**4**

votes

**1**answer

143 views

### When is the corner algebra $PM_n(A)P$ isomorphic to $A$?

In the algebra of matrices $M_n(A)$ over a $C^*$ algebra $A$, consider the corner algebra $PM_n(A)P$ for a Hermitian projection $P\in M_n(A)$. Is there any condition known for $P$ to make $PM_n(A)P$ ...

**4**

votes

**2**answers

271 views

### Topological K-theory for commutative C*-algebras

It is in some sense folklore that given two arbitrary abelian groups $G,H$ one can find a $C^*$ algebra $A$ such that $K_0(A)=G$ and $K_1(A)=H$. My question is the following: what is known in the case ...

**2**

votes

**0**answers

120 views

### Extension of a bounded operator on manifold

I have a problem, which is quite urgent, as I have only today discovered an error in a proof i had in a thesis which is to be handed in tomorrow.
The problem, if stated in as full generality as ...

**3**

votes

**1**answer

133 views

### Pullbacks of $C^*$-algebras

I am reading the paper of Pedersen: "Pullback and Pushout Constructions in C^*-Algebra Theory". I try to work out the arguments from Proposition $3.1$ of his paper (you can find this article in the ...

**8**

votes

**2**answers

172 views

### Universal unital $C^*$ algebra generated by the relations of an n by n projection

Take generators $a_{ij}$ for $1\le i,j\le n$. Impose relations which come from the matrix
\begin{eqnarray*}
P=\left(\begin{array}{ccc}a_{11} & \dots & a_{1n} \\\vdots & \ddots & ...

**5**

votes

**2**answers

241 views

### When is the group algebra $L^1(G)$ semisimple?

Let $G$ be locally compact group. Define group algebra as
$$L^1(G)=\{f\colon G\to\Bbb{C}\mid\int\lvert f(x)\rvert\, dx<\infty\}$$
with convolution product. When is the group algebra $L^1(G)$ ...

**1**

vote

**2**answers

217 views

### Continuous linear functionals in strong operator and $\sigma$-strong topologies

It was mentioned in the comments to http://math.stackexchange.com/questions/517369/comparison-of-strong-operator-and-weak-topologies-on-bh that continuous linear functionals on ...

**1**

vote

**0**answers

82 views

### Differentiable Path of Operators and their Inverses

Let $\mathcal{H}$ be a separable Hilbert space. Consider a differentiable map $\mathbb{R} \rightarrow \mathcal{B}(\mathcal{H}), t \mapsto A(t)$, where $\mathcal{B}(\mathcal{H})$ is the space of ...

**8**

votes

**0**answers

215 views

### How many ideals are there in $B(H)^{**}$?

It is well-known (and easy to prove) that the only closed ideals of $B(\ell_2)$ are $\{0\}$, $B(\ell_2)$ and $K(\ell_2)$, the ideal of compact operators on $\ell_2$. I am curious whether we know what ...

**2**

votes

**1**answer

121 views

### positive maps and bimodules

Given a positive (or completely positive map) $\phi:A\to B$ between C* algebras, is there a way to construct an $A-B$ bimodule? This would more or less generalise the following construction: If $\phi$ ...

**11**

votes

**1**answer

163 views

### Understanding the computation of the center of Tambara-Yamagami fusion categories when realized as C* categories

Recall that the Tambara-Yamagami categories are those with fusion ring $\mathbb{Z}[A \cup m]$ where $A$ is an abelian group and $m$ is a non-invertible (simple) object such that $ma = am = m$ for all ...

**4**

votes

**1**answer

183 views

### von Neumann algebras generated by commutators

Let $A$ be a UHF-algebra of type $n^{\infty}$ and denote its unique and faithful trace by $\tau$. Let $L^2(A)$ be the Hilbert space of the GNS-representation associated to $\tau$. We have two ...

**1**

vote

**0**answers

124 views

### Number of connected components of a $C^{*}$ algebra

Motivating by the concept in the following post
What are these compact sets called?
We introduce the following concept:
Let $A$ be a unital $C^{*}$ algebra. We consider the unitary equivalent ...

**0**

votes

**1**answer

158 views

### Injective element of a commutative Banach algebra

A revision:
According to the comment of Nate Eldredge, in order to avoid the triviality, we revise the property $P$.
Assume that $A$ is a commutative unital Banach algebra. Its maximal ideal ...

**2**

votes

**0**answers

181 views

### Predual of a von Neumann algebra in terms of trace class operators

For a von Neumann algebra $\mathcal{A} \subseteq \mathcal{B(H)}$ where $\mathcal{B(H)}$ is the space of all bounded linear operators on the Hilbert space $\mathcal{H}$, there is a Banach space $ ...

**0**

votes

**0**answers

154 views

### Can I define Fredholm Index using $\dim \ker ST - \dim \ker TS$?

$X$, $Y$ are Banach spaces.
Let $S \in L(X, Y)$, $T \in L(Y, X)$, where $L(X, Y)$ denotes the Banach algebra of bounded linear operators from $X$ to $Y$. If we have that $Id_Y - ST \in \mathbb{K}(Y)$ ...

**0**

votes

**0**answers

253 views

### A noncommutative vector bundle

We know that a noncommutative vector bundle is a finitely generated projective $A$-module where $A$ is a non commutative $C^{*}$ algebra. In this question we introduce a particular non commutative ...

**1**

vote

**1**answer

61 views

### Square of Pierce 1/2 elements

Let $p$ be a nontrivial idempotent in a JB-algebra $A$ with Pierce decomposition $A = A_1 \oplus A_{1/2} \oplus A_0$. Then the projection onto $A_1$ (resp. $A_0$) is given by $U_p$ (resp. $U_{p'}$). ...

**1**

vote

**0**answers

177 views

### Bootstrap subcategory abelian?

In the book "K-Theory for Operator Algebras" by Bruce Blackadar, the exercise 23.15.8. on page 246 says:
"Let KKN be the full subcategory of KK with objects in N. Show that KKN is abelian category by ...

**5**

votes

**0**answers

146 views

### Dense ideals in C*-algebras and K-theory

Let $A$ be a nonunital C*-algebra and let $I \subset A$ be a dense, $*$-closed, 2-sided ideal. I was under the impression that there existed some "obvious" argument proving that $I$ carries all the ...

**7**

votes

**1**answer

216 views

### What is the general form of the duality transform for the Fock space?

I am interested in properties of the symmetric Fock space, looked at via the associated Wiener space. It is well known that for a Hilbert space $k$, the symmetric Fock space ...

**5**

votes

**1**answer

254 views

### NonCommutative Baire theorem

The classical Baire theorem says that the intersection of a sequence of open dense subsets of $X$, is dense, if the space is compact Hausdorff. In the language of $C^{*}$ algebras this is ...

**1**

vote

**0**answers

71 views

### tensor product of the disc algebra with itself

Let $A=\mathcal{A}(\mathbb{D})$ be the disc algebra. Is there a cross norm on $A\otimes A$, which is compatible to the Banach algebra multiplication, such that the resulting Banach algebra has a ...

**1**

vote

**0**answers

81 views

### Existence of orthogonal projections generating Von Neumann algebras

Let $V$ and $V'$ be Abelian Von Neumann algebras of projections on some Hilbert space $H$, and let $V_{1}$ and $V_{2}$ be minimal sub-algebras of $V$ and $V'$ generated by projections $P_{1} \in V$ ...

**7**

votes

**0**answers

297 views

### Unique maximal ideal in group C*-algebras

Let $G$ be a discrete group. Let $C^*(G)$ denote the full group C*-algebra of $G.$ Let $\pi:C^*(G)\rightarrow \mathbb{C}$ be the *-homomorphism associated with the trivial representation of $G.$
...

**7**

votes

**2**answers

365 views

### Projective modules over noncommutative tori?

It is a theorem of Rieffel that for any simple noncommutative tori ($\mathcal{A}$) of dimension $n$, every projective module over it is isomorphic to direct sum of $\mathcal{S}(M)$, Schwartz class ...

**3**

votes

**0**answers

73 views

### Closure of pseudodifferential operators of order 0 on compact manifolds

Let $M$ be a compact manifold. If we have a pseudodifferential operator $P$ of order $0$ on $M$, then $P$ is pseudolocal, i.e., every commutator $[f,P]$ with a continuous function $f \in C(M)$ is a ...

**9**

votes

**1**answer

278 views

### K-theory of ultrapowers

It may well be a trivial question but I was wondering if there is any relation between $K$-groups and ultrapowers of $C^*$-algebras. For instance, if $A$ is a $C^*$-algebra does $K_0(A^U)$ depend on ...

**7**

votes

**2**answers

234 views

### Literature on “real” $C^*$-algebras

I am trying to get a better understanding of "real" $C^*$-algebras. I encountered them in the paper
D. Voiculescu, Dual algebraic structures, J. Operator Theory 17(1987), 85-98,
which cites
G.G. ...

**5**

votes

**2**answers

439 views

### How to check whether a matrix is completely positive or not?

The definition:
cone of completely positive matrices
$\mathcal{C}=\{\sum_{i=1}^kx_ix_i^T:x_i\in\mathbb{R}^n_+\ for \ i=1,2,...,k\}$.
I just don't knwo how to check whether a matrix belongs to ...

**12**

votes

**3**answers

438 views

### Is the space of *-homomorphisms between two $C^*$-algebras locally path connected

Given the set of *-homomorphisms between two $C^*$-algebras $A$ and $B$, we may define a metric on it by setting $d(f,g):= \sup_{0<\|a\|\le 1}\|f(a)-g(a)\|$. Could it be true that, for each ...

**2**

votes

**0**answers

614 views

### The link between the subfactors and the motives as enriched Galois theories? [closed]

On 29th March 2007, at the "École normale supérieure" of Paris, the mathematician Vaughan Jones, gave a conference (in French) entitled "Les sous-facteurs : une théorie de Galois enrichie" (see here), ...

**1**

vote

**1**answer

224 views

### Existence of homogeneous single chain compositions of a given maximal subfactor?

All the subfactors here are irreducible inclusion of hyperfinite II$_1$ factors.
A subfactor $(N \subset M)$ is Homogeneous Single Chain ($HSC$) if its lattice of intermediate subfactors is a ...

**3**

votes

**0**answers

98 views

### Noncommutative Poincare inequalities

This is a question on how (or if) people in the community think about the Poincare inequality in noncommutative geometry. In geometry, the Poincare inequality (when it exists) gives a bound on a ...

**7**

votes

**1**answer

258 views

### Infinite tensor product of states

Tensor products of finite number of different objects are always well described in the literature. However, the situation of infinite tensor products seems to be much tougher.
Even in the simplest ...

**2**

votes

**1**answer

119 views

### What is the multiplicative unitary for SU_q(2) (or other quantum groups)?

Consider a (von Neumann algebraic) locally compact quantum group $(M, \Delta, \phi, \psi)$ where the von Neumann algebra $M$ is realized as operators on the Hilbert space $H$. There is a ...

**5**

votes

**0**answers

115 views

### When is an inner derivation a Fredholm operator?

Let $\mathcal{B}(H)$ denote the algebra of bounded operators on a Hilbert space $H$. I'm interested in inner derivations acting on the Schatten ideals $L^p\subseteq\mathcal{B}(H)$ (defined by ...

**2**

votes

**0**answers

263 views

### Are the homogeneous single chain subfactors, Dedekind?

Background: See here and there.
Recall that a subfactor is Dedekind if all its intermediate subfactors are normal.
A subfactor $(N \subset M)$ is Homogeneous Single Chain (HSC) if its lattice ...

**2**

votes

**1**answer

123 views

### Continuous depedence of the spectrum on elements

Suppose $a_n \to a$ in a unital C*-algebra $A$. If $\lambda_n \in \sigma(a_n)$ and $\lambda_n \to \lambda$, then $\lambda \in \sigma(a)$. Does the converse hold?
So if $\lambda \in \sigma(a)$, does ...

**3**

votes

**1**answer

202 views

### Strange (?) definition of the spectrum

Suppose that $A$ is a commutative, unital $C^*$-algebra. Then it is isomorphic to $C(X)$ for some compact Hausdorff topological space $X$. $X$ can be identified as the space of all unital ...

**3**

votes

**0**answers

81 views

### States and extremal states of quantum SU(2) and the Podleś sphere

Is there any description (preferably somehow related to the original generators) for the state space (as in C*-algebras) of quantum SU(2) and the Podleś sphere? If so (this is pushing my luck) are the ...

**2**

votes

**2**answers

240 views

### Lower bounds for norms of commutators

For various reasons I became interested in bounds on the norm of commutators of operators. For instance, if $B(H)$ is the algebra of bounded operators on a Hilbert space, one may ask for a lower bound ...

**12**

votes

**0**answers

364 views

### Does anybody know if the Fourier algebra of SL(3,Z) has an approximate identity?

(Note to those who like to tidy LaTeX, or ${\rm \LaTeX}$: I kindly request that you don't put any LaTeX in the title of this question, nor change the bolds below to blackboard ...

**2**

votes

**0**answers

139 views

### A section from subfactors to transitive groups

A finite group-subgroup subfactor is a subfactor $(N \subset M)$ isomorphic to $(R^G \subset R^H)$ with $(H \subset G)$ an inclusion of finite groups acting as outer automorphism on the hyperfinite ...

**5**

votes

**1**answer

476 views

### Noncommutative geometry and category theory

The point in which one starts to talk about noncommutative geometry is the Gelfand Najmark theorem. It establishes an equivalence of the catgeories of commutative (non)unital $C^*$-algebras and the ...

**13**

votes

**2**answers

626 views

### Groups which are only defined up to conjugation

I'm trying to understand what the right way is to think about "groups which are only well-defined up to conjugation." Since this is somewhat vague let me clarify it by pointing out the main examples ...

**1**

vote

**1**answer

204 views

### On the relation between the set of extreme points of the unit ball of $M(X)$ and $M(X)^{**}$

Suppose that $X$ is a locally compact topological space. Let $M(X)$ denote the Banach space of regular Borel measures on $X$. It is known that the bidual of $C_0(X)$ is a commutative $C^*-$algebra. ...

**3**

votes

**2**answers

168 views

### Strong Morita Equivalence and Morphisms Between $ C^{*} $-Algebras

If $ A $ and $ B $ are $ C^{*} $-algebras, then they are strongly Morita equivalent if there exist a $ (B,A) $-bimodule $ E $ and an $ (A,B) $-bimodule $ F $ such that
$$
E \otimes_{A} F \cong B \quad ...

**6**

votes

**0**answers

251 views

### C* algebras of free semicircular systems

It was shown by Pimsner and Voiculescu in 1982 that the reduced group $C^{*}$-algebras $C^{*}_{r}(\mathbb{F}_{n})$ and $C^{*}_{r}(\mathbb{F}_{m})$ are isomorphic if and only if $n = m$ (here, ...

**5**

votes

**2**answers

145 views

### Extension of $C^*$ isomorphism to $W^*$ isomorphism

Let $\mathfrak{A}$ be $C^*$algebra, and $\pi$ its faithful representation on Hilbert space $\mathcal{H}$. Bicommutant $\mathfrak{B}=\pi(\mathfrak{A})''$ is the von Neumann algebra generated by ...