**3**

votes

**2**answers

214 views

### When is a groupoid the path groupoid of a graph?

I am actually interested in the $C^*$-algebras, so perhaps my question should be: How can you recognize whether a $C^*$-algebra $A$ is isomorphic to $C^*(\Lambda)$ for some (higher-rank) graph ...

**7**

votes

**0**answers

325 views

### The approximation property of group C*-algebras

Let $G$ be a discrete group. Then the group C*-algebra $C^*(G)$ is nuclear if and only if $G$ is amenable. I am wondering whether nuclearity of $C^*(G)$ can fail for a Banach-space reason, namely due ...

**1**

vote

**0**answers

214 views

### Is an exact operator, unitary equivalent to a banded operator?

Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators.
Definition : Let $(e_{n})_{n \in \mathbb{N}}$ be an orthonormal basis.
$T \in B(H)$ is ...

**11**

votes

**1**answer

513 views

### Is there a proof that the $C^{*}$-algebras don't see the invariant subspace problem?

This post is an appendix of this one.
Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators.
Invariant subspace problem: Let $T \in B(H)$. Is ...

**3**

votes

**0**answers

363 views

### About generator and isomorphism problems for free groups operator algebras

Let $H$ be an infinite dimensional separable Hilbert space.
The $C^{*}$-algebras and von Neumann here unital and subalgebras of $B(H)$.
Definition : Let $\mathcal{A}$ be $C^{*}$-algebra (resp. a ...

**15**

votes

**2**answers

431 views

### automorphisms of C*-algebras and partial isometries

Let $A$ be a $C^*$-algebra, let $p$ and $q$ be Murray-von Neumann equivalent projections in $A$, i.e. there is a partial isometry $v$ such that $v^*v = p$ and
$vv^* = q$. Suppose $\alpha \in Aut(A)$ ...

**2**

votes

**1**answer

337 views

### Is there an operator algebraic reformulation of the invariant subspace problem?

Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators.
Invariant subspace problem: Let $T \in B(H)$. Is there a non-trivial closed $T$-invariant ...

**4**

votes

**0**answers

457 views

### Do the banded operators check the invariant subspace problem?

Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators.
Invariant subspace problem: Let $T \in B(H)$. Is there a non-trivial closed $T$-invariant ...

**6**

votes

**2**answers

600 views

### Structure theorem for finite dimensional $C^*$-algebras and their representations

I would like a source for some Artin-Wedderburn type facts about these algebras which seem to have easy proofs, and are probably written somewhere.
Let $\mathcal{A} \subset M_n(\mathbb{C})$ be an ...

**5**

votes

**0**answers

169 views

### Does the suspension isomorphism $K_1(A) \to K_0(SA)$ descend from a more refined invariant?

If $A$ is a C*-algebra, denote its minimal unitization by $\tilde A$ and its suspension by $SA$, thought of as all continuous $a:[0,1] \to A$ with $a(0)=a(1)=0$. The unitized suspension ...

**5**

votes

**1**answer

350 views

### Is there an irreducible, noncompact commuting, nonnormal operator, with spectrum strictly continuous?

Let $H$ be an infinite dimensional separable Hilbert space.
Definition: The commutant $\mathcal{S}'$ of a subset $\mathcal{S} \subset B(H)$ is $ \{A \in B(H) : AB=BA \ , \ \forall B \in \mathcal{S} ...

**5**

votes

**0**answers

99 views

### Equivariant zero dimensional extension recovering a given measure

Let $X$ be a compact metrizable space and $\alpha: \mathbb{Z}^d\curvearrowright X$ a continuous group action. Then it is well known that there exists a zero dimensional compact space $Y$, an action ...

**3**

votes

**2**answers

229 views

### Tensoring with a CAR-algebra

Let $A$ and $B$ be two unital infinite-dimensional simple separable nuclear $C^{\ast}$-algebras and let $C$ be a CAR-algebra. When does $A\otimes C \simeq B\otimes C$, imply $A\simeq B$?
The answer ...

**4**

votes

**2**answers

352 views

### Projections in a W*-algebra as a continuous lattice?

A continuous lattice is a complete lattice $L$ in which every element $y$ is equal to $\bigvee${$x \in L \mid x \ll y$} where $x \ll y$ ("x approximates y" or "x is way below y") if for any directed ...

**0**

votes

**0**answers

103 views

### All AI-algebras are AT-algebras

It is known that every AI-algebra (i.e. inductive limit of interval algebras) is an AT-algebra (i.e. inductive limit of circle algebras)?
This seems a little bit odd because a building block of an ...

**2**

votes

**1**answer

103 views

### References on countable W*-algebras

In "Operator algebras with a faithful weakly-closed representation" (1955), Kadison describes a countable W*-algebra as a C*-algebra which has a faithful representation as a countably decomposable ...

**3**

votes

**1**answer

444 views

### C* Algebras, Foliations and Dynamical Systems

I am a Ph.D student involved in topics like integrability of foliations arising from center stable bundles of partially hyperbolic dynamical systems. These are generally only continuous bundles so one ...

**2**

votes

**1**answer

225 views

### Inner automorphisms and $K$-theory

It is known that any inner automorphism of a unital $C^{\ast}$-algebra $A$ induces the identity map on $K_{0}(A)$ because unitary equivalence implies Murray-von Neumann equivalence. What is known ...

**0**

votes

**1**answer

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

**2**

votes

**1**answer

251 views

### Tensor product of C*-algebras of bounded, uniformly continuous functions on metric spaces

This is a follow up question to this one.
If $X$ is a metric space, denote by $C_u(X)$ the $C^\ast$-algebra of all bounded, uniformly continuous functions on $X$ (with the sup-norm).
Do we have ...

**22**

votes

**1**answer

456 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' ...

**1**

vote

**0**answers

309 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 ...

**7**

votes

**2**answers

267 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 ...

**3**

votes

**1**answer

292 views

### Amenability at infinity

I have a few questions about amenability at infinity for locally compact, second countable, Hausdorff topological groups. Recall that a locally compact group $G$ is said to be amenable at infinity if ...

**7**

votes

**1**answer

358 views

### Trace Class Functions on locally compact groups

Let $G$ be a locally compact subgroup, $\mu$ a Haar-measure.
For $f \in L^1(G)$, and for $\pi$ a unitary, topology irreducible, representation of $G$ on
an Hilbert space $H_\pi$, it is customary to ...

**6**

votes

**1**answer

251 views

### What are the sub $C^*$-algebras of $C(X,M_n)$?

Let $X$ be a locally compact Hausdorff topological space, denote by $M_n$ the $C^*$-algebra of complex $n\times n$ matrices, by $C_0(X,M_n)$ the $C^*$-algebra of continuous functions on $X$ with ...

**7**

votes

**2**answers

450 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

959 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 ...

**4**

votes

**1**answer

322 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

397 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. ...

**8**

votes

**2**answers

318 views

### Injectivity of the Baum-Connes assembly map for locally compact groups

Skandalis, Tu and Yu in "The coarse Baum-Connes conjecture and groupoids" proved that:
Let $\Gamma$ be a countable group with a proper left-invariant metric $d$. If $\Gamma$ admits a uniform ...

**2**

votes

**0**answers

104 views

### Is the Poincare action on the Klein-Gordon quantum field strongly continuous?

I am interested in checking continuity property of the Poincare group action on the Klein-Gordon quantum field theory defined over the Minkowski spacetime. Maybe the simplest example of QFT out there.
...

**2**

votes

**1**answer

181 views

### Ideal spanned by matrix units isomorphic to compact operators

Hello,
Assume we have $(n+1)$ isometries $S_1,...,S_{n+1}$ in the separable Hilbert space $H$ with the properties that $\sum_{i=1}^{n+1}S_iS_i^*=I, S_i^*S_j=0$ (i.e. $S_i$ are the generators of the ...

**0**

votes

**1**answer

280 views

### Commutant of a von Neumann algebra as the linear span of unitaries.

I'm reading chapter 4 of Gerard Murphy's C*-algebras book and am confused by a statement in his proof of theorem 4.1.10. In his proof, he says, "$A'$ is the linear span of its unitaries" (where $A'$ ...

**10**

votes

**2**answers

380 views

### C*-algebras with no nontrivial endomorphisms

Pick a C*-algebra $A$ and call a (*-)endomorphism $\alpha:A\to A$ nontrivial if it is injective and $\alpha(A)\neq A$.
Question: Do there exist infinite dimensional C*-algebras with no nontrivial ...

**9**

votes

**0**answers

344 views

### Tensorial decomposition of $B(H)$

Let $H$ be an infinite-dimensional Hilbert space and let $\mathcal{B}(H)$ be the (C*/W*-)algebra of bounded operators on it. Actually, you may forget about the involution in $\mathcal{B}(H)$ because I ...

**11**

votes

**3**answers

509 views

### Von Neumann algebra associated to the infinite Cuntz algebra

The Cuntz algebra $\mathcal{O}_{\infty}$ is the universal $C^*$-algebra generated by countably infinitely many isometries $s_i$ satisfying the relations $s_i^*s_j = \delta_{ij}$ (there is no condition ...

**10**

votes

**1**answer

243 views

### Which W*-algebras are the duals of C*-coalgebras?

A Banach algebra (assumed associative and unital) is precisely a monoid object in the monoidal category of Banach spaces, short linear maps, and the projective tensor product. A Banach coalgebra is ...

**3**

votes

**2**answers

478 views

### Do semi-continuous functions generate bounded Borel measurable functions as a $C^*$-algebra?

This question is related to Question 2 of my previous posting.
Question. Let $\mu$ be a Radon measure on a compact Hausdorff space $\Omega$ and $L^{\infty}(\Omega,\mu)$ the set of essentially bounded ...

**4**

votes

**0**answers

216 views

### Extensions of completely positive mappings

I would like to ask the following two questions.
Let $1_{\mathcal{H}}\in \mathcal{A}\subset\mathcal{B}\subset\overline{\mathcal{A}}^{SOT}\subset\mathbb{B}(\mathcal{H})$ be a sequence of ...

**6**

votes

**2**answers

517 views

### The monotone closure of a $C^*$-algebra

Related to Jon's question, I have two questions. Let $\mathcal{A}$ be a concrete $C^*$-algebra on a Hilbert space $\mathcal{H}$. For any selfadjoint subset $S$ of $\mathbb{B}(\mathcal{H})$, let $S^m$ ...

**2**

votes

**1**answer

209 views

### Function spaces over pseudocompact spaces

Let $K$ be a pseudocompact Tychonoff space that is, a Hausdorff $T_{3.5}$ space for which every continuous function $\varphi \colon K\to \mathbb{C}$ is bounded. Let $\beta$ be the Stone-Cech extension ...

**5**

votes

**2**answers

414 views

### Bounded linear functionals and representations

Suppose that $A$ is a unital C$^*$-algebra and that $\varphi: A \to \mathbb{C}$ is a bounded linear functional. Then there exists a Hilbert space $H$, a representation $\pi: A \to B(H)$ and vectors ...

**5**

votes

**1**answer

203 views

### States/functionals on crossed product C*-algebras

Let $A$ be a C*-algebra, $\alpha$ a strongly continuous automorphic action by a locally compact group $G$ on $A$, and consider the crossed product $A\rtimes_\alpha G$. I am looking for references ...

**11**

votes

**1**answer

423 views

### Kuiper's theorem via approximation

Kuiper's theorem says that the unitary group $U(H)$ of a separable infinite dimensional Hilbert space $H$ is contractible, if it is equipped with the norm topology.
Let's suppose, I do not know this ...

**3**

votes

**1**answer

311 views

### When an AW*-algebra is a W*-algebra

In a very old book of Kaplansky "Rings of operators", on p. 123 one can find the following sentence:
It is a standing conjecture that an AW${}^\ast$-algebra is W${}^\ast$ if its center is W${}^\ast$.
...

**0**

votes

**1**answer

201 views

### simultaneously Approximated by self-adjoint elements.

We know that a $C^*$-algebra $A$ has real rank zero iff every self-adjoint element of $A$ can be approximated in norm by self-adjoint element with finite spectra. My question is:
If we have two ...

**3**

votes

**2**answers

389 views

### Polar decomposition in C*-algebras

A very nice feature of W*-algebras is the following:
once you have an element $a$ of a W*-algebra $M$, and $a=u|a|$ (the polar decomposition), then $u\in M$.
It seems that it carries over to ...

**4**

votes

**1**answer

235 views

### Abelian sub-W*-algebras

Let $M$ be a von Neumann algebra which acts faithfully on a Hilbert space of density character $\kappa$ but does not on a Hilbert space of density character $\lambda<\kappa$ (that is, the density ...

**10**

votes

**1**answer

650 views

### Does the hyperfinite II_1 factor admit two irreducible representations that are not unitarily equivalent?

Regarding the hyperfinite $II_{1}$ factor $R$ as $C^{*}$-algebra, is it known whether any two irreducible representations of $R$ are unitarily equivalent? If it is known that there exists a pair of ...