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: [banach-algebras], [von-neumann-algebras], [operator-algebras], [spectral-theory].

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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\...
Ali Taghavi's user avatar
7 votes
1 answer
207 views

$*$-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 $...
Max Schattman's user avatar
5 votes
1 answer
267 views

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 ...
user avatar
2 votes
0 answers
97 views

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 \...
Darman's user avatar
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4 votes
1 answer
147 views

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 ...
Darman's user avatar
  • 327
8 votes
1 answer
473 views

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 ...
André Henriques's user avatar
6 votes
0 answers
225 views

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 ...
Matthias Ludewig's user avatar
6 votes
0 answers
235 views

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 ...
Mark Roelands's user avatar
31 votes
3 answers
2k 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)...
André Henriques's user avatar
5 votes
1 answer
252 views

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 ...
Math Lover's user avatar
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3 votes
1 answer
139 views

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 ...
Ruy's user avatar
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0 votes
1 answer
243 views

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 ...
Andi Bauer's user avatar
  • 2,901
1 vote
0 answers
191 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 ...
Josh Lackman's user avatar
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2 votes
1 answer
89 views

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 ...
user avatar
1 vote
0 answers
92 views

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 ...
ABB's user avatar
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6 votes
0 answers
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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}(\...
Adrián González Pérez's user avatar
0 votes
1 answer
178 views

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 ...
C. Ding's user avatar
  • 135
2 votes
1 answer
124 views

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 ...
ervx's user avatar
  • 267
3 votes
1 answer
292 views

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 ...
Meisam Soleimani Malekan's user avatar
6 votes
0 answers
131 views

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 ...
Matthew Daws's user avatar
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4 votes
1 answer
293 views

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 ...
Max Schattman's user avatar
1 vote
1 answer
674 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?
ABB's user avatar
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6 votes
1 answer
573 views

Is the conditional expectation faithful?

Let $G$ be a locally compact group and let $H$ be an 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\...
Sabrina Gemsa's user avatar
7 votes
1 answer
327 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}...
Julio Cáceres's user avatar
5 votes
0 answers
49 views

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 ...
geometricK's user avatar
  • 1,851
4 votes
1 answer
264 views

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, ...
Targaryen's user avatar
4 votes
1 answer
257 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 ...
Ali Taghavi's user avatar
3 votes
2 answers
395 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$^*$-...
Masayoshi Kaneda's user avatar
4 votes
1 answer
477 views

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^...
geometricK's user avatar
  • 1,851
3 votes
1 answer
296 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?)
Ali Taghavi's user avatar
1 vote
1 answer
296 views

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?
Ali Taghavi's user avatar
7 votes
1 answer
420 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 $\...
Masayoshi Kaneda's user avatar
1 vote
0 answers
119 views

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}=...
Ali Taghavi's user avatar
3 votes
1 answer
142 views

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 ...
Ali Taghavi's user avatar
3 votes
1 answer
250 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 $\|a\|$?
user31459's user avatar
  • 175
5 votes
0 answers
124 views

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 ...
Severino Melo's user avatar
1 vote
1 answer
143 views

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 $\...
C. Ding's user avatar
  • 135
3 votes
1 answer
130 views

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}:=...
worldreporter's user avatar
4 votes
1 answer
282 views

Full spectrum positive elements of a $C^*$-algebra

I would like to know criteria for a C*-algebra $A$ to have a positive contraction $a$ with full spectrum, ie $\sigma(a) = [0,1]$. I am particularly interested in the simple case. I believe that if a C*...
Aldo van Baerle's user avatar
5 votes
0 answers
77 views

C*-algebra of a singular surface foliation

Noncommutative geometry associates a $C^*$-algebra $C^*(S,{\cal F})$ with a foliation $\cal F$ on a manifold $S.$ Did somebody study this construction for noncompact surfaces $S$? What I am really ...
Adam's user avatar
  • 2,370
7 votes
1 answer
301 views

Homomorphism to multiplier algebra of groupoid $C^\ast$-algebra

If I have a functor $X\to Y$ between topological groupoids with appropriate Haar measures, such that $X_0 \to Y_0$ is injective and a homeomorphism onto its image, then I should have (or rather, I ...
David Roberts's user avatar
  • 33.8k
3 votes
1 answer
119 views

an inverse semigroup (and perhaps a $C^*\!$-algebra) associated with a directed graph

The following inverse semigroup associated to a directed graph came up in my research. I've read that from an inverse semigroup one may derive a $C^*\!$-algebra whose generators are partial isometries ...
David Hillman's user avatar
8 votes
1 answer
264 views

Factor traces of the Temperley-Lieb algebra

Given $\delta\in\mathbb C$, let $A(\delta)$ denote the complex unital $*$-algebra generated by an identity $1$ and selfadjoint elements $e_k$, $k\in\mathbb N$, satisfying $e_k^2=\delta e_k$, $e_ke_l=...
Gandalf Lechner's user avatar
19 votes
1 answer
756 views

Are algebraically isomorphic $C^*$-algebras $*$-isomorphic?

If A and B are C^*-algebras that are algebraically isomorphic to each other, does this imply that they are *-isomorphic to each other?
Doc Matrix's user avatar
2 votes
2 answers
358 views

Finitely generated $K_0$ of $C^*$-algebras

Let $A$ and $B$ be $C^*$-algebras. Let $A_0$ be a dense *-subalgebra of $A$, and let $B_0$ a dense *-subalgebra of $B$. Assume that $A_0$ is isomorphic to $B_0$. Finally, assume that $K_0(A)$ is ...
Doc Matrix's user avatar
0 votes
1 answer
142 views

A property of compact topological space via certain $C^*$ embedding in operator algebras

Assume that $A$ is a unital $C^*$ algebra. Is there a $C^*$ embedding of $A$ in some $B(H)$ whose image is a hereditary $C^*$ subalgebra of $B(H)$? If not, is the answer affirmative when $A$ is ...
Ali Taghavi's user avatar
5 votes
0 answers
130 views

Modules of algebras with idempotents and the Stone-von Neumann theorem

The Stone-von Neumann theorem tells us that all unitary irreducible representations of the integrated/exponentiated/Weyl form of the canonical commutation relations (CCR) algebra in finite dimensions ...
Carlos's user avatar
  • 633
3 votes
1 answer
181 views

Algebraic tensor product of C*-algebras extends via ideals? Application to restriction theorem?

Is the following assertion and the proof below correct, or am I missing something very important? Moreover, would the corollaries be correct then? Besides, I would also appreciate a lot any comment, ...
C-star-W-star's user avatar
2 votes
0 answers
163 views

An operator valued Egoroff's theorem

The following statements suggests $B(H)$-valued Egoroff's theorem when $H$ is a separable Hilbert space. Probably it will be hold even if a von Neumann algebra $M$ whose predual is separable is ...
ABB's user avatar
  • 3,898
2 votes
1 answer
585 views

Enveloping $C^*$-algebra

Consider a finite dimensional $C^*$-algebra $\cal{A}$. Is there any enveloping $C^*$-algebra $\cal{C^*(G)}$ such that $\cal{A}\cong C^*(G)$ for some locally compact group $\cal{G}$? (Note that "$\...
Albert harold's user avatar

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