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

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### How is the shape of $A+R+T$?

Put $C=C_{0}(\mathbb{R})$
According to Gelfand duality between locally compact spaces and commutative $C^{*}$ algebras and motivated by the following question, we construct some new ...

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### $V(A)\cong {\mathbb N}\cup\{0\}$ and $$V(A_+)\cong\{(m,n)\in {\mathbb Z}^2 \mid m,n \geq 0, \hbox{ $m+n$ even}\}. $$ [closed]

In Professor Blackadar's book "K theory for operator algebras", there is an example in Chapter 3, $K_0$-theory and order:
Let
$$ A=\{f :[0,1]\to M_2 \mid f(0)={\rm diag}(x,0), f(1)={\rm ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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, ...

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### 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 = ...

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### 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 = ...

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### 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 ...

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### 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 ...

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### 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 ...

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

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

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### 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 ...

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### 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 ...

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### 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, ...

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### 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 ...

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### 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 ...

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### 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
...

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### 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 ...

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### 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 ...

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### 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: ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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

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

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### 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" ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### Morita Equivalence of Full Corners in $C^*$-algebras

Suppose $\mathcal{A}$ is a $C^*$-algebra with a unique normalized trace and $p \in \mathcal{A}$ is a projection so that $\mathcal{B} = p\mathcal{A}p$ is a full corner.
Does $\mathcal{B}$ have a ...

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### Definition of homotopy between Kasparov modules

I'm trying to understand the definition of homotopy between Kasparov modules as presented in Blackadar's book on K-theory for operator algebras. $A,B$ will be C*-algebras, while $E$ will denote a ...

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### Why does infinite tensor product associated with some vectors in the operator algebras?

I notice that in the definition of infinite tensor product in the operator algebras, such as C*-algebras and W*-algebras, every component in the product is associated with a vector(or s state) and the ...

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### Status of the analog of the Haar measure on quantum groups

In (Masuda, Nakagami, Woronowicz)'s paper, in the introduction, the authors mentioned the deficiency common to both their and (Kusterman, Vaes)'s approach regarding the Haar state (or Haar measure ...

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### A perturbation question for the intersection of C*-subalgebras

This feels like something I may have asked before (in which case, apologies) and it also might be some kind of "standard counterexample in a book on C* algebras".
Let M be a unital C*-algebra and let ...

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### One dimensional foliation of surfaces with prescribed graph of foliation

According to the definition of the graph of a foliation by Winkelnkemper we ask the following questions:
Let $G$ be one of the following non hausdorff 3 dim manifold
1) $G$ is a ...

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### Separability of the C*-algebra in the definition of K-homology

There are (at least) two approaches to K-homoology: one is via the so called dual algebra which is due to Paschke. The second is via the Fredholm modules and is due to Kasparov. In Nigel Higson's book ...

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### Commuting nets for commuting projections

I think this should not be too difficult, but I am not an expert. I did not get an answer on stackexchange.
Let $A$ be a $C$*-algebra and let $p,q\in A^{**}$ be two commuting projections. Then there ...

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### The convolution on a semisimple finite quantum groupoid

Let $\mathbb{A}$ be a finite dim. weak Hopf $C^*$-algebra (or semisimple finite quantum groupoid) and $\hat{\mathbb{A}}$ its dual.
Let the Fourier transform $\mathcal{F}: \mathbb{A} \to ...

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### Reference request on operator matrices [closed]

I'm looking for a reference on linear, bounded, self-adjoint operators defined on the product space, $T:E\times F\to E\times F$ such that
$$Tx = \begin{pmatrix}A & B \\
C & D
...

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### An unconventional definition of the $ C^{*} $-algebraic reduced crossed product

Let $ (A,G,\alpha) $ be a $ C^{*} $-dynamical system, i.e., $ A $ is a $ C^{*} $-algebra, $ G $ is a locally compact Hausdorff group and $ \alpha $ is a strongly continuous action of $ G $ on $ A $ by ...

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### Morphisms associated to measured spaces [duplicate]

In a previous discussion (von neumann algebras and measurable spaces), the connexion between von Neumann algebras and localized measured spaces was clarified. I would like to have a category theory ...

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### Is this left ideal of C*-algebra principal?

This is a follow up of this question. Let $I$ be closed left ideal of $C^*$-algebra $A$.
Assume we are given a sequence of left $A$-module morphisms $R_n:I\to A$ with $\sum_n \Vert ...

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### Is a finite depth-index irreducible subfactor, intermediate of a depth ≤ 3 one?

Let $(N \subset M)$ be a finite depth-index irreducible subfactor.
Main question: Is $(N \subset M)$ the intermediate of a finite index depth $\le 3$ irreducible subfactor?
(In others words, is ...