**6**

votes

**2**answers

524 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

420 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 ...

**6**

votes

**1**answer

205 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

426 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

313 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

397 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

236 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

657 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 ...

**11**

votes

**0**answers

457 views

### Properties of orthogonality-preserving c.p. maps between $C^*$-algebras

Suppose that $A,C$ are $C^*$-algebras and $\phi:A \to C$ is a completely positive, orthogonality-preserving linear map.
(Orthogonality preserving means: if $a,b \in A$ satisfy $ab=0$ then ...

**8**

votes

**1**answer

373 views

### The Haar state on compact quantum groups $A_u(Q)$ and $A_o(Q)$

Let $Q\in GL_n(\mathbb{C})$. The free unitary quantum group is the universal $C^*$-algebra $A_u(Q)$ with generators $u_{ij},1\leq i,j\leq n$ and relations making $u=(u_{ij})$ as well as ...

**2**

votes

**1**answer

286 views

### when does a $C^*$-algebra have no nonzero unital quotient?

In their paper: "Addition of $C^*$-algebra extensions", G. A. Elliott and D. E. Handelman have discussed some relation between traces and equivalence of projections in $M(A)$, where $M(A)$ is the ...

**1**

vote

**2**answers

358 views

### Questions about special $C^*$-subalgebras and ideals.

Let $A$ be a $C^*$-algebra and $I$ be a two side closed (essential) ideal of $A$. Suppose that $p \in A\backslash I$ is a non trivial projection. Let $B=pIp$. My questions are:
(1) Is $B$ a ...

**4**

votes

**1**answer

268 views

### What's the link between the Toeplitz operators on H^2 and those used to define Cuntz-Pimsner algebras?

An alternate way to phrase this question might be, "How did the Toeplitz operators used in the definition of the Cuntz-Pimsner algebra come by their name?" or, "What's the relationship between the ...

**8**

votes

**1**answer

773 views

### Matrices with entries in a $C^*$-algebra

Let $\mathcal{A}$ be a $C^\ast$-algebra. Consider vector space of matrices of size $n\times n$ whose entries in $\mathcal{A}$. Denote this vector space $M_{n,n}(\mathcal{A})$. We can define involution ...

**10**

votes

**7**answers

1k views

### Positive but not completely positive?

The only example I know of a positive map which is not completely positive is the transpose map on $M_n(\mathbb{C})$. Of course, one can come up with minor perturbations of this (compose it with, or ...

**10**

votes

**2**answers

570 views

### C*-algebras with bizzarre structure of projections

This is probably well-known to the experts but I could not find any answer neither in my head nor in the literature: Is there a (unital) C*-algebra such that its projections do not form a lattice ...

**2**

votes

**2**answers

310 views

### ED compact $K$ such that $C(K)$ is not a dual Banach space

Almost every mathematician working in von Neumann algebras says that there are certainly extremely disconnected compact spaces $K$ such that $C(K)$ is not a dual space (they are not hyperstonean). ...

**2**

votes

**1**answer

652 views

### Centralizers in C*-algebra

Let $a,b\in A$ be self-adjoint elements in $C^*$-algebra $A$ with equal centralizers, $\{x\in A; [a,x]=0\}=\{x\in A; [b,x]=0\}$. Can we say anything about the correspondence between $a$ and $b$?
For ...

**7**

votes

**1**answer

306 views

### Are the Drinfeld compact quantum groups simply connected ?

To fix notations : let G be simply connected simple compact group, and $U_q(\mathcal{G})$ the Drinfeld-Jimbo universal algebra quantization of its complexified algebra defined as usual, with q not ...

**6**

votes

**3**answers

869 views

### A variant of the Stone-Weierstrass theorem?

I would like to ask specialists in C*-algebras if the following variant of the Stone-Weierstrass theorem is true.
Suppose $A$ is a C*-algebra and $C$ is its center. Since $C$ is a commutative ...

**6**

votes

**1**answer

391 views

### $c_0$-direct sum of $\mathcal{K}(\mathcal{H})$

Let $\mathcal{K}(\mathcal{H})$ be the C*-algebra of compact operators on a Hilbert space $\mathcal{H}$. I am interested in the ($c_0$-)sum
$A=\sum \mathcal{K}(\mathcal{H})$
of countably many ...

**7**

votes

**5**answers

961 views

### Do subalgebras of C(X) admit a description in terms of the compact Hausdorff space X?

In light of the well-known theorem of Gelfand that, bluntly put, ends up saying that unital abelian C*-algebras are the 'same' as compact Hausdorff topological spaces, I tried to compile a dictionary ...

**6**

votes

**0**answers

392 views

### References for “folklore” on strong amenability of (group) C*-algebras?

[Apologies in advance for the prolixity - but I was unsure how much of the story is familiar.]
$\newcommand{\ptp}{\widehat{\otimes}}
\newcommand{\co}{\operatorname{co}}
...

**1**

vote

**1**answer

357 views

### Algebraically simple Banach algebras

There are plenty of semi-simple Banach algebras - this broad class includes C*-algebras and algebras of bounded operators on a given Banach space. On the other hand, it seems unlikely to me that there ...

**3**

votes

**2**answers

426 views

### Galois cover via C star algebras

Hello to all, here's my question, I hope it's not too trivial. I haven't found reference for it so far.
We know that abelian C star algebras are the same as locally compact spaces.
Now what is the ...

**36**

votes

**1**answer

2k views

### What is the current status of the Kaplansky zero-divisor conjecture for group rings?

Let $K$ be a field and $G$ a group. The so called zero-divisor conjecture for group rings asserts that the group ring $K[G]$ is a domain if and only if $G$ is a torsion-free group.
A couple of good ...

**7**

votes

**0**answers

306 views

### Quantum Braid Group

Recently I learned the definition of the quantum permutation group $A_s(n)$, which starts from the fundamental representation by permutation matrices and exchanges the entries by noncommuting ...

**20**

votes

**2**answers

828 views

### Does left-invertible imply invertible in full group C*-algebras (discrete case)?

The following question/problem has been bugging me on and off for some time now: so I thought it might be worth broaching here on MO, as a case of "ask the experts".
Let $G$ be a discrete group. ...

**2**

votes

**1**answer

395 views

### proper action and amenable action

We say that an action of a (discrete) group G on a locally compact space X is called proper if the map from $G\times X$ to $X\times X$ defined by $(g,x)\mapsto (gx,x)$ is proper. Why is a proper ...

**2**

votes

**0**answers

240 views

### Six term exact sequence In E-theory

I just want to know whether the two six term exact sequences in E-theory is true for nonseparable C*-algebras. We know already if the first varible is complex number, then we get six term exact ...

**4**

votes

**1**answer

402 views

### Crossed product of a non unital C*-algebra

Let $X$ be a locally compact space, and let $T:X\rightarrow X$ be a homeomorphism. Then \begin{align*}
&\alpha:C_0(X)\rightarrow C_0(X)\\\
&\alpha(f)=f\circ T
\end{align*}
is an automorphism. ...

**1**

vote

**1**answer

471 views

### Pairwise orthogonal projections in C*-algebras

Is every non-zero projection in a C*-algebra $A$ a supremum or infimum (at least majorized by / majorizes) a family of pairwise orthogonal non-zero projections in $A$?
PS. Are there any cheap ways ...

**8**

votes

**2**answers

670 views

### Solving the equation $xax=b$ in a C*-algebra.

Let $a, b\in A_+$ be positive elements of some C*-algebra $A$.
Assume furthermore that $a$ is invertible.
Is it true that
$$
\exists! x\in A_+\quad:\quad xax=b\quad ?
$$
Already in the case ...

**1**

vote

**1**answer

255 views

### Products on the K-theory of graded C*-algebras

One can define products on the K-theory of graded C*-algebras as in
http://web.me.com/ndh2/math/Papers_files/Higson,%20Guentner%20-%202004%20-%20Group%20C*-algebras%20and%20K-theory.pdf
on page 152, ...

**6**

votes

**2**answers

983 views

### Space of compact operators

I am interested in the Banach space $\mathcal{K}=\mathcal{K}(\ell^2)$ of compact operators on $\ell^2$, however my questions can be stated for any $\mathcal{K}(E)$, where $E$ is an arbitrary Banach ...

**18**

votes

**2**answers

801 views

### Commutators in the reduced C*-algebra of the free group

Is it known whether any element of trace 0 in the reduced $C^*$-algebra of a non-abelian free group, is a limit of sums of (additive) commutators?

**0**

votes

**1**answer

261 views

### The stabilized homotopy category of graded C* algebra

Hi everyone
On page 147 of the note "Group C*-Algebras and K-theory" by N.Higson and E.Guentner there are something about the stabilized homotopy category of graded C* algebra, which is a category ...

**2**

votes

**1**answer

349 views

### When C(K) is closed in sigma strong topology?

Fix a compact Hausdorff space $K$ and think about $C(K)$ as a C*-algebra acting on a Hilbert space $H$. Suppose that $C(K)$ is closed in $\mathcal{B}(H)$ in:
$\sigma$-strong
$\sigma$-strong*
...

**0**

votes

**0**answers

170 views

### Grading on Multiplier Algebras

A graded C*-algebra A is inner-graded if there exists a self-adjoint unitary $\varepsilon$ in the multiplier algebra M(A) of A which implements the grading automorphism $\alpha$ on A: ...

**3**

votes

**1**answer

300 views

### Graded $C^*$-algebras can be faithfully represented on a graded Hilbert space

Hi everyone
I try to use GNS-construction to show every graded C*-algebras can be faithfully represented on a graded Hilbert space.
If $A$ is a graded C*-algebra with grading automorphism $\alpha$ of ...

**1**

vote

**2**answers

659 views

### Why is this a conditional expectation into the fixed point algebra?

Let $A$ be a C*-algebra and let $\alpha$ be an action of the circle group $S_1$ on $A$ (Gauge action).
We define the following map:
$$E:A\rightarrow A;\quad E(a):=\int\alpha_t(a)\textrm{d} t.$$
My ...

**7**

votes

**1**answer

603 views

### When are certain group C*-algebras exact?

This is somewhere between a "reference request" and "ask an expert", but I hope it is not too trivial or off-topic.
Anyway. There has been a lot of attention given to showing that for certain ...

**1**

vote

**1**answer

622 views

### Strict positivity in dense subalgebras of $C^{*}$-algebras

Let $A$ be a $C^{*}$-algebra, represented on a Hilbert space $H$, and $D$ a selfadjoint unbounded operator on $H$ (note that we do not impose that $D$ have compact resolvent). Let
...

**5**

votes

**1**answer

283 views

### Is the unitary group of $l^2(A)$ with the strict topology contractible?

Let $A$ be a $C^*$-algebra with countable approximate unit. Let $\mathbb{K}$ denote the compact operators on a separable Hilbert space. Mingo and later Cuntz and Higson have shown that the unitary ...

**4**

votes

**2**answers

1k views

### Non degenerate representations for C*-algebras

Hi!
While studying C*-algebras I found 2 different definitions for non degenerate representations (-homomorphisms $\pi:\mathcal{A} \rightarrow B(\mathcal{h})$ where $\mathcal{A}$ is a C-algebra and ...

**4**

votes

**0**answers

358 views

### What is the spectrum of the commutative C*-algebra I have constructed here?

Let $B$ and $F$ be compact Hausdorff spaces.
Let $E\to B$ be a fiber bundle with fibre $F$ and structure group $\mathrm{Homeo}(F)$, the group of homeomorphisms of $F$.
I think this induces a fiber ...

**12**

votes

**4**answers

2k views

### Reference: Learning noncommutative geometry and C^* algebras

I am starting to study noncommutative geometry and C^* algebras so my question is
Does anyone knows a good reference on this subject?
I would like a basic book with intuitions for definitions and ...

**1**

vote

**0**answers

175 views

### Fredholmness and invertibility in a C* algebra generated convolution-type operators

Let $PC$ be the algebra of complex-valued, piecewise-continuous functions from $[-\infty,+\infty]$, $SO$ be the algebra of bounded, continuous, complex-valued functions on $\mathbb R$ which are slowly ...