As of May 31, 2023, we have updated our Code of Conduct.

# Questions tagged [oa.operator-algebras]

Algebras of operators on Hilbert space, $C^*-$algebras, von Neumann algebras, non-commutative geometry

2,002 questions
Filter by
Sorted by
Tagged with
36 views

### Value of Plancherel weight on convolutions (Takesaki VII Section 3 Theorem 3.4)

In this post, I follow conventions and notations from Takesaki's second volume "Theory of operator algebras" (chapter VII sections 2 and 3). Let $G$ be a locally compact group with left Haar ...
191 views

### Is the category of $Z^*$ algebra equivalent to the category of $C^*$ algebras

A $Z^*$ algebra is a $C^*$ algebra whose all positive elements are zero divisor. They form a category with usual structures. Question. Is this category equivalent to the category of $C^*$ algebras? ...
1 vote
96 views

### What is the status of The Halmos Similarity Problem?

What is the general status of "The Halmos Similarity Problem"(HSP) in Operator theory?For What conditions ,HSP has been solved?
101 views

### Tensor product of operator values weights (in the theory of locally compact quantum groups)

Let $M$ be a von Neumann algebra and $\psi$ a normal faithful semifinite weight on $M$. Then one should be able to form the object $$\iota \otimes \psi: (M\overline{\otimes} M)_+ \to \widehat{M_+}.$$ ...
83 views

### Given $\sigma(AB-BA) = \{0\}$, what can be said about $\sigma(A)$ and $\sigma(B)$?

Let $\mathcal H$ be a separable Hilbert space, and $\mathfrak B(\mathcal H)$ denote the algebra of bounded linear operators on $\mathcal H$. Furthermore, let $A,B \in \mathfrak B(\mathcal H)$ be two ...
94 views

### Extreme points in the space of ucp maps

Suppose $M$ and $N$ are $\mathrm{II}_1$ factors. Let $\tau\mathrm{UCP}(M,N)$ be the convex space of trace-preserving UCP maps from $M$ to $N$, equipped with the topology of pointwise weak* convergence....
1 vote
247 views

### A subalgebra of $B(H)$ which does not contain a commutator element

Is there a commutative subalgebra $A\subset B(H)$ containing the 1 dimensional scalars with the following property: The algebra $A$ has trivial intersection with the set of commutator ...
1 vote
78 views

### An example of a non rigid Banach algebra

A Banach algebra $A$ is called a rigid Banach algebra if for every injective Banach algebra morphism $J:A\to A$ we have either $\overline{J(A)}$ is ismorphic to $A$ or it does not contain ...
1 vote
120 views

### Intersection of von-Neumann algebra factors

Given two von-Neumann algebra factors $\mathcal M,\mathcal N$, is $\mathcal M\cap\mathcal N$ a factor? And how about the intersection of infinitely many factors? Notes: I know that the intersection ...
169 views

106 views

211 views

### Why are projectionless $C^*$-algebras important (Kadison's conjecture)

It was considered an important result for a long time to show that the reduced $C^*$-algebra of the free group $C^*_r(F_2)$ has no nontrivial projections. I believe this is also known as Kadison's ...
507 views

### Integration in Banach algebra

Let $\mu$ be a Borel measure on the real line $\mathbb{R}$ taking values in a separable Banach algebra $A$. Assume that $\mu$ is such that the total variation measure $|\mu|$ is finite. Let $f$ be a ...
309 views

### Takesaki II Lemma 1.13: stuck in proof

Consider the following fragment from Takesaki's book "Theory of operator algebras II" (Lemma 1.13 on p8, in chapter VI "Left Hilbert algebras"): Here, we associate with an ...
227 views

A classic result, of Murray and Von Neumann I believe, is that if $\mathcal M\subseteq B(H)$ is a factor then the $*$-homomorphism $\pi : \mathcal M \odot \mathcal M' \rightarrow B(H)$ given by $\pi(... 3 votes 0 answers 92 views ### subfactors and trace For every factor$M$there is an (essentially unique) dimension function (or trace)$\tau_M : M_p \to [0,\infty]$(where$M_p$is the lattice of orthogonal projections in$M$). My question is: Under ... 1 vote 2 answers 152 views ### Limit of a countable separable is countable separable? Let$\rho$be a positive trace class operator on$H\otimes H$, where$H$is a separable Hilbert space (not necessarily finite dimensional). We say that$\rho$is countable separable if$\rho=\sum_{i=...
We refer to [JS97] for the notion of subfactor. Yasuo Watatani proved the following result [W96, Theorem 2.2]: Theorem: An irreducible finite index subfactor of a type $\mathrm{II}_1$ factor has ...