The c-star-algebras tag has no wiki summary.

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### 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). ...

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### 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 ...

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### 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 ...

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### 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 ...

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### $c_0$-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 ...

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### 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 ...

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### 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}}
...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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. ...

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### 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 ...

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### 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 ...

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### 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. ...

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### 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 ...

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### 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 ...

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### 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, ...

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### 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 ...

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### 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 ...

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### 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*
...

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### 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: ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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
...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### topology on the automorphism group of a C* algebra

Let $A$ be a $C^*$-algebra. The group $Aut(A)$ of $\ast$-automorphisms of $A$ is usually equipped either with the pointwise norm topology, i.e. the topology generated by the semi-norms $\lVert \varphi ...

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### Do separable $C^*$-algebras form a set?

The question is in subject.
Update: See Andreas Thom's answer.

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### Is every action from an amenable group amenable on a unital $C^*$-algebra?

Is every action from an amenable group amenable on a unital $C^*$-algebra?

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### Range of completely positive projection

Let $A$ be a C*-algebra. Suppose that $P:A \rightarrow A$ is a contractive completely positive projection. Does the range $P(A)$ is completely order isomorphic to a $C^*$-algebra?
In the case where ...

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### Continuity of functors under inductive sequence of $C^*$-algebras.

We know the fact that $K_0(-)$ and $K_1(-)$ are continuous under inductive sequence of $C^*$-algebras (in fact inductive system), i.e. $K_0(\varinjlim A_n)=\varinjlim K_0(A_n)$ similar for $K_1(-)$. ...

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### Given a C-star dynamical system and a subgroup of the acting group, is the corresponding map on crossed product algebras necessarily an injection

Let $(A,\alpha, G)$ be a $C^*$-dynamical system, where $G$ is a discrete group. Let $\Gamma$ be a subgroup of $G$, then we can form two universal crossed products $A\rtimes_\alpha \Gamma$ and ...

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### k_0 group for $M(J\otimes K)=0$

Let J be any C*-algebra and K be the C*-algbra of compact operators on a separable, infinite dimensional Hibert space.
How to show $K_0(M(J\otimes K))=0$, where M denotes Multiplier algebra

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### Finite dimensionality of certain $C^{\star}$-algebras

In the discussion about the question Finite-dimensional subalgebras of $C^{\star}$-algebras the following separate question came up:
Let $H$ be a Hilbert space and $a_1, \dots, a_n \in B(H)$ be ...

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### What does the representation theory of the reduced C*-algebra correspond to?

Let $G$ be a locally compact group. The group C*-algebra $C^* (G)$ is designed to come with a natural bijection between its (nondegenerate) representations and the (strongly continuous, unitary) ...

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### What is the commutative analogue of a C*-subalgebra?

Using the duality between locally compact Hausdorff spaces and commutative $C^*$-algebras one can write down a vocabulary list translating topological notions regarding a locally compact Hausdorff ...

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### Ideal of “Compact Operators” in a W*-algebra which gives the sigma-strong-* topology.

In the case of bounded operators on a Hilbert space $\mathcal{H}$, $L(\mathcal{H})$, there are multiple descriptions of the $\sigma$-strong-* topology, namely:
1) As given by seminorms ...

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### Finite-dimensional subalgebras of $C^\star$-algebras

Let $A$ be a unital $C^\star$-algebra and let $a_1,\dots,a_n$ be a finite list of normal elements in $A$ which (together with their adjoints) generate a norm-dense $\star$-subalgebra $B \subset A$. ...

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### Conjugacy classes and reduced group $C^*$-algebra of an amenable group

The reduced $C^*$-algebra of a non-abelian free group $G$ has a unique trace. Hence, there is no chance to separate conjugacy classes of group elements using traces on $C^\star_{red} G$. On the other ...

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### Surjective *-homs between multiplier algebras

Let A and B be C*-algebras, and let $\phi:A\rightarrow B$ be a surjective *-homomorphism. Then $\phi$ is non-degenerate, and so we can extend it to *-homomorphism between the multiplier algebras: ...

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### The functoriality of group C* algebra structure

Hi!
Let G,H be discrete groups and f:G->H be any homomorphism of these groups. I have three questions about it:
1) how to prove the functoriality of the construction of universal C*-algebra of ...

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### Is there a good reference for studying the ideal structure of group C* algebras?

Is there a good reference for studying the ideal structure of group C* algebras?
Thanks.

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### Relative K-theory and split exact sequences of C* algebras

Let $A$ be a C* algebra, $J$ an ideal, $\pi: A \to A/J$ the quotient map. Recall that the relative K theory group $K_0(A, A/J)$ consists of equivalence classes of triples $(p,q,x)$ where $p$ and $q$ ...

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### Must we close weakly to apply the spectral theorem?

Let $H$ be an infinite dimensional separable complex Hilbert space. All C*-subalgebras of $B(H)$ are assumed to be non-degenerate.
The spectral projections of a self-adjoint element $T$ of $B(H)$ lie ...

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### Simple inequality in C*-algebras

Sorry the title is a bit vague. Let A be a C*-algebra, and let x and y be positive elements in A. Is it true that $$ \|x-y\|^2 \leq \|x^2-y^2\|? $$
Well, yes. But the proof I have is a bit of a ...