# Tagged Questions

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, ...

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### Semi-finite trace on a von Neumann algebra: Equivalent definitions

Let $(N,\tau)$ be a semi-finite von Neumann algebra. This means that $\tau$ is a normal, faithful and semi-finite trace. Normality means that $\tau(x) = \sup_i \tau(x_i)$ if $x \in N_+$ is the limit ...
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### If $\|p-q\|<{1\over2}$ then $p$ is homotopy equivalent to $q$

Let $A$ be a $C^*$ algebra, $p,q \in A$ projections, such that $\|p-q\|< {1 \over 2}$. Show that $p$ homotopy equivalent to $q$. Proof. Let $a_t=(1-t)p+tq$, then $a_t$ is positive (self-adjoint ...
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### $A\in B(H)$ a unital abelian $C^*$-algebra with cyclic vector then $A'$ is abelian as well

Let $A$ be a unital abelian $C^*$-subalgebra of $B(H)$ (with the same unit as that of $B(H)$), and assume there exists a vector $\xi \in H$ which is cyclic for $A$ (that is, $\{a\xi | a\in A \}$ ...
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### non abelian von Neumann algebras

I'm not familiar with von Neumann algebras, but I need the following fact (if it's true) for an other proof. Let $H$ be a Hilbert space, $A\subseteq L(H)$ a non abelian von Neumann algebra. Must $A$ ...
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### Minimal projections and Type II von Neumann Algebras.

Let $M \subseteq B(H)$ be a type $II_1$ factor. Can it contain a minimal projection? If it can't, what would go wrong? I assume something about the trace being faithful?
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### Unique trace on a type $II_1$ von Neumann Algebra

Let $M \subseteq B(H)$ be a type $II_1$ von Neumann Algebra. Then any two non-zero ultraweakly continuous normalised traces $Tr,tr : \rightarrow \mathbb{C}$ are equal. I'm trying to understand this ...
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### Abstract Von Neumann Algebras

I have just read this question Is a von Neumann algebra just a C*-algebra which is generated by its projections? and am wondering about Robert Israel's answer when he says that a subalgebra of $C(X)$ ...
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### if $A_n$ weakly converges to $A$, does $|A_n| \rightarrow _{wo} |A|$?

Suppose that $A_n,A$ are self-adjoint operators in $B(H)$. If $A_n$ weakly converges to $A$, does $|A_n| \rightarrow _{wo} |A|$? From Proposition. 2.3.2 of Pederson'book, I know the result holds in ...
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### predual of von Neumann algebra

I know that each separable finite von Neumann has separable predual. Can some one give me an example of a non-separable finite von Neumann algebra with separable predual?
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### Strong convergence of Spectral Projection

Let $H$ be a Hilbert space and $B(H)$ be the space of all bounded linear operators on $H$. Assume that $\{A_n\in B(H)\}_n$ strongly converges to $A$. $E^{|A|}(1,\infty)$ is a spectral projection of ...
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### separable weakly dense subalgebra of a von Neumann algebra

This is an exercise of "A short course on spectral theory" by William Arveson. For any von Neumann algebra $\mathcal{M}$ on a Hilbert space $H$, I need to show there exists a unital $C^\ast$-...
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### What is the motivation for Murray-von Neumann equivalence

Definition: If $p,q$ are projections in a $C^*$-algebra $A$, we say that they are Murray-von Neumann equivalent, and we write $p\sim q$, if there exist $u\in A$ such that $p=u^*u$ and $q=uu^*$. I ...
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### Representations of C*/W*-algebras and projections

I have come across the two following relations between subrepresentations and projections in two slightly different situations and want to clarify the differences: Let $(\mathcal{H},\pi)$ be a ...
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### Two approaches on the support of functionals

I am writing concerning support of functionals on C*-algebras. I feel there are two different definitions for this notion. Let $A$ be a C*-algebra and $\phi$ be a positive linear functional on $A$. ...
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### A certain *-isomorphism

let $A$ be a C*-algebra and $z\in A^{**}$ the supremum of all the minimal projections in $A^{**}$. How can show $* -$ homomorphism $A\to zA\subset zA^{**}$ is injective?
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Let $A$ be a C*-algebra and $\phi$ be a positive linear functional on $A$. We let $\tilde{\phi}$ be its unique $w^*$-continuous extension on $A^{**}$. It is supposed to focus on the left kernel of $\... 1answer 19 views ### Equivalent non-degenerate representations of C*-algebras For two non-degenerate representations$\pi_j:A\to B(H_{\pi_j})$($j=1,2$), we write$\pi_1\sim\pi_2$if there exists a$w^*$-continuous isometrically isomorphism from$\pi_1(A)''$onto$\pi_2(A)''$... 1answer 50 views ### The point-wise closure of the space of continuous functions Let$X$be a locally compact space and consider$C_0(X)$. We denote$b(X)$, by the set of all bounded functions on$X$. It is easy to be checked that$b(X)$may be considered as a C*-sub algebra of$...
Let $A$ be a C*-algebra. Let $\phi$ be a positive linear functional on $A$. The support of $\phi$ is defined by the smallest projection $q\in A^{**}$ with $\phi(q)=||\phi||$ and in the literature ...