# Tagged Questions

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

**0**

votes

**1**answer

164 views

### 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 $Spec(uv)\subsetneqq\mathbb{T}$?

**0**

votes

**1**answer

158 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 ...

**2**

votes

**2**answers

133 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 ...

**3**

votes

**1**answer

346 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 ...

**1**

vote

**1**answer

133 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 $\alpha_{1}, \alpha_{2}$ ...

**6**

votes

**1**answer

226 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?

**2**

votes

**1**answer

366 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, ...

**0**

votes

**0**answers

131 views

### 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**

votes

**1**answer

179 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 ...

**0**

votes

**0**answers

178 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 ...

**6**

votes

**1**answer

281 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
...

**22**

votes

**1**answer

383 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' ...

**4**

votes

**2**answers

329 views

### 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 ...

**1**

vote

**0**answers

157 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 ...

**0**

votes

**0**answers

128 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 ...

**12**

votes

**0**answers

318 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 ...

**5**

votes

**0**answers

207 views

### 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 ...

**1**

vote

**0**answers

297 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 ...

**6**

votes

**2**answers

229 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 ...

**2**

votes

**1**answer

152 views

### 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 ...

**5**

votes

**0**answers

244 views

### 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 ...

**5**

votes

**1**answer

175 views

### 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 ...

**1**

vote

**2**answers

142 views

### 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 ...

**0**

votes

**0**answers

161 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 ...

**1**

vote

**1**answer

799 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 ...

**9**

votes

**2**answers

363 views

### 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 ...

**2**

votes

**1**answer

195 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 ...

**4**

votes

**1**answer

216 views

### Taking direct sums in $K$-theory in Kirchberg-Phillips classification

A theorem by Kirchberg and Phillips states that two unital separable nuclear simple purely infinite $C^*$-algebras (so called Kirchberg algebras) are isomorphic if and only if their topological ...

**3**

votes

**0**answers

103 views

### reference for direct finiteness of the ring of affiliated operators

Let $\Gamma$ be a group, $N(\Gamma)$ its group von Neumann algebra,
$\newcommand{\cUG}{{\mathcal U}(\Gamma)}$
and $\cUG$ the ring of all densely-defined, closed operators ...

**3**

votes

**1**answer

136 views

### Local cross sections for Unitary group in a hilbert space

Let $U(\mathcal{H})$ be the group of unitary operators on a Hilbert space with the norm topology. Let $H\subset U(\mathcal{H})$ be a closed subgroup.
Under which condidions (on the ...

**4**

votes

**2**answers

445 views

### When is spectral norm of AB equal to that of BA?

I have $A^{1/2} B A^{1/2} \preceq I$ for two PSD matrices $A$ and $B$, and I'd like to know if that implies $\|AB\|_2 \leq 1.$
The argument I was using to show this is that for any two square ...

**1**

vote

**1**answer

415 views

### Is this result of Spain correct?

Let us have a look on the proof of Theorem 2 in [P. G. Spain, Boolean algebras of projections, Proceedings of the Edinburgh Mathematical Society (Series 2) 19, 03, March 1975, 287-289]
The author ...

**6**

votes

**1**answer

213 views

### Masas in second duals of Banach algebras

Lel $B$ be a Banach algebra and give $B^{**}$ one of the Arens products in order to make it a Banach algebra. Then the canonical embedding $\kappa\colon B\to B^{\ast\ast}$ is a homomorphic embedding ...

**3**

votes

**1**answer

151 views

### When is a $*$-homomorphism between multiplier algebras strictly continuous?

(This question was posted on MSE here but didn't get any answers.)
The strict topology on the multiplier algebra M(A) of a C*-algebra A is that generated by the seminorms
$$ x\mapsto \|ax\|\quad ...

**1**

vote

**0**answers

183 views

### Factorization through Hilbert Space

I am working on my Masters thesis which is on the Grothendieck's inequality. My aim is to study its formulation in different contexts starting from commutative case and then covering non-commutative ...

**8**

votes

**3**answers

480 views

### Compact subgroups of the unitary group of operators in a hilbert space

Is there a characterization for the compact subgroups of the unitary operators in a Hilbert space, where the unitaries are furnished with the norm topology? What about other topologies?

**3**

votes

**1**answer

157 views

### Dense subspaces in primitive ideals of C-star algebras

Let $G$ be a unimodular locally compact group (my main examples are algebraic groups over local fields. Thefore we can assume $G$ is Type I, if necessary). Then there are at least three group algebras ...

**4**

votes

**2**answers

242 views

### system of homogeneous matrix equations

Edit: Maybe I should make it clear that by a solution I mean a pair of matrices $(A,B)$ such that the identity below holds for any complex numbers $x,y$.
One of my friend asked me the following ...

**7**

votes

**2**answers

349 views

### General recipe for building C*-algebras out of combinatorial object

I want to ask what should be a nice way to build C*-algebras out of objects like groups, inverse-semigroups, semigroups, ringgs or graphs. I know there are well known construction of C*-algebras out ...

**14**

votes

**3**answers

684 views

### Realizing universal C*-algebras as concrete C*-algebras

How do I in general realize a universal C*-algebra generated by some generators and relation as concrete C*-algebras? For example, I know that universal C*-algebra generated by a single unitary is ...

**3**

votes

**2**answers

270 views

### When is a Pseudo-differential operator trace class or in Dixmier ideal?

Let's denote the set of all Pseudo-differential operators with symbol of “order” $d$ by $\Psi_d(M)$ and Sobolev space on $M$ by $H_s(M)$. It is known that
If $P\in\Psi_d(M)$ Then $P$ extends to a ...

**1**

vote

**1**answer

167 views

### Cyclic vectors for C* algebras

Let A be a C* algebra of operators on a Hilbert space H. Can it happen that for some x in H the set Ax is dense in H but it is not the whole H?

**3**

votes

**1**answer

140 views

### Second quantization of partial isometry

If we have a unitary map from Hilbert space $H$ to $H$, we get a unitary map from $e^{H}$ to
$e^{H}$, where $e^{H}$ is the symmetric Fock space of $H$. But if we replace the unitary with partial ...

**3**

votes

**0**answers

154 views

### Is the construction of ring C*-algebra functorial?

Cunz and Li defined defined C*-algebras for arbitrary rings with of course some condition. One can look at their article (http://arxiv.org/abs/0905.4861). My question: is the construction functorial? ...

**0**

votes

**0**answers

179 views

### What methods have been used to study AW*-algebras up to now?

I am interested mainly in ring theory and homological algebra. Now I want to know about the research methods of AW*-algebras. So I want to know the answer to the question:"what methods have been used ...

**2**

votes

**2**answers

301 views

### Stabilization in Banach algebras

In $C^\ast$-algebras we use $K(H)$, the algebra of compact operators on a separable Hilbert space, for stabilization of a $C^\ast$-algebra, i.e. $S(A):=A\otimes K(H)$. Is there any similar ...

**4**

votes

**1**answer

281 views

### Kadison-Singer problem in exotic Hilbert spaces

The Kadison-Singer problem is considered in relation to the separable Hilbert space:
KS: Does every pure state on the diagonal (atomic) masa of $B(\ell_2)$ has a unique extension to $B(\ell_2)$?
...

**9**

votes

**2**answers

419 views

### Calkin Algebra and the embedding

Let $H$ be a separable, infinite dimensional Hilbert Space and $Calk(H):=B(H)/K(H)$ denotes
the Calkin algebra. There is obvious surjection $\pi: B(H) \to Calk(H)$ but I'm interested
in somehow ...

**1**

vote

**1**answer

204 views

### crossed product

on Williams Crossed product book,on page 198, it is mentioned that there is only one regular representation for C_c(G), and that is the left regular representation.
I know that this representation is ...

**9**

votes

**2**answers

382 views

### Seeing topological (geom.) properties of the space via corresponding C^*-algebra

Compact Hausdorff spaces bijectively correspond to C^*-algebras with identity. One needs to consider the algebra of continuous functions C(X) to go in one direction and spectrum to go in the other. ...