**1**

vote

**1**answer

53 views

### Comparision of two $C^{*}$ algebras associated to a non vanishing vector field on a compact manifold

Let $X$ be a non vanishing vector field on a compact manifold $M$ so we have a one dimensional foliation $F$ of $M$ with orbits of $X$.
This foliation defines a $C^{*}$ algebra $C^{*}(F)$. On the ...

**1**

vote

**1**answer

109 views

### A $C^{*}$ algebra associated to a graded $C^{*}$ algebra

A $C^{*}$ algebra $A$ is graded by $\mathbb{Z}_{n}$ iff it can be acted by $\mathbb{Z}_{n}$. So we associate the $C^{*}$ algebra $A\rtimes \mathbb{Z}_{n}$ to a $\mathbb{Z}_{n}$-graded $C^{*}$ ...

**8**

votes

**1**answer

267 views

### For discrete groups, does the Haagerup property imply the AP of Haagerup-Kraus?

I don't expect to find an explicit counterexample to my question, because any example which was known to have the Haagerup property yet not have AP would have given an exact group without AP, and the ...

**2**

votes

**0**answers

94 views

### Arveson spectrum for a unitary representation of a group on a Hilbert space

Although this is not research, I think the question is a little bit too specific for math.stackexchange
Let $G = \mathbb{R}$. By Stone's theorem, $U(t)\in\mathcal{B}(\mathcal{H})$ is generated by a ...

**1**

vote

**0**answers

60 views

### Why does one only consider one-parameter groups in Borchers-Arveson theorem?

(question from math.stackexchange)
The theorem (Operator algebras and Quantum statistical mechanics vol. 1, Bratteli, Robinson, Thm. 3.2.46 p.261) roughly says that if one has a one parameter ...

**1**

vote

**0**answers

55 views

### Are there infinitely many amenable Hadamard-Petrescu subfactors?

The complex Hadamard matrices of dimension $n$ are used to build index $n$ subfactors through the commuting square construction. For more details, see the paper Subfactors and Hadamard Matrices by ...

**5**

votes

**1**answer

133 views

### When are countably generated Hilbert modules generated by c.p.c. order zero maps?

Throughout let $B$ be a stable C*-algebra, i.e. $B\cong B\otimes K$, where $K$ is the C*-algebra of compact operators on an infinite dimensional separable Hilbert space. It is well-known that any ...

**3**

votes

**0**answers

100 views

### Two questions on topological and geometric structure of projections in a simple $C^{*}$ algebra

Let $A$ be a simple $C^{*}$ algebra. Assume that the space of projections has a connected component homeomorphic to the complex Grassmanian $G(k,n)$. Is it true to say that, for all $k'<k$, the ...

**0**

votes

**0**answers

55 views

### Is $KK^G(\mathbb{C}^n,B)$ countably additive in $B$ and countable?

Let $G$ be a finite discrete groupoid, $A=\mathbb{C}^n$ a finite dimensional, commutative $C^*$-algebra and assume we have given a $G$-action on $A$. Note that the action of $G$ on ...

**0**

votes

**1**answer

154 views

### A $C^{*}$ algebra associated to a group [closed]

Let $G$ be a compact group which act on a Hilbert space $H$. We define a linear map $T$ on the dual space $H^{*}$ with $$T(\phi)(x)=\int_{G} \phi(g.x)$$ The integration is based on the Haar ...

**14**

votes

**2**answers

849 views

### What are the applications of operator algebras to other areas?

Question: What are the applications of operator algebras to other areas?
More precisely, I would like to know the results in mathematical areas outside of operator algebras which were proved by ...

**0**

votes

**0**answers

78 views

### How to generalize multiplication and addition to cyclic subfactors?

Let $(N \subset M)$ be a finite index irreducible subfactor and $P=P(N \subset M)$ its subfactor planar algebra.
Definition: $(N \subset M)$ is cyclic if its lattice of intermediate subfactors ...

**0**

votes

**0**answers

125 views

### Range of a trace preserving completely positive projection

I'm working in $M_n(\mathbb{C})$, the algebra of complex $n\times n$ matrices. I managed to build a completely positive, trace preserving, star preserving, projection $P$. That is
$$\text{Tr}(P(A)) = ...

**6**

votes

**1**answer

318 views

### Formula for the distance in noncommutative geometry

Probably the most famous formula in noncommutative geometry is the following formula allowing one to compute distance of two points using the operator theoretic data:
$$(1) \ \ ...

**5**

votes

**2**answers

236 views

### fixpoint algebras of a permutation action

Let $D$ be an infinite UHF algebra, e.g. the infinite tensor product of the matrix algebra $M_k(\mathbb{C})$. The permutation group $\Sigma_n$ acts on the $n$-fold tensor product $D^{\otimes n}$ in a ...

**3**

votes

**0**answers

119 views

### Dimension of Birman-Murakami-Wenzl Algebra

I was reading the paper Braids, Link Polynomials and A New Algebra by J. S. Birman and H. Wenzl, and I was wondering is there a combinatorial way to compute the dimension of the algebras ...

**3**

votes

**1**answer

128 views

### Conjugacy of circle actions on UHF C*-algebras

Consider pointwise continuous actions of the unit circle on the $2^{\infty}$-UHF C*-algebra A by *-automorphisms. Assume that two such actions have the same fixed point algebra, i.e., elements that ...

**3**

votes

**1**answer

165 views

### Do c.p.c. order zero maps between local C*-algebras map C*-subalgebras to C*-subalgebras?

For the sake of this question I will assume the following
Definition A pre-C*-algebra $A$ is local in the sense of [1], i.e. if there is a family of C*-subalgebras $\{A_i\}$ of $A$ with the property ...

**4**

votes

**2**answers

322 views

### Question about projections in von Neumann algebras

Let $M$ be a von Neumann algebra, and let $\mathcal{P}$ be the set of nontrivial (not equal to $0$ or $e$) projections of $M$. Define $p,q \in \mathcal{P}$ to be equivalent if there exist projections ...

**5**

votes

**0**answers

119 views

### Approximation in the tensor square of a weakly exact von Neumann algebra

Background. I think I can prove something about a certain construction definition for Fourier algebras of discrete groups, under the assumption that the group is exact (well, really I use Yu's ...

**6**

votes

**0**answers

299 views

### Does the Approximation Property (AP) pass to quotients by amenable subgroups?

Given a countable group $G$ with the AP, and an normal, amenable subgroup $N$ of $G$, does $G/N$ have the AP?
In particular, does there exist a group $G$ with the AP and a surjective group ...

**5**

votes

**1**answer

202 views

### Are isometric homorphisms of C* algebras *-homorphisms

Here is my precise question:
Let $A$ and $B$ be two $C^*$ algebras. Let $f: A \rightarrow B$ a homomorphism of $C^*$ algebras which is isometric (on its image). Is $f$ necessary a $*$-homomorphism ?
...

**11**

votes

**1**answer

426 views

### A generalization of the Powers-Stormer inequality

The well-known Powers-Stormer inequality says the following: for positive semidefinite operators $A, B$, we have that $\mathrm{Tr}((A - B)(A - B)) \leq \| A^2 - B^2 \|_1$, where $\| \cdot \|_1$ ...

**5**

votes

**1**answer

174 views

### Comparing cardinalities of the spectrum of two masas in $B(H)$

If I imagine that (the self-adjoint part of) a C*-algebra $A$ represents the algebra of observables of some quantum system, then certain perspectives on algebraic quantum theory would ask me to ...

**0**

votes

**1**answer

94 views

### Is there an irreducible subfactor with an infinite homogeneous single chain lattice?

We know that we can build an irreducible subfactor realizing a finite single chain lattice containing any finite index irreducible maximal subfactors, by using the free composition (see here).
Now ...

**1**

vote

**0**answers

114 views

### A continuous choice of invertible elements

Let $A$ be a simple unital $C^{*}$ algebra with invertible elements $G(A)$. Assume that $A^{*}$ is its dual space, which is equipped with the weak star topology.
Is there a continuous map ...

**3**

votes

**2**answers

255 views

### Does this C*-algebra embed into a simple nuclear C*-algebra?

Let $\mathcal K$ denote the C*-algebra of compact operators, and fix an embedding $\phi_n:M_n(\mathbb C) \to \mathcal K$ for each $n\in \mathbb N$. Define the C*-algebra
$A := \{(a_n)_{n=1}^\infty ...

**0**

votes

**0**answers

67 views

### Specific optimization problem solution procedures

Is there a standard procedure to solve following two optimization problems?
$$\mathsf{Problem\mbox{ }I}:\mbox{ }\min_{A\in\{0,1\}^{n\times n}:rk(A)=r}\mbox{ }\max_{R,S\in\Bbb R^{n\times ...

**8**

votes

**1**answer

263 views

### Embedding the group von Neumann algebra into an injective von Neumann algebra on the same Hilbert space

Let $\Gamma$ be a discrete group, $\newcommand{\VN}{\rm VN}$
and let $\VN(\Gamma)$ denote its von Neumann algebra, regarded as a subalgebra of ${\sf B}(\ell^2(\Gamma))$. It is well known that ...

**3**

votes

**1**answer

231 views

### Finite codimensional subvector space of $C^{*}$ algebras which contains no invertible elements

Assume that $A$ is a unital $C^{*}$ algebra. Is there a subvector space $Y\subset A$ of finite codimension which does not contain any invertible element?
Let $n(A)$ be the infimum of such ...

**7**

votes

**0**answers

200 views

### Unital $C^{*}$ algebras which all elements have path connected spectrum

A unital $C^{*}$ algebra is called "Path connected" if the spectrum of all its elements are path connected.
Is the tensor product of two path connected algebra, a path connected algebra?(For ...

**0**

votes

**0**answers

84 views

### A possible characterization of finite dimensional $C^{*}$ algebras

Is there an infinite dimensional unital $C^{*}$ algebra $A$ with invertible group $U$ such that for every finite dimensional subvector space $Y\subset A$, the number of connected components of ...

**1**

vote

**0**answers

131 views

### The planar algebra generated by the biprojections

Let $(N_1 \subset M_1)$ and $(N_2 \subset M_2)$ be two irreducible finite index subfactors.
Let $\mathcal{B}_i$ be the set of all the biprojections of $\mathcal{P}_{2+}(N_i \subset M_i)$.
Let ...

**0**

votes

**1**answer

172 views

### Totally non hereditary $C^{*}$-subalgebras

Assume that $B$ is a $C^{*}$ subalgebra of $A$. We say $B$ is totally non hereditary subalgebra of $A$ if not only $B$ is not a hereditary subalgebra but also it is not isomorphic to any ...

**2**

votes

**1**answer

154 views

### Has a subfactor with lattice $B_3$, a singly generated identity biprojection?

Let $(N \subset M)$ be an irreducible finite index subfactor.
If its lattice of intermediate subfactors is equivalent to $B_3$ (the lattice of divisors of $n=p_1p_2p_3$ square free):
...

**3**

votes

**1**answer

178 views

### K-homology of Cantor set and abelian AF-algebras

This may be a standard question answered in a book, or article. I don't know. I know that there exist related results with $\lim^1$-sequences (Rosenberg and Schochet).
What is
...

**1**

vote

**0**answers

75 views

### A reasonable framework to study properties of operator $A \mapsto KAK$ on Banach space

Let $K$ be a continuous linear operator on $C[0,1]$ (more, precisely, it is a linear integral operator). Then $K$ defines a continous linear operator $\widehat K$ on $\mathcal L(C[0,1])$ by the rule
...

**2**

votes

**1**answer

214 views

### Integration in C^* algebra

Let $\mathfrak{A}$ be a C${}^*$ algebra and $\mathbb{R}\ni s \mapsto \alpha_s$ a continuous family of its automorphisms. Is it true that
$$
\int d s \, f(s)\, \alpha_s(A)
$$
is well defined as a ...

**5**

votes

**2**answers

188 views

### Two-sided ideals of $B(H)$ are hereditary

I seem to recall that (not necessarily closed) two-sided ideals of $B(H)$ are hereditary. Is that true?
If it is, can anyone post a proof/reference?

**4**

votes

**2**answers

221 views

### The closure of all periodic homeomorphisms of circle

Let $X$ be the space of all continuous maps from $S^{1}$ to $S^{1}$. $X$ is equipped with $C^{0}$ topology. Let $Y\subset X$ be the union of all finite order homeomorphisms i.e: all ...

**2**

votes

**0**answers

209 views

### The kernel of $C^{*}(G)\to C_{r}^{*}(G)$

Let $G$ be a locally compact group. Put $I(G)=\ker: C^{*}(G) \to C_{r}^{*} (G)$, the kernel of the canonical morphism.
What type of $C^{*}$ algebras can not be isomorphic to $I(G)$, for some ...

**3**

votes

**1**answer

204 views

### Can the full and reduced group $C^*$-algebras be “noncanonically” isomorphic?

Is there a locally compact group $G$ such that the canonical map from $C^{*}(G)$ to $C^{*}_{red} G$ is not isomorphism, hence $G$ is not amenable but these two $C^{*}$ algebras are isomorphic ...

**10**

votes

**1**answer

246 views

### Inner and extendible automorphisms of C*-algebras

If an automorphism $\alpha$ of a C*-algebra $A$ is inner then whenever $A$ is a subalgebra of another C*-algebra $B$, $\alpha$ obviously extends to $B$.
Is the converse true: if an automorphism ...

**2**

votes

**0**answers

213 views

### Is there a non-trivial maximal Hopf algebra?

Let $H$ be a Hopf algebra over an algebraically
closed field $\mathbb{K}$ of characteristic $0$.
Maximal means without left coideal subalgebra $I$ (i.e. $\Delta(I) \subset H \otimes I$) other than ...

**6**

votes

**0**answers

162 views

### Correspondences as generalized morphism between $C^*$-algebras

While reading about Morita equivalence in the category of $C^*$-algebras I met also the following notion: a correspondence between two $C^*$-algebras $A,B$ is a pair $(X,\varphi)$ where $X$ is a ...

**13**

votes

**0**answers

307 views

### $C^*$-algebra generated by those operators that are bounded on every $\ell_p$

Suppose $T: c_{00} \to c_{00}$ is a linear map such that, when regarded as an infinite matrix, there is a uniform bound on the $\ell_1$-norms of its columns, and a uniform bound on the $\ell_1$-norms ...

**5**

votes

**1**answer

119 views

### Isomorphism of Hilbert modules

In Hilbert space theory if we have a linear bijection $U:H \to K$ which is an isometry, then it preserves inner products. Suppose now that you have two (right) Hilbert modules $X,Y$ (say, over the ...

**8**

votes

**0**answers

164 views

### Commutative spectral triples not coming from manifolds

There is a very deep and remarkable theorem by Connes (the so called reconstruction theorem) which states that from a commutative spectral triple obeying certain axioms one can reconstruct a smooth ...

**3**

votes

**0**answers

78 views

### How the modular theory of von Neumann algebras, deal with generating C*-algebras?

Let $H$ be a separable infinite dimensional Hilbert space, $M \subset B(H)$ a von Neumann algebra and $A \subset M$ a separable $C^*$-algebra such that $A''=M$. Suppose the existence of a bicyclic ...

**0**

votes

**1**answer

234 views

### Murray–von Neumann equivalence on C$^*$-algebra and von Neumann algebra

Let $H$ be a separable infinite dimensional Hilbert space, $M \subset B(H)$ a von Neumann algebra and $A \subset M$ a separable $C^*$-algebra such that $A''=M$.
Let $p,q \in M_{\infty}(A)$ be ...