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

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not commutative symmetric vs strong symmetric spaces

Let M be a von Neumann algebra with semi-finite normal faithful trace $\tau$, $S(M)$ is space of all measurable operators introduced by I.Segal. For the self-adjoint measurable operator $X\eta M$ ...
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509 views

Why did Voiculescu develop free probability?

I was recently asked why Voiculescu developed free probability theory. I am not very expert in this and the only answer I was able to provide is the classical one: he was challenging the isomorphism ...
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239 views

Is a C* completion of a nuclear Fréchet algebra a nuclear C* algebra?

I am sure that this is well known in the right places, but: Is the C* completion of a star nuclear Fréchet algebra a nuclear C* algebra? (Suppose that the C* norm is continuous with respect to the ...
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201 views

How to determine there exists a unique invariant subspace for a set of matrices

Hi everyone, Ive been looking at the following problem, but its not entirely in my area and some potential solutions seem to rely on algebraic geometry. Maybe thats just a complicated way to solve ...
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98 views

Reduced C*-algebras of locally compact etale Hausdorff groupoids

Let $G$ be an étale locally compact Hausdorff groupoid (possibly second-countable) and let $a\in C_{\textrm{red}}^*(G)$. Is it true that for all $\varepsilon>0$ there is $s\in C_c(G)$ satisfying ...
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when is an algebra map conjugate to a star algebra map

Take a (unital) algebra map $f:A\to B$ between two unital C* algebras - not necessarily star preserving. Under what circumstances is there a $b\in B$ so that $g(a)=b\ f(a)\ b^{-1}$ is a star algebra ...
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104 views

Is there a maximal finite depth infinite index irreducible subfactor?

A subfactor $N \subset M $ is irreducible if $N' \cap M = \mathbb{C} $. It's maximal if it admits no non-trivial intermediate subfactors $N \subset P \subset M $. It's cyclic if its lattice of ...
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199 views

Double ultrapower of the hyperfinite $II_1$-factor

Let $\omega$ be a free ultrafilter on the natural numbers and $R$ be the hyperfinite $II_1$-factor (the definition of $R$ is recalled in the comments). Question: Does there exist another free ...
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227 views

Tensoring with a CAR-algebra

Let $A$ and $B$ be two unital infinite-dimensional simple separable nuclear $C^{\ast}$-algebras and let $C$ be a CAR-algebra. When does $A\otimes C \simeq B\otimes C$, imply $A\simeq B$? The answer ...
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118 views

About the classification of infinite depth irreducible finite index maximal subfactors

The Temperley Lieb subfactors $A_{\infty}$ are the first examples of infinite depth irreducible finite index maximal subfactors. We can see these subfactors as coming from the simple Lie group ...
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What is about nonassociative geometry?

At the end of a conference given by Alain Connes in 2000 (here is a video in French), a member of the audience asked a question. I transcribed and translated it for you below: Audience: You showed ...
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The cyclic subfactors theory: a quantum arithmetic?

Context: First recall some results: - Actions of finite groups on the hyperfinite type $II_{1}$ factor $R$ (Jones 1980). - A Galois correspondence for depth 2 irreducible subfactors ...
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Projections in a W*-algebra as a continuous lattice?

A continuous lattice is a complete lattice $L$ in which every element $y$ is equal to $\bigvee${$x \in L \mid x \ll y$} where $x \ll y$ ("x approximates y" or "x is way below y") if for any directed ...
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102 views

All AI-algebras are AT-algebras

It is known that every AI-algebra (i.e. inductive limit of interval algebras) is an AT-algebra (i.e. inductive limit of circle algebras)? This seems a little bit odd because a building block of an ...
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104 views

Is there an infinite depth irreducible finite index maximal subfactor (other than Temperley Lieb) ?

A subfactor $N \subset M$ is maximal if it admits no non-trivial intermediate subfactors $N \subset P \subset M$. Is there an infinite depth irreducible finite index maximal subfactor (other than ...
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235 views

Are there only finitely many maximal subfactors of a fixed finite index ?

A subfactor $N \subset M $ is maximal if it admits no non-trivial intermediate subfactors $N \subset P \subset M $. Question: are there only finitely many maximal subfactors of a fixed finite ...
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660 views

Non-“weakly group theoretical” integral fusion categories?

Is there an integral fusion category of global dimension $210$, such that the simple objects have dimensions $\{1,5,5,5,6,7,7\}$ and the following fusion matrices? $\small{\begin{smallmatrix} 1 & ...
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186 views

Fredholmness of an operator-valued Toeplitz operator

Let $f$ be an invertible element of $C({\mathbb{T}}; C_b(r,1))$, that is, there exists a $f^{-1}\in C({\mathbb{T}}; C_b(r,1))$ such that for all $z\in {\mathbb{T}}$, $f(z)f^{-1}(z)=1$ in $C_b(r,1)$. ...
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320 views

Defining holomorphic functions in terms of Banach algebras, and similarly for C*-algebras

Let $C$ be the category of commutative Banach algebras and let $U : C \to \text{Set}$ be the usual forgetful functor. The holomorphic functional calculus guarantees that every holomorphic function $f ...
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131 views

Nonlinear Operators(with the group property?)

Let V be a finitely generated vector space with dimension(V) = $n \in \mathbb{N}>1$. Now let T: $ V \to V$ be a map such that $\forall \hat{v},\hat{w} \in V$, $\; T(\hat{v}+\hat{w}) \neq ...
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173 views

Is the extension of full free group c^* algebra a group?

It is know that the extension of reduced free group C*-algebra is not a group.(By Haagerup and Thorbjørnsen). How about the extension of the full group C-*algebra by compact operators?
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218 views

Inner automorphisms and $K$-theory

It is known that any inner automorphism of a unital $C^{\ast}$-algebra $A$ induces the identity map on $K_{0}(A)$ because unitary equivalence implies Murray-von Neumann equivalence. What is known ...
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A question about multiplier algebra of $C_0(G)\otimes C_b(G)$ for a locally compact group $G$

Let $G$ be locally compact group. How we can show that $$ M(C_0(G)\otimes C_b(G))=C_b(G,C_b(G)). $$ ($M(C_0(G)\otimes C_b(G))$ is the multiplier algebra of $C_0(G)\otimes C_b(G)$)
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Unitary with full spectrum

I have a unitary element $u\in C(\mathbb{T},M_{n}(\mathbb{C}))$ such that $Spec(u)=\mathbb{T}$. Does there exist a unitary $v\in C(\mathbb{T},\mathbb{C})$ such that $Spec(uv)\subsetneqq\mathbb{T}$?
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161 views

Spectral decomposition function [closed]

Once I met a notation of "spectral decomposition function" (for a self-adjoint operator). No definition was given. Could someone give me a clue what can that be, cause I can't find this exact phrase ...
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2answers
138 views

Homomorphisms preserving constant functions

Assume we have a homomorphism $\phi: C(S^{1},M_{n}(\mathbb{C}))\rightarrow C(S^{1},M_{m}(\mathbb{C}))$ where $n$ divides $m$. Under what conditions does $\phi$ send constant functions to constant ...
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442 views

Algorithm to find exponential map of differential operators acting on function

I am trying to write a computer program which computes the action of the exponential of a differential operator on a function, for any given differential operator. Examples: $\exp(\varepsilon ...
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1answer
134 views

Inductive limit of mapping tori

I have two mapping tori $A_{n}$={$f\in C([0,1], M_{n}(\mathbb{R}))\mid f(1)=\alpha_{1}(f(0))$}, $B_{n}$={$f\in C([0,1], M_{n}(\mathbb{R}))\mid f(1)=\alpha_{2}(f(0))$} where $\alpha_{1}, \alpha_{2}$ ...
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238 views

Inductive limit of C*-algebras

It is known that an intertwining (or even approximately intertwining) diagram implies the isomorphism of the limit algebras. Under what conditions the converse holds?
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386 views

When does a $W^*$-algebra have a standard Borel spectrum?

EDIT: André Henriques has commented below that the correct separability condition is not weak-* separability as I have written below, but separability of the predual. This post came out a bit long, ...
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Need help determining whether a certain map is a $C^\ast$ homomorphism

Hello, I need help determining whether the map I defined between two algebras is a well-defined homomorphism of $C^\ast$-algebras. I ran into this problem while trying to define a "rotation map" ...
3
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1answer
257 views

Metrics and completions on the direct limit of matrices of all sizes over arbitrary fields

Let $M_n$ denote the $n$ by $n$ matrices. Consider the homomorphisms $$\phi_{n,kn}: M_n \rightarrow M_{kn}$$ which takes a matrix $A \in M_n$ to $A \otimes I_k \in M_{kn}.$ This gives a sensible way ...
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181 views

Weights on Von Neuman factors

Let A be a type $I$ factor on a Hilbert space H. Let $\varphi$ be a semi-finite normal weight on $A^{+}$ is it possible to say that then there exist Hilbert spaces $H_2 \subset H_1$ and an isomorphism ...
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313 views

Tannaka duality for C*-algebras?

Tannaka-Krein duality shows how to recover a group $G$ from its category $\mathbf{Rep}(G)$ of finite-dimensional complex representations and the forgetful functor $F:\mathbf{Rep}(G)\to ...
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433 views

Can nuclearity be determined by tensoring with a single C*-algebra?

A C*-algebra is nuclear if the algebraic tensor product $A\odot B$ ($B$ is any other C*-algebra) admits a unique C*-norm. This definition requires testing the condition for nuclearity with `all' ...
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Are the groups $C( \mathbb{R} ; U(n) )$ isomorphic?

Let $C(\mathbb{R};{U}(n))$ denote the topological group of continuous functions $\mathbb{R}\to {U}(n)$ with pointwise multiplication and compact-open topology. My question is: Are these groups ...
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166 views

From positive definite function to Følner sequence --— a question on amenability and nuclearity

We know that amenability of countable discrete group $\Gamma$ has many equivalent characterizations. In particular, there are two: a) there is a sequence of finitely supported positive definite ...
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130 views

isomophism, commutator

Let X be a Banach space. $B(X)$ is the algebra of all bounded linear operators on X. $\phi: B(X)\rightarrow B(X)$ is a isomoprphisn, $\varphi: B(X)\rightarrow B(X)$ is a isomorphism or negative ...
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341 views

Noncommutative arithmetic mean geometric mean inequality and symmetric polynomials

While analyzing convergence speed of stochastic-gradient methods for convex optimization problems, Recht et al (2011) posed a tantalizing conjecture. It seems quite tricky, so after having struggled a ...
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Find a lower bound for a pre-invariant $Fol(L(F_m), X_m)$

In the paper of Bannon and Ravichandran, A Folner invariant for type $\rm{II}_1$ factors, they defined an invariant $Fol(M)$ for a separable type $\rm{II}_1$ factor $M$, especially for the free group ...
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306 views

Hans Saar's thesis

I would love to have a look on some results which are claimed by some people to be in Saar's thesis: H. Saar, Kompakte, vollständig beschränkte Abbildungen mit Werten in einer nuklearen ...
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247 views

Quasinilpotent elements of group C-star algebras

If $G$ is a discrete torsion-free group, can its (reduced or full) group C-star algebra contain non-zero quasinilpotent elements? I've seen various examples in the group von Neumann algebra setting ...
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Arveson index and curvature

Can someone explain me what is the intuitive idea behind Arveson Index and curvature of $E_0$ semigroups. I was reading the standard paper of Arveson, but is lost and yet to get intuition about it. An ...
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What information about a locally compact group $G$ is encoded in $C_r^\ast(G)$ which is not in $L^1(G)$?

Let $G$ be a locally compact group and let $ C_r^\ast(G) $ denote its reduced group $C^\ast$-algebra. Many features of a $G$ can be realized from $L^1(G)$ or $C_r^\ast(G)$. For example, $G$ is ...
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Hyperfiniteness of CCR algebra

Hi, It is known that the double commutant of the CCR algebra in it's GNS space with respect to some quasi-free states are always type III factors. My question is; Will some of them be hyperfinite ...
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What is the dual of an semidefinitely representable (SDR) cone?

The Question Let $V\simeq \mathbb{R}^r$ be an $r$-dimensional vector space with the usual Euclidean inner product. Let $\mathcal K\subset V$ be a cone defined as $$ \mathcal K=\Big\{x\in ...
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167 views

abstract algebra for component wise operations on “vectors” or what it might be called

I have a quite tough problem to solve and need an algebra that allows to "vectors" following operations: - multiplication between two vectors are componentwise that means v=(v1, v2, v3,...) multiplied ...
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1answer
890 views

PhD in operator algebras and non-commutative geometry [closed]

I do not know whether it is a good place to ask this question or not. I want to PhD in operator algebras and non-commutative geometry. What are the best places in the world for that? I want a good ...
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B(H) as a direct sum of a closed two sided ideal and a subalgebra

Let $B(H)$ is the C*-algebra of all bounded linear operators on Hilbert space $H$. Are there a closed two-sided ideal $I$ and a subalgebra $A$ of $B(H)$ such that $B(H)=I \oplus A$ (direct sum I and ...
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1answer
206 views

Is the image of a vNA under a SOT-continuous morphism still a vNA?

It looks very easy but I must admit I am struggling with this problem. Okay, let $M$ be a von Neumann algebra acting on a Hilbert space $H$ and let $K$ be another Hilbert space. Suppose $h\colon M\to ...