# Tagged Questions

The subject of operator algebras is primarily C*-algebras and von Neumann algebras, and associated topics. It also includes more general algebras of operators on Hilbert space, and may include algebras of operators on other topological vector spaces. It is related to but distinct from the subjects ...

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### Group von Neumann algebras

I have a question about group von Neumann algebras structure. If $L(G)$ is a subset of $L(H)$, can we find a subgroup $G_1$ of $H$ such that $L(G_1)$ is isomorphic to $L(G)$? I appreciate any help.
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### Continuous family of subalgebras in a C* algebra

Let A be a separable C* algebra. Fix $n\in\mathbb N$. For t $\in$ $\mathbb{R}$ let $A_t$ be a subalgebra of $A$ such that: $A_t \cong \mathcal{O}_n$ (Cuntz algebra). Generators of $A_t$ depend ...
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### noncommutative algebra

If we let $B$ be a noncommutative Banach algebra with unit $e$, then, obviously, $xy\not=yx$, but are they related? I suspect that the spectral radii of $xy$ and $yx$ are the same, but I couldn't ...
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### Does tensor product with $L_p$ operator algebra preserve exact sequences?

By $L_p$ operator algebra I mean a closed subalgebra of the algebra of bounded linear operators on some $L_p$ space where $p\in(1,\infty)$. There is a notion of tensor product of $L_p$ spaces (as ...
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### Tensor product of algebras of operators on $SQ_p$ spaces

I am aware that there is a canonical tensor product for $L_p$ spaces that allows one to talk about tensor products of algebras of bounded linear operators on $L_p$ spaces. Is there an analogous notion ...
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### Uncountable sequence $R_1 \subset R_2 \subset …$ of von Neumann algebras acting on separable $H$?

Can there exist an uncountable sequence $R_1 \subset R_2 ...$ of von Neumann algebras all acting on the same separable Hilbert space $H$, with a "limit" algebra $R$ such that $R_\alpha \subset R$ for ...
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### All nonzero singular values of $A$ are equal to $1$ iff $A^*=A^*AA^*$ and $A=AA^*A$

I want to show that all the non-zero s-numbers, i.e. singular values $s_j(A):=(\lambda_j(A^*A))^{1/2}$, of A (a bounded linear operator of finite rank acting on a separable Hilbert space $H$) are ...
Let $A$ be a bounded linear (compact) operator acting on a separable Hilbert space $H$, and let $\varphi_1,\varphi_2,\ldots$ be an orthonormal basis of $H$. I Assume that $|\left< A\varphi_j,\... 1answer 30 views ### Equivalence of some characterization of pure states I'm looking for a reference or a proof for these well-known facts in$C^*$-algebras theory for which, however, I havent found any clearly written proof of the same type of ones I will sketch. ... 1answer 33 views ### A certain limit of a state is zero Let A be a C$^*$-algebra,$a\in A$strictly positive (this means: for every state$\varphi$of A is$\varphi(a)>0$). Let$u_n=(\frac{1}{n}+a)^{-1}$. Then for all$b\in A$and all states$\varphi$... 1answer 17 views ### Semi-finite trace on a von Neumann algebra: Equivalent definitions Let$(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_+$is the limit ... 0answers 25 views ### modular operator I could calculate a modular operator$Δ$for a state$τ_α(x_{ij})= αx_{11}+(1-α)x_{22}(0<α<1)$. But, I cannot understand an automorphism$σ_t(x)＝Δ^{it}(x)Δ^{-it}$of$M_2(\mathbb{C})$. For ... 1answer 41 views ### Compute the positive part of$K_0(A)$where$A$is a simple AF algebra I'm trying to understand the following example from my lecture notes: Define$A_n=M_{F_n}(\Bbb{C})\oplus M_{F_{n+1}}(\Bbb{C})$where$F_n$defined by$F_1=1, \ F_2=2, \ F_{n+2}=F_{n}+F_{n+1}$, i.e., ... 1answer 24 views ### Countable weighted shift has no invariant subspace. Suppose I have$T(e_n)=w_ne_{n+1}$where$w_n>0$(and are bounded) and$\{e_n\}$denotes the canonical basis of$l^{2}(\mathbb{N})$. I would like to prove that the only (closed) invariant ... 1answer 69 views ### If$\|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$is positive (self-adjoint ... 3answers 72 views ### Nontrivial closed ideal of$\mathbb{B(H)}$,$\mathbb{H}$is a non-separable Hilbert space.$\mathbb{H}$is a non-separable Hilbert space. Give an example of nontrivial closed ideal$I$of$\mathbb{B(H)}$, that is different from$\mathbb{B_0(H)}$which is the ideal of compact operators. Any ... 1answer 38 views ### Generalized polar decomposition Let$x\in B(H)$. We say$(x,v,y)$is a polar decomposition for$x$if,$\bullety$is positive.$\bulletv$is a partial isometry with$x=vy$.$\bullet$Ker$(x)$=Ker$(y)$=Ker($v$) The polar ... 0answers 34 views ### Nonunital C*-Algebra: Proper Ideals Given a C*-algebra without unit. Does there exist a nontrivial proper ideal that does not lie in a maximally nontrivial proper ideal? (For the unital case this follows easily by Zorn's lemma.) ... 0answers 10 views ### Bott projection as$K_1$class Consider the Bott projection (described in Exercise 5.I of Wegge-Olsen's book$K$-theory and$C^*$-algebras) given by$b(z)=\frac{1}{1+|z|^2}\begin{pmatrix} 1 & \bar{z} \\ z & |z|^2 \end{...
Let $A_0$ be a C*-subalgebra in a C*-algebra $A$. Let $\phi_0$ be a bounded linear functional on $A_0$ and assume $\phi$ is an extension of $\phi_0$ on $A$. I mean $\phi\in A^*$ with \$\phi_{|_{A_0}}=\...