# Questions tagged [c-star-algebras]

A C*-algebra is a complex Banach algebra together with an isometric antilinear involution satisfying (a b)* = b* a* and the C*-identity ‖a* a‖ = ‖a‖². Related tags: [tag:banach-algebras], [tag:von-neumann-algebras], [tag:operator-algebras], [tag:spectral-theory].

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### center of a $C^*$-algebra

Does there exist a $C^*$-algebra $A$ such that the center of $A$ is $0$ and $A$ also has a tracial state?
I know the fact that the center of $\mathcal{K}(H)$ is $0$, but $\mathcal{K}(H)$ has no ...

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### A relation between Hochschild cohomology of a $C^*$ algebra and its bidual

Let $A$ be a $C^*$ algebra and $A''$ be its bidual with the Arens product.
Is there any relation between the Hochschild cohomolgy of $A$ with complex coefficients and the Hochschild cohomology of $A''$...

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121 views

### irreducible representation of a $C^*$ algebra

Suppose we have a $C^*$ algebra $A=\{(x_n)\in \prod M_n(\Bbb C),lim_n tr_n(x_n^*x_n)=0\}$.
If $B$ is any nonzero $C^*$-sub algebra of $A$,does there exist a finite dimensional irreducible ...

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### tracial states of a corona algebra

Suppose $I=\bigoplus_nM_{k(n)}(\Bbb C)$,there are many tracial states on $M(I)/I$,where $M(I)$ is a multiplier algebra of $I$. For any free ultrafilter $\omega$ on $\Bbb N$ ,we can construct a tracial ...

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### If $\ $ $yx_n\to 0 $ for all $y$ in a C$^*$-algebra, Is it true that $x_n$ is weakly convergent to $0$?

$A$ is a C$^*\! $-algebra and $(x_n)_{n\in \mathbb{N}} \subseteq A $.
If $\ $ $yx_n\to 0 $ for all $y\in A$, Is it true that $x_n$ is weakly
convergent to $0$ ?
For unitals this is trivial. ...

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120 views

### Cartan subalgebras in the group algebras of virtually abelian groups

Let $G$ be a virtually abelian group. Are there any general results on the existence or non-existence of Cartan subalgebras in the generated group $C^*$-algebra or group von Neumann algebra?

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### construct a non-unital nuclear $C^*$ algebra

Let $I=\bigoplus_n M_n(\Bbb C)$,can we construct a non-unital $C^*$ algebra $A$ such that $I$ is essential in $A$ and $A/I\cong K(H)$ for some separable infinite Hilbert space.
[note added by YC: ...

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110 views

### construct a nuclear $C^*$ algebra

Can we construct a non-unital nuclear $C^*$ algebra $A$ such that $I=\bigoplus_n M_n(\Bbb C)$ is an essential proper ideal in $A$?

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### Examples of non-isomorphic $C^\ast$ algebras with isomorphic quasi-state spaces

Let $A$ (resp. $B$) be a unital $C^\ast$-algebra, $\mathcal{Q}(A)$ (resp. $\mathcal{Q}(B)$) the compact convex subset of $A^\ast$ equipped with the $\sigma(A^\ast, A)$ (resp. $\sigma(B^\ast, B)$) ...

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### Semi-continuous fields of C*-algebras having dimension one on a dense set

Given a Hausdorff, locally compact space $X$, let us consider a semi-continuous field $\{A_x\}_{x\in X}$ of C*-algebras over $X$, such that $A_x$ is one-dimensional for every $x$ in a dense subset $D$ ...

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### Topological amenability vs amenability of an action

Let $G$ be a discrete group and let $X$ be a compact, Hausdorff space.
Assume that $G$ acts on $X$ by homeomorphisms.
Consider the following two definitions:
[$C^*$-algebras and finite dimensional ...

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### Extension of a von Neumann algebra by a von Neumann algebra

I asked this question at MSE now I repeat it at MO:
Let $A,B,C$ be $3$ unital $C^*$ algebras. Assume that we have the following short exact sequence of $C^*$-algebras:
$$0\to A\to C\to B\...

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### $*$-algebras, completions, and $K$-theory

What is an example of a $*$-algebra $\cal{A}$, which admits two non-equivalent norms $\| \cdot \|_1$ and $\| \cdot \|_2$, with respect to which we can complete $\cal{A}$ to give two $C^*$-algebras $...

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### example of a non-amenable l.c. group such that $C_r^*(G)$ satisfies the UCT

Are there known any examples of non-amenable locally compact (or more restrictive, non-amenable discrete) groups $G$ for which the reduced group $C^*$-algebra $C_r^*(G)$ satisfies the universal ...

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### Is $\ell_2(A)$ a Hilbert C$^*$-module with Opial property?

If $A=Mat_{n\times n}(\mathbb{C}) $, Is $\ell_2(A)$ a Hilbert $A$-module with Opial property?
Opial property: If ($w-\lim x_n=0 $) then $
(\liminf \lVert x_n\rVert<\liminf \lVert x_n-y \...

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### Does a closed right ideal of a C$^*$-algebra have a C$^*$-algebra?

$A$ is an infinite dimensional C$^*$-algebra and $J\subset A$ is a closed right ideal. $A$ and $J$ are infinite dimensional(as a vector space). I want to find an infinite dimensional C$^*$-algebra ...

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### Maps which are both completely positive and positive

Definition:A linear map $f:\mathbb C^n\to \mathbb C^n$ is called positive if $\langle fa,a\rangle\ge0$ for all $a\in \mathbb C^n$. Equivalently, $f\in M_{n}(\mathbb C)$ is positive if it can be ...

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### Group $C^*$ vs group von-Neumann algebras

Let $\Gamma$ be a countable (discrete) group (in what follows, make additional assumptions as you wish). Let $C^*_r(\Gamma)$ and $W^*_r(\Gamma)$ be the reduced $C^*$-algebra respectively the reduced ...

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### A characterisation of certain $C^*$-algebras

I was wondering if there is a characterisation for $C^*$-algebras (unital) for which the bidual does not have any central atoms. It is not sufficient for example to demand that the $C^*$-algebra does ...

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651 views

### What does it mean for a category to admit direct integrals?

Given an infinite countable group $G$, the category of unitary representations of $G$ admits direct integrals.
Namely, given a measure space $(X,\mu)$ and a measurable family of unitary $G$-reps $(H_x)...

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### Behaviour of direct limit with matrices

I am trying to understand direct limits in the category of $C^*$-algebras by self reading. My last question was also related to direct limits. Here is another of my doubts:
Let $(A_n,f_n)$ be a ...

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### Continuity of functions on étale groupoids

Let $\mathcal G$ be an étale groupoid with a locally compact, Hausdorff unit space $\mathcal G^{(0)}$. If $U⊆\mathcal G$ is an open subset, which is Hausdorff in the induced topology, and if $f$ is ...

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### Classification of finite-dimensional real super C*-algebras

The title says it all. I feel like one should be able to find this somewhere, but every time I try to google, I just get results for "super Lie algebras". Does anybody know a reference? I am not so ...

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### Hilbert space separability for spectral triples

A spectral triple $({\cal A},{\cal H},D)$ consists of a unital $*$-algebra ${\cal A}$ represented as bounded operators on a Hilbert space ${\cal H}$, together with an unbounded operator $D$ having ...

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130 views

### Idempotents in Group Algebras

What is known about idempotents in Lie group algebras (such as on the classical Lie groups)? Specifically the self-adjoint ones. Is there anything interesting to say? I haven't been able to find much ...

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### Approximation of unity by projectors

Let $A$ be a $\sigma$-unital $C^*$-algebra and $A_s:=A\otimes K$ its stabilization (where $K$ is the algebra of compact operators on a separable Hilbert space). Is it true that there exist an ...

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### An example of a sequence of finite projections

Let $A$ be a vn-algebra. Suppose that $x$ is an isometry with $\inf_{n\geq1} x^nx^{*n}=0$ (Note that $x^nx^{*n}$ are all projections). Let $e$ be a (non-zero) finite projection and put $q_n$ to be ...

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### Characterizing fullness of a von Neumann algebra by the topology of its bimodules

Let $\mathcal{M}$ be a $\mathrm{II}_1$ factor. Among other characterizations, it is said to be full iff the adjoint map:
$$
\mathrm{Ad}: U(\mathcal{M})/\mathbb{T} \longrightarrow \mathrm{Aut}(\...

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### Does Borel functional calculus commute with *-isomorphism?

I am confused with the underlined equation in the following picture.
I know that a *-isomorphism commutes with continuous functional calculus since every continuous functions on the compact subset of ...

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### Closeness of points in the irreducible decomposition of a C$^{*}$-algebra representation

Suppose $X$ and $Y$ are compact metric spaces. Let $\varphi\colon C(X)\to M_{n}(C(Y))$ be any $*$-homomorphism. If $\pi$ is an irreducible representation of $M_{n}(C(Y))$, then $\pi$ is unitarily ...

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### How rich the group of unitary elements in a von Neumann algebra to get “Murray-von Neumann” equivalence?

Denote by $\sim$ the "Murray-von Neumann" equivalence in the projection lattice of a von Neumann algebra. It is well known that in a finite von Neumann algebra the equivalence of projections can be ...

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### Schröder–Bernstein for representations of operator algebras

This is claimed in a Wikipedia Article:
If two representations (of a $C^*$-algebra $A$) $\rho$ and $\sigma$, on Hilbert spaces $H$ and $G$ respectively, are each unitarily equivalent to a ...

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### Extending maps from dense $*$-algebras of $C^*$-algebras

Given $\cal{A},\cal{B}$ two dense $*$-algebras of two $C^*$-algebras $A$ and $B$ respectively, together with a $*$-algebra homomorphism $f:\cal{A} \to \cal{B}$, is it clear that $f$ extends to a ...

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267 views

### finite dimensional C*-algebras

Let $A$ be a C*-algebra. Suppose that every cyclic representation of $A$ is finite dimensional.
Q. Is $A$ finite dimensional?

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### is the conditional expectation faithful?

Let $G$ be locally compact group and let $H$ be a open subgroup in $G$.
Then the full group $C^*$-algebra of $H$, $C^*(H)$, is a subalgebra of $C^*(G)$ and there is a conditional expectation $$E\colon ...

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155 views

### Generator of $K_0(C_0(\mathbb{C}))$

$\newcommand{\C}{\mathbb{C}}\newcommand{\Z}{\mathbb{Z}}$
I know from Bott-periodicity that $K_0(C_0(\mathbb{C}))\simeq \Z$, is there any easy way to compute an explicit generator of $K_0(C_0(\mathbb{C}...

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### Non-existence of projections in crossed product

If $X$ is a smooth manifold on which a Lie group $G$ acts properly and cocompactly (meaning $X/G$ is compact), then one can find a compactly support cut-off function $c:X\rightarrow\mathbb{R}$ such ...

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### Pure infiniteness of tensor product $C^\ast$-algebras

I found the following theorem (Proposition 4.5) in the paper of E. Kirchberg and M. Rørdam: Non-simple purely infinite $C^\ast$-algebras. Amer. J. Math. 122(3) (2000), 637-666 (MR1759891, jstor, ...

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210 views

### A precise definition of contractible Banach algebras

I asked this question at MSE but I did not received any answer. So I ask it here at MO
I am sorry if this question is elementary:
What is a precise definition of a contractible Banach ...

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356 views

### Is the ideal property of $X^{**}$ inheritable to $X$?

Let $X$ be an operator space such that there is a weak$^*$-continuous complete isometry $\phi$ from its second dual $X^{**}$ into a $W^*$-algebra $M$ in which $\phi(X^{**})$ is a (necessarily weak$^*$-...

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### Relation between maximal and reduced group $C^*$-algebras

Let $G$ be a Lie group and $C_r^*(G)$ and $C^*(G)$ be its reduced and maximal group $C^*$-algebras respectively. The left-regular representation of a group $G$ induces a surjective map
$$\lambda_G:C^...

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222 views

### Is every nontrivial idempotent in the Cuntz algebra, a commutator element?

Is it true to say that every nontrivial idempotent in the Cuntz algebra $\mathcal{O}(n)$ is a commutator element?(Or a linear combination of commutator elements?)

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### Simple $C^*$ algebras whose all commutator elements have scalar square

Is there a simple $C^*$ algebra $A$, not isomorphic to $M_2(\mathbb{C})$, such that for every commutator element $x=ab-ba$, $x^2$ is an scalar element?

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314 views

### Open projections and Murray-von Neumann equivalence

Let $\mathcal{A}$ be a $C^*$-algebra and $p\in\mathcal{A}^{**}$ be an open projection, that is, $p=p^*=p^2$ and $p\in\overline{(p\mathcal{A}^{**}p\cap\hat{\mathcal{A}})}^{\operatorname{w}^*}$, where $\...

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### The groupoid $C^*$ algebra associated to a certain groupoid

Let $\mathbb{N}$ be the set of all natural numbers. We define a groupoid structure on $\mathbb{N}^{\mathbb{N}}$ as follows:
We put $G^1=\mathbb{N}^{\mathbb{N}},\;G^0 =\{(a_n)\in G^1\mid a_{2n-1}=...

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### Solvability of a certain functional equation in simple $C^*$ algebras

For which simple unital $C^*$ algebras does the following functional equation have a solution:
$$ d^2=0,\;{(d+d^*)}^2=1$$
The Calkin algebra and $M_{2n}(\mathbb{C})$ are some examples. It is not ...

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165 views

### Regarding spectral radius

Let $A$ be a $C^*$ algebra. Let $a\in A$ be such that $a^*a-aa^*\geq 0$. Doe this imply that the spectral radius of $a$ is equal to $\|a\|$?

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### When is K0 of a C* algebra finitely generated?

Are there workable conditions that imply that $K_0(A)$ is finitely generated, for a noncommutative unital C* algebra $A$. My actual question is in fact much more specific than this:
Is it possible ...

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### The product of two controlled operators is also a controlled operator

The following picture is lemma 4.23 in Lectures on Coarse Geometry by John Roe:
I guess the $E_i$ in the centered formula is $X_i$. Does Roe mean that $X_j\cap \mathrm{Supp}(u)=\emptyset $ implies $\...

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### C*-algebras: Existence of an element inducing an injective map

I'm wondering if the following statement is true: Let $A$ be a $C^*$-algebra and $\phi: A\rightarrow A$ be a $*$-homomorphism. Is there always an element $a\in A$ such that the map
$\left\{ \phi^{n}:=...