All Questions
Tagged with oa.operator-algebras c-star-algebras
597 questions
6
votes
0
answers
168
views
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:
$$
\mathrm{Ad}: U(\mathcal{M})/\mathbb{T} \longrightarrow \mathrm{Aut}(\...
- 1,090
2
votes
1
answer
131
views
Closeness of points in the irreducible decomposition of a C$^{*}$-algebra representation
Suppose $X$ and $Y$ are compact metric spaces. Let $\varphi\colon C(X)\to M_{n}(C(Y))$ be any $*$-homomorphism. If $\pi$ is an irreducible representation of $M_{n}(C(Y))$, then $\pi$ is unitarily ...
- 267
3
votes
1
answer
309
views
How rich the group of unitary elements in a von Neumann algebra to get "Murray-von Neumann" equivalence?
Denote by $\sim$ the "Murray-von Neumann" equivalence in the projection lattice of a von Neumann algebra. It is well known that in a finite von Neumann algebra the equivalence of projections can be ...
- 2,118
6
votes
0
answers
132
views
Schröder–Bernstein for representations of operator algebras
This is claimed in a Wikipedia Article:
If two representations (of a $C^*$-algebra $A$) $\rho$ and $\sigma$, on Hilbert spaces $H$ and $G$ respectively, are each unitarily equivalent to a ...
- 18.7k
4
votes
1
answer
357
views
Extending maps from dense $*$-algebras of $C^*$-algebras
Given $\cal{A},\cal{B}$ two dense $*$-algebras of two $C^*$-algebras $A$ and $B$ respectively, together with a $*$-algebra homomorphism $f:\cal{A} \to \cal{B}$, is it clear that $f$ extends to a ...
- 993
1
vote
1
answer
786
views
finite dimensional C*-algebras
Let $A$ be a C*-algebra. Suppose that every cyclic representation of $A$ is finite dimensional.
Q. Is $A$ finite dimensional?
- 4,058
6
votes
1
answer
623
views
Is the conditional expectation faithful?
Let $G$ be a locally compact group and let $H$ be an open subgroup in $G$.
Then the full group $C^*$-algebra of $H$, $C^*(H)$, is a subalgebra of $C^*(G)$ and there is a conditional expectation $$E\...
- 1,188
7
votes
1
answer
373
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}...
- 173
5
votes
0
answers
50
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 ...
- 1,903
4
votes
1
answer
279
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, ...
- 181
4
votes
1
answer
262
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 ...
- 356
3
votes
2
answers
397
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$^*$-...
- 619
4
votes
1
answer
540
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
$$\lambda_G:C^...
- 1,903
3
votes
1
answer
303
views
Is every nontrivial idempotent in the Cuntz algebra, a commutator element?
Is it true to say that every nontrivial idempotent in the Cuntz algebra $\mathcal{O}(n)$ is a commutator element?(Or a linear combination of commutator elements?)
- 356
1
vote
1
answer
306
views
Simple $C^*$ algebras whose all commutator elements have scalar square
Is there a simple $C^*$ algebra $A$, not isomorphic to $M_2(\mathbb{C})$, such that for every commutator element $x=ab-ba$, $x^2$ is an scalar element?
- 356
7
votes
1
answer
429
views
Open projections and Murray-von Neumann equivalence
Let $\mathcal{A}$ be a $C^*$-algebra and $p\in\mathcal{A}^{**}$ be an open projection, that is, $p=p^*=p^2$ and $p\in\overline{(p\mathcal{A}^{**}p\cap\hat{\mathcal{A}})}^{\operatorname{w}^*}$, where $\...
- 619
3
votes
1
answer
143
views
Solvability of a certain functional equation in simple $C^*$ algebras
For which simple unital $C^*$ algebras does the following functional equation have a solution:
$$ d^2=0,\;{(d+d^*)}^2=1$$
The Calkin algebra and $M_{2n}(\mathbb{C})$ are some examples. It is not ...
- 356
5
votes
0
answers
125
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 ...
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}:=...
- 741
4
votes
1
answer
313
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*...
5
votes
0
answers
81
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 ...
- 2,390
8
votes
1
answer
281
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=...
- 1,017
19
votes
1
answer
773
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?
- 213
2
votes
2
answers
373
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 ...
- 213
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 ...
- 356
3
votes
1
answer
187
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, ...
- 598
2
votes
0
answers
164
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 ...
- 4,058
2
votes
1
answer
341
views
Closed two-sided ideals in $C(X,M_n)$
As is known (see Kadison-Ringrose, 3.4.1) each closed ideal $I$ in the $C^*$-algebra $C(X)$ of continuous functions on a compact space $X$ has the form
$$
I=\{f\in C(X): \ \forall x\in S\quad f(x)=0 \}...
- 7,426
4
votes
0
answers
384
views
Extension of Coburn's theorem on isometry and Toeplitz algebra
$\newcommand{\id}{\mathrm{id}}$Let $H$ be a Hilbert space, and $X \in B(H)$ a proper isometry (i.e. $X^{\star}X = \id$ and $XX^{\star} \neq \id$). Coburn's theorem states that ${\rm C}^{\star}(X)$, ...
3
votes
1
answer
218
views
Extensions of sub-homogeneous $C^*$-algebras are subhomogeneous
Definition: A $C^*$-algebra $A$ is called sub-homogeneous if there exists $n\in\mathbb{N}$ such that every irreducible representation of $A$ has dimension at most $n$.
I could not find a proof or a ...
- 33
2
votes
0
answers
125
views
Computing the $K$-theory of the free inverse semigroup $C^*$-algebra
A typical example where I know the $K$-theory of the quotient, and wonder if this could help to compute the $K$-theory of the extension. (Or other way round: knowing one part out of three ones.)
I ...
- 685
10
votes
0
answers
201
views
Masas in SAW*-algebras
I asked this question three years ago at MSe but it has no response; let me try here.
Pedersen distilled the following class of C*-algebras which he termed SAW*-algebras (Journal of Operator Theory, ...
- 11.3k
0
votes
1
answer
144
views
Dualizing the trivial action on a $C^*$-algebra
Let $G$ be a finite abelian group (cosidered as a discrete topological group), $A$ a unital separable $C^*$-algebra. Let $T\colon G\to \operatorname{Aut}(A)$, $T_g(a)=a$ for all $g\in G$ the trivial ...
- 1,188
6
votes
1
answer
272
views
Non-invertible version of unitary intertwiners between correspondences of $C^\ast$-algebras
There is a version of the Eilenberg-Watts theorem for $C^\ast$-algebras, where functors between appropriate module categories correspond to what are called 'correspondences' of $C^\ast$-algebras. ...
- 35.4k
0
votes
1
answer
153
views
C*-algebra of free monogenic inverse semigroup
Let us take the right shift operator $S$ acting on the Hilbert space $l^2(\mathbb{N})$. Consider the C*-algebra generated the operator
$
\begin{pmatrix}
S & 0 \\
0 & S^*
\end{pmatrix}
$ ...
- 493
8
votes
1
answer
724
views
Role of the UCT problem in classification theory for C*-algebras
Elliott's program for nuclear C*-algebras deals with the problem of classifying nuclear C*-algebras by K-theoretical invariants. A major open question in this context is the UCT problem.
A separable ...
- 741
7
votes
0
answers
222
views
Can C*/W*-algebras be realized as (involutive?) monoid/co-monoid objects?
I would like to know how close one can get to realizing the category of C*-algebras as a category of monoid objects. Related (almost, but not quite, duplicate) questions are:
"Recovering a monoidal ...
- 146
4
votes
1
answer
157
views
Geometric Motivation for Hilbert $C^*$-Bimodules
I'm trying to get an understanding of Hilbert $C^*$-bi-modules from a geometric point of view. As is well-known, we have that
i) Commutative unital $C^*$-algebras correspond to compact Hausdorff ...
- 167
3
votes
1
answer
232
views
unital embedding into the coner $C^*$-algebra
Let $A$ be a $C^*$-algebra, we denote with $V(A)$ the semigroup of Murray-von Neumann equivalence classes of projections in matrices over $A$ (as usual). In https://arxiv.org/pdf/math/0310340.pdf, ...
- 1,188
4
votes
1
answer
201
views
closure of a separating set of pure states
Let $A$ be a unital C*-algebra, and let $\mathcal R$ be a separating family of irreducible representations of $A$. Each vector state of a representation in $\mathcal R$ is a pure state, and the span ...
- 974
2
votes
1
answer
168
views
Crossed products and unitaries implementing $\mathbb{Z}_n$-actions
I'm working through Li's and Barlak's Cartan Subalgebras and the UCT Problem but I'm stuck at one of the simpler proofs of the paper. On page 9 they deal with masas (maximal abelian subalgebras) of a ...
- 741
1
vote
1
answer
269
views
description of a map in KK-theory
The following situation is given: Let $A$ be a unital, separable, nuclear $C^*$-Algebra, $i:\mathbb{C}\to A$ the unital embedding. All $C^*$-algebras are considered as trivially graded. Consider the ...
user62639
3
votes
0
answers
141
views
Existence of a unique cyclic and separating vector in a *-representation
I'm interested in knowing the requirements for a $*$-representation, $\pi_{\omega}$, of a C*-algebra, $\mathbb{C}(\mathcal{G})$, (or equivalently the requirements for the unitary representation, $U_{\...
- 31
2
votes
0
answers
715
views
Universal $C^*$ algebra generated by two self adjoint elements with $x^2+y^2=1+{(xy-yx)}^2$
Is there universal $C^*$ algebra generated by self adjoint elements $x,y$ subject to relation $x^2+y^2=1+{(xy-yx)}^2$?
What is a precise description of such algebra?
- 356
1
vote
4
answers
367
views
Classification of $C^*$ algebras whose all non scalar elements have disconnected spectrum
To what extent have all unital $C^*$ algebras $A$ with the following property been classified? Is there a simple $C^*$ algebra with this property? Does $C(K)$ satisfy this property, where $K$ is an ...
- 356
7
votes
0
answers
555
views
maximal tensor product commutes with inductive limits
Let $(A_n, \phi_n)$ be an inductive system of $C^*$ algebras and let $B$ be an arbitary $C^*$ algebra.
I want to prove $(\varinjlim A_n)\otimes_{max} B \cong \varinjlim (A_n \otimes_{max} B)$. This ...
- 1,188
4
votes
0
answers
96
views
Approximate unit of specific form in a crossed product by $\Bbb {R} $ algebra
The following lemma appears in the paper "Rokhlin dimension for flows"- by Winter, Hirshberg, Szábo, Wu.
Lemma 6.5: Let $A $ be a $\sigma $-unital $C^*$-algebra with a flow $\alpha:\Bbb {R}\to Aut (A)...
- 41
0
votes
1
answer
132
views
Is there an embedding $v(\mathcal{O}_\infty\otimes \mathcal{K})v^*\to \mathcal{O}_\infty$?
Following situation: If $\mathcal{O}_\infty$ is the Cuntz algebra in infinitely many generators and $\mathcal{K}$ the compact operators on a separable Hilbert space, let $v\in \mathcal{O}_\infty\...
3
votes
1
answer
159
views
K-group properties of quasi-diagonal $C^*$-algebras
Let $A$ be a separable unital quasidiagonal $C^*$-algebra.
What can be said about the $K$-theory of $A$, for example some properties? Especially, are there some criterions to decide whether or not $K_*...
30
votes
2
answers
3k
views
Is every maximal ideal in a C*-algebra always closed?
I wonder if every maximal two-sided (self-adjoint) ideal in a C*-algebra is automatically closed. It is a very basic fact of C*-algebra theory that it holds true for the unital case. In the non-unital ...
- 10.1k