\varphi$ is continuous in unit ...
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0answers
101 views

Properly outer automorphisms on type II$_1$ von Neumann algebras

Let $M$ be a von Neumann algebra with separable predual. Let us assume that $M$ is of type II$_1$, meaning that it is finite but has no type I part. Let $\tau$ be a faithful normal tracial state on $M$...
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0answers
110 views

Obstruction to the existence of a complex-valued determinant function

Let $\mathcal A$ be a Type $II_1$ von Neumann algebra, equipped with a finite trace $\tau: \mathcal A \to \mathbb C$. Further, denote by $\mathcal A^\times \subset \mathcal A$ the group of invertible ...
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0answers
77 views

“Adding” a projection to a von Neumann algebra

This is a question about what happens when you "add" a new projection $p$ to a von Neumann algebra $\mathcal{R}$ to generate a larger v.N. algebra $(\mathcal{R} \cup \{p\})''$. Suppose that $\mathcal{...
3
votes
2answers
353 views

Is the ideal property of $X^{**}$ inheritable to $X$?

Let $X$ be an operator space such that there is a weak$^*$-continuous complete isometry $\phi$ from its second dual $X^{**}$ into a $W^*$-algebra $M$ in which $\phi(X^{**})$ is a (necessarily weak$^*$-...
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0answers
111 views

Type III factor examples?

How to prove the crossed product of $G$ and von Neumann algebra $M$, where $G$ is locally compact group acting on $M$ via free ergodic action and $M$ is type $II_{\infty}$ factor, is type $III$ factor,...
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1answer
129 views

On projection theory for inseparable Hilbert spaces

How can one see that $I$ is an infinite projection in $B(\mathcal{H})$, where $\mathcal{H}$ is an inseparable Hilbert space?
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0answers
77 views

What is spectral multiplicity for multiplication operators in general von Neumann algebra set up?

When two multiplication operators $M_{f}$ and $M_{g}$ acting on $L^2(X,\mu) $and $L^2(Y,\nu)$ are unitary equivalent? How multiplicity function look like here? What is the spectral multiplicity in ...
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0answers
63 views

Projections in properly infinite factor

Denote by $\sim$ the "Murray-von Neumann" equivalence in the projection lattice of a von Numann algebra. Let $e$ be a projection in a properly infinite factor. Is it always true that $e\sim 1$ or $1-e\...