# Questions tagged [von-neumann-algebras]

A von Neumann algebra is a unital *-subalgebra of the algebra of bounded operators on a Hilbert space, closed in the weak operator topology. Also called a $W^*$-algebra, and may be regarded as a non-commutative generalization of $L^\infty$ space. These algebras are extensively used in knot theory, non-commutative geometry, and quantum field theory.

589 questions
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### Free probability and projections

Let $(\mathcal{A},\varphi)$ be a free probability space, where $\mathcal{A}$ is a von Neumann algebra and $\varphi$ a finite and faithful trace. Let furthermore $p\in\mathcal{A}$ be a projection. ...
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### Inverse of compression (von Neumann algebra)

I am stuck with this seemingly easy problem but I am having trouble showing this: Let $\mathcal{A}\subseteq\mathcal{B}(\mathcal{H})$ be a von Neumann algebra realized inside a subalgebra of the ...
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### dimension of a von Neumann algebra

Is there any dimension on a von Neumann algebra? Is there any relationship between finite von Neumann algebras and finite dimensional von Neumann algebras?
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### Example of norm separable c-star algebras [closed]

I want to know enough examples of norm separable $C^{*}$-algebras which are neither finite dimensional nor commutative.
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### examples of non-unital commutative $C^*$-algebras

I know that all the non-unital commutative $C^*$ algebras are isomorphic to $C_0(\Omega)$,where $\Omega$ is a locally compact space. Can anyone show me some common non-unital commutative examples.I ...
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### non isomorphic finite dimensional $C*$ algebras

How many non isomorphic finite dimensional $C^*$ algebras if the dimensions without a bound? Is it countable or uncountable?
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### spectrum of $C^*$ algebras

When $A=\bigoplus B(\Bbb C^n)$ ($c_0$ direct sum),how to compute the spectrum of $A$ ?What about the conclusion If we replace the $\ell ^\infty$ direct sum with $c_0$ direct sum?
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### arbitrary $n$-dimensional matrix algebras in II$_1$ factor

I'm struggling to show that in a type II$_1$ and any $n$ $\exists$ a subfactor $M$ such that $M \cong M_n$ . I suppose it should follow from the isomorphism between the equivalence classes of ...
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### approximate identity element

Let $I\subset \prod_n B(H_{m_n})$ be a separable $C^*$ algebra,where $\prod B(H_{m_n})$ denotes the $\ell ^{\infty}$direct sum of $B(H_n)$ and $dim(H_{m_n})<\infty$. We suppose that there exits a ...
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### equivalent projections in finite factors are unitarily equivalent

Why are two Murray von Neuman equivalent projections $p$ and $q$ in a finite factor unitarily equivalent?
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### image of some ideal under the quotient map

! I am still confused about the range of $\sigma(I_{\omega})$.Since $\|x_n\|_2\to 0,$we have $\|x_n\|\to 0$,$\sigma(I_{\omega})=0,$then $J=0$,it is trivial. Is my understanding correct?
### Equality of tracial states on dense $C^*$-subalgebra implies equality on generated von Neumann algebra?
Maybe this is a simple question, but I'm not sure about the following: Let $\cal M$, $\cal N$ be von Neumann algebras and $X\subseteq \cal M$ a weakly dense (possibly separable) $C^*$-subalgebra. Let ...