**1**

vote

**0**answers

128 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

154 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

151 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

172 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

71 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

210 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

180 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

209 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

207 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

187 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

241 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

205 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

155 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

274 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

114 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

160 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

72 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

194 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 ...

**3**

votes

**1**answer

279 views

### Strong Morita equivalence and representation theory

In the context of pure algebra we say that two algebras (in general: rings) $A,B$ are Morita equivalence when there are bimodules $_AP_B,_BQ_A$ such that $P \otimes_B Q \cong _AA_A$ and $Q \otimes_A P ...

**15**

votes

**2**answers

1k views

### The letters of the word “ART”

Edit: According to the Gelfand duality between topological spaces and commutative $C^{*}$algebras, I add some new tags. So the question is that what is the structure of $ Ext (A,A)$ where $A$ is ...

**0**

votes

**1**answer

184 views

### Coaction of a group

Suppose $G$ is a finite group which acts on a $C^*-$algebra which we denote by $A$. I was wondering if there is a naturally induced coaction on $A\otimes C(G)$, here $C(G)$ denotes functions on $G$.
I ...

**13**

votes

**0**answers

558 views

### What is operator tmf?

One of the many wonderful things about K-theory, relative to other generalized cohomology theories, is that it can be defined for not-necessarily-commutative C*-algebras. The resulting construction, ...

**1**

vote

**1**answer

100 views

### Is there a Frobenius reciprocity for the coproduct?

Let $\mathcal{P}$ be an irreducible finite index-depth subfactor planar algebra. The $2$-boxes space $\mathcal{P}_{2,+}$ is equipped with the coproduct $(a,b) \mapsto a*b = ...

**1**

vote

**1**answer

100 views

### Is the coproduct of central operators, also central?

Let $\mathcal{P}$ be an irreducible finite index-depth subfactor planar algebra. The $2$-boxes space $\mathcal{P}_{2,+}$ is equipped with the coproduct $(a,b) \mapsto a*b = ...

**2**

votes

**1**answer

221 views

### A commutative Banach algebra with an abundance of discountinuous functions

Let $A$ be the algebra of all bounded functions from $[0,\;1]$ to $\mathbb{C}$.
For $f\in A,\;$ $\omega_{f}$ is the standard oscillation function.. Each of the following two (equivalent) norms ...

**0**

votes

**1**answer

191 views

### A question on K- theory of non commutative $C^\star$ algebra

Edit: According to the comment of Andre Henriques I revise the question:
What is an example of a noncommutative unital $C^\star$ algebra $A$, which is not Morita equivalent to a commutative ...

**2**

votes

**1**answer

116 views

### Obstructions for $C^\star$ algebras to contain a $Z^\star$ algebra

As the comment of Andreas Thom indicated here, a separable $C^\star$ algebra $A$ can not contain a $Z^\star$ algebra.(A $Z^\star$ algebra is a $C^\star$ algebra which all elements are zero ...

**3**

votes

**1**answer

121 views

### Simple $Z^{*}$ algebra

What is an example of a simple $C^{*}$ algebra which all elements are (two sided or equivalently one sided) zero divisor?

**2**

votes

**0**answers

70 views

### A question on extension of $Z^{*}$ algebras

A $Z^{*}$ algebra is a $C^{*}$ algebra which all elements are(two sided or equivalently one sided) zero divisor.
Are there two $Z^{*}$ algebras $A,B$ such that for every short exact sequence ...

**4**

votes

**1**answer

177 views

### Nuclearity noncommutative torus

I read that the Noncommutative torus (rotation algebra) is nuclear when $\theta\in\mathbb{R}\setminus\mathbb{Q}$. Unfortunately, I haven't found a proof. Could someone give me a reference and/or an ...

**5**

votes

**0**answers

196 views

### Unitary representations of Tarski Monsters and other beasts

Did people study the unitary representations of Tarsky Monsters, for example the ones constructed by Ol'shanskii? Are there any exotic representations, ie. except the ones related to the left regular, ...

**4**

votes

**1**answer

216 views

### C*-Algebras: Dynamics vs. Derivations

Problem
Given a C*-algebra $\mathcal{A}$.
Consider dynamics $\tau:\mathbb{R}\to\mathrm{Aut}(\mathcal{A})$ and $\tau':\mathbb{R}\to\mathrm{Aut}(\mathcal{A})$.
(More precisely, strongly continuous ...

**6**

votes

**0**answers

217 views

### Does every commutative $*$-algebra of operators on a prehilbert space have a character?

My question can be equivalently stated as follows:
Let $A$ be a complex commutative algebra with involution, and assume there exists a non-zero homomorphism $\pi:A\to L(V)$ to the algebra of ...

**4**

votes

**1**answer

154 views

### Noncommutative version of Littlewood's First Principle

There are definitely noncommutative analogues for Lusin's theorem and Egoroff's theorem (found in Blackadar for example). I'm curious if there is a version of the first principle:
Every Lebesgue
...

**3**

votes

**1**answer

108 views

### Strong and weak equivalence of C$^∗$-extensions by compacts

Let $A$ be a $C^*$-algebra. An extension of $A$ by the compact operators $K$ is an embedding $\epsilon$ of $A$ into the Calkin algebra $B(H)/K$.
Two embeddings $\epsilon_1$ and $\epsilon_2$ are ...

**3**

votes

**1**answer

164 views

### A nilpotency question on $C^{*}$ algebras

What is an example of a $C^{*}$ algebra $A$ with the property that: for every nilpotent(Quasi nilpotent) $a$ and for every $n\in \mathbb{N}$, there is a $b$ with $b^{n}=a$.
To what extent ...

**1**

vote

**0**answers

117 views

### Noncommutativization of fixed point theory

What papers or references have been devoted for a noncommutativization of "Fixed point theory". Here the terminology Noncommutativiztion, as usual, indicates to that famous table with 2 columns: ...

**5**

votes

**1**answer

161 views

### A coalgebra structure on compact operators

Is there a coalgebra structure $\Delta_{n}$ on $M_{n}(\mathbb{C})$ which is compatible with the natural embedding $i_{n:}M_{n}(\mathbb{C})\to M_{n+1}(\mathbb{C})$ with $i_{n}(A)= A\oplus ...

**2**

votes

**1**answer

230 views

### Non commutative topological manifolds

Assume that $A$ is a Banach algebra with two closed two sided ideals $I$ and $J$ such that $I$ and $J$ are commutative and $A=I+J$. Does this implies that $A$ is commutative? For the ...

**3**

votes

**2**answers

114 views

### States with a unique state extension

I guess that the answer to the following question is both well known and easy. But I was unable to solve the exercise.
Consider a unital $C^*$-$\,$algebra $\mathcal A$ and and a proper unital ...

**12**

votes

**2**answers

2k views

### Mysterious quotes (at least for me)

I heard two quotes, one from Alain Connes and an other one from Orlov.
Alain Connes was talking about noncommutative geometry and he said the following:
" a noncommutative algebra creates its own ...

**4**

votes

**0**answers

229 views

### Faithful and weakly-mixing representations of Property (T) groups in relation to left regular rep

Is it known that: Any countable Property (T) group (or more generally, a non-amenable group) has a faithful, weakly-mixing representation which is NOT weakly included in its left regular ...

**1**

vote

**0**answers

116 views

### Clebsch Gordan coefficients of compact quantum groups

Consider a compact quantum group $G$. Let $a, b$ and $c$ be irreducible unitary corepresentations and assume that $c$ is contained in $a \otimes b$. Let $U$ be the intertwiner from the representation ...

**7**

votes

**2**answers

300 views

### Triangle inequality for $L^1$-norm with respect to a state

It is well-known that the naive construction of non-commutative $L^p$-spaces is performed only in tracial case. I would like to know if it is really a necessity.
To wit, let $\varphi$ be a normal ...

**4**

votes

**1**answer

84 views

### CB-norm of mappings from a matrix space

The following result of Roger Smith is well known to operator algebraists:
$$\| \phi: E \rightarrow M_n\|_{cb} =\| \phi^{(n)} \otimes id_{M_n}: E \otimes_{min}M_n \rightarrow M_n \otimes M_n\|,$$
...

**3**

votes

**1**answer

112 views

### Is the square diagram of index and exponential maps in $K$-theory of $C^*$-algebras anti-commutative?

Assume we have a $3\times 3$ grid with rows and columns being short exact sequences of $C^*$-algebras.
This gives a grid of 6-term exact sequences: 3 "horizontal" sequences and 3 "vertical" ...

**2**

votes

**1**answer

81 views

### A Possible characterization of F.D or AF commutative $C^{*}$ algebras

By F.D or AF $C^{*}$ algebra,we mean finite dimensional or approximately finite dimensional $C^{*}$ algebra.
Let $A$ be a unital commutative $C^{*}$ algebra with the property that for every ...

**4**

votes

**1**answer

110 views

### A Hilbert-space completion of a Hilbert $ C^{*} $-module over a separable $ C^{*} $-algebra

Let $ B $ be a separable $ C^{*} $-algebra and $ \mathcal{E} $ a Hilbert $ B $-module. We know that $ B $ has a faithful state $ \phi $. Using $ \phi $, we can construct a $ \mathbb{C} $-valued ...

**2**

votes

**1**answer

111 views

### Eigenvalues and Compact Resolvent

For $A$ an unbounded (densely defined) operator on a separable Hilbert space, what conditions on its eigenvalues will show that, for $\lambda \notin $spec$(A)$, we have that $(A-\lambda)^{-1}$ is a ...

**0**

votes

**0**answers

107 views

### Bounded operators with infinite matrix representations

I asked this question on StackExchange originally, but I'm giving it a go here as well.
Suppose that $A$ is a unital $C^*$-algebra, $\varphi\colon A\to B(H)$ is a unital, completely positive map and ...