All Questions
Tagged with oa.operator-algebras c-star-algebras
597 questions
6
votes
1
answer
446
views
A Question About the Elliott-Natsume-Nest Proof of Bott Periodicity
In Wegge-Olsen’s book K-Theory and C$ ^{*} $-Algebras, there is an outline of a proof of Bott Periodicity (the proof is due to George Elliott, Toshikazu Natsume and Ryszard Nest). The first step of ...
user36116
13
votes
2
answers
897
views
Can non-central projections still commute with all other projections?
Let $A$ be a C*-algebra and let $\mathcal{P}(A)$ denote the set of projections in $A$. If $p\in\mathcal{P}(A)$ commutes with everything in $\mathcal{P}(A)$ does it necessarily commute with everything ...
- 1,307
18
votes
2
answers
924
views
Which groups are the unitary group of a $C^*$-algebra
Which groups are the unitary group of a $C^*$-algebra?
Does anyone know anything in this direction?
- 221
6
votes
2
answers
429
views
Metrics on the space of $C^{*}$ algebras
I think that there is a metric on the huge space of all $C^{*}$ algebras. What is the explicit
definition of this metric?may you introduce me a reference?
Moreover is the restriction of this ...
- 356
1
vote
0
answers
198
views
Connected component of the identity in graded Banach algebras
I search for a noncommutative idempotent-less Banach algebra $A$ which is graded by a finite abelian group $G$ such that a nontrivial homogenous element lies in the same connected component as $1_{...
- 356
4
votes
1
answer
126
views
Automorphisms of "rational" Kirchberg algebras
Let $M_{\mathbb{Q}}$ be the universal UHF-algebra and let $\mathcal{O}_{\infty}$ be the infinite Cuntz algebra. Let $A$ be a Kirchberg algebra that satisfies the UCT with $K_0(A) \cong \mathbb{Q}^n$ ...
- 7,637
4
votes
1
answer
148
views
Crossed Products by Permutation Groups
What can be said about the following crossed product $C^*$-algebra?
Let $A$ be a Kirchberg algebra with $K_0(A) = \mathbb{Q}$ and $K_1(A) = 0$. Consider the direct sum of $n$ copies of $A$, i.e. $B = ...
- 7,637
5
votes
1
answer
303
views
$Z_{2}$- graded structures for $C_{red} ^{*} (F_{2})$
Let $F_{2}$ be the free group with two generators.
Then $F_{2}=\{\text{odd words}\}\sqcup\{\text{even words}\}$. This gives us a $Z_{2}$ graded structure for $C^{*}_{red} (F_{2})$, in a natural way.
...
- 356
2
votes
1
answer
171
views
One-parameter semigroup of operators of a $C^*$-algebra applied to positive self-adjoint element
Let $X$ denote a complex $C^*$-algebra and $\{Z(t)\}_{t\geq 0}$ is a $C_0$-semigroup of operators on $X$.
Let $x\in X$ satisfy have $x=x^*$ (x is self-adjoint), such that its spectrum satisfies $\...
- 153
10
votes
1
answer
533
views
Who first identified the universal $C^*$-algebra generated by an idempotent of norm at most $C$?
So much is known about hermitian and non-hermitian idempotents in a $C^*$-algebra, that someone must have written down the following.
Theorem The universal $C^*$-algebra generated by one element $x$...
- 1,749
15
votes
4
answers
3k
views
Universal $C^*$-algebra with generators and relations
We say that the $C^*$-algebra $A$ generated by $a_1,...,a_n$ is universal subject to relations $R_1,...,R_m$ if for every $C^*$-algebra $B$ with elements $b_1,...,b_n$ satisfying relations $R_1,...,...
- 9,330
17
votes
5
answers
2k
views
If two projections are close, then they are unitarily equivalent
Given two projections $p,q\in B(H)$, it is well-known that if $\|p-q\|<1$, then there exists a unitary $u\in B(H)$ with $q=upu^*$.
The proof that immediately occurs to me uses comparison of ...
- 2,793
7
votes
1
answer
701
views
A Question About Pure States, Support Projections and Central Covers
I am trying to study the paper Consistency of a Counterexample to Naimark’s Problem by Charles Akemann and Nik Weaver, and there is a claim in Lemma 1 of the paper that I am stuck at, which is as ...
user36116
6
votes
1
answer
449
views
Weakly amenability and exactness for discrete groups
A countable discrete group $\Gamma$ is said to be weakly amenable with Cowling-Haagerup constant 1 if there exists a sequence of finitely supported functions $(\phi_n)$ on $\Gamma$ such that $\phi_n\...
- 1,702
7
votes
0
answers
189
views
Replacing commutative C*-algebras by simple ones
I am looking for functorial ways of replacing a commutative $C^*$-algebra $C$ by a simple one, say $A$ , such that the $K$-theory remains unchanged, i.e. $K_*(C) \cong K_*(A)$.
I am particularly ...
- 7,637
6
votes
2
answers
366
views
Expression of a non-orthogonal projection in a $C^*$ algebra via an orthogonal one
A paper I'm currently reading uses the following fact. If $A$ is a unital $C^*$-algebra, $P=P^2\in A$, then there are $T, F\in A$ s.t. $F$ is an orthogonal projection ($F=F^*=F^2$) and
$$P=F+FT(1-F).\...
- 1,284
8
votes
2
answers
465
views
Extending a $ * $-Representation of $ ({C_{c}}(G,\mathscr{A}),\star,^{*}) $ to a $ * $-Representation of $ \mathscr{A} \rtimes_{\alpha} G $
Let $ (\mathscr{A},G,\alpha) $ be a $ C^{*} $-dynamical system, and consider the twisted convolution $ * $-algebra $ ({L^{1}}(G,\mathscr{A}),\star,^{*}) $ defined by
\begin{align*}
\forall \phi,\psi \...
user36116
9
votes
2
answers
883
views
Are the reduced group Von Neumann algebra/ Group $C^{\ast}$ algebra functorial in the case of LCH groups
Let $G$ be a LCH group and $\mu$ be its left Haar measure. Call $\lambda_G : G \to U(L_2(G,\mu))$ the left regular representation. We can define the reduced $C^{\ast}$ algebra and reduced Von Neumann ...
- 1,090
0
votes
0
answers
293
views
Lifting triangles in K-theory to KL-groups
Let $X$ and $Y$ be finite simplicial complexes (or $CW$-complexes) so that $Y\subseteq X$. Let $s\colon C(X)\to C(Y)$ be the map given by restriction. In particular $K_{*}(C(X))$ and $K_{*}(C(Y))$ are ...
3
votes
1
answer
301
views
‘Non-Induced’ Left Regular Representations of $ C^{*} $-Dynamical Systems
In what follows, a ‘$ * $-representation’ always means a non-degenerate $ * $-representation.
Let $ (\mathscr{A},G,\alpha) $ be a $ C^{*} $-dynamical system, and let $ \pi: \mathscr{A} \to B(\mathcal{...
user36116
3
votes
1
answer
481
views
Question on structure of von Neumann algebras, clarification in Conway's "A course in operator theory"
I was reading the section on the structure of type I von Neumann algebras in John B. Conway's "A course in operator theory" and a few questions about certain definitions and references arose, I was ...
- 417
5
votes
2
answers
337
views
When is a groupoid the path groupoid of a graph?
I am actually interested in the $C^*$-algebras, so perhaps my question should be: How can you recognize whether a $C^*$-algebra $A$ is isomorphic to $C^*(\Lambda)$ for some (higher-rank) graph $\...
- 151
9
votes
0
answers
483
views
The approximation property of group C*-algebras
Let $G$ be a discrete group. Then the group C*-algebra $C^*(G)$ is nuclear if and only if $G$ is amenable. I am wondering whether nuclearity of $C^*(G)$ can fail for a Banach-space reason, namely due ...
- 11.3k
12
votes
1
answer
901
views
Is there a proof that the $C^{*}$-algebras don't see the invariant subspace problem?
This post is an appendix of this one.
Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators.
Invariant subspace problem: Let $T \in B(H)$. Is ...
8
votes
0
answers
952
views
About generator and isomorphism problems for free groups operator algebras
Let $H$ be an infinite dimensional separable Hilbert space.
The $C^{*}$-algebras and von Neumann here unital and subalgebras of $B(H)$.
Definition : Let $\mathcal{A}$ be $C^{*}$-algebra (resp. a von ...
15
votes
2
answers
778
views
automorphisms of C*-algebras and partial isometries
Let $A$ be a $C^*$-algebra, let $p$ and $q$ be Murray-von Neumann equivalent projections in $A$, i.e. there is a partial isometry $v$ such that $v^*v = p$ and
$vv^* = q$. Suppose $\alpha \in Aut(A)$ ...
- 7,637
6
votes
1
answer
680
views
Is there an operator algebraic reformulation of the invariant subspace problem?
Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators.
Invariant subspace problem: Let $T \in B(H)$. Is there a non-trivial closed $T$-invariant ...
6
votes
1
answer
643
views
Is there an irreducible, noncompact commuting, nonnormal operator, with spectrum strictly continuous?
Let $H$ be an infinite dimensional separable Hilbert space.
Definition: The commutant $\mathcal{S}'$ of a subset $\mathcal{S} \subset B(H)$ is $ \{A \in B(H) : AB=BA \ , \ \forall B \in \mathcal{S} \}...
3
votes
2
answers
553
views
Tensoring with a CAR-algebra
Let $A$ and $B$ be two unital infinite-dimensional simple separable nuclear $C^{\ast}$-algebras and let $C$ be a CAR-algebra. When does $A\otimes C \simeq B\otimes C$, imply $A\simeq B$?
The answer ...
- 169
5
votes
2
answers
988
views
Projections in a W*-algebra as a continuous lattice?
A continuous lattice is a complete lattice $L$ in which every element $y$ is equal to $\bigvee${$x \in L \mid x \ll y$} where $x \ll y$ ("x approximates y" or "x is way below y") if for any directed ...
- 85
0
votes
1
answer
287
views
All AI-algebras are AT-algebras
It is known that every AI-algebra (i.e. inductive limit of interval algebras) is an AT-algebra (i.e. inductive limit of circle algebras)?
This seems a little bit odd because a building block of an AT-...
- 169
3
votes
1
answer
427
views
Inner automorphisms and $K$-theory
It is known that any inner automorphism of a unital $C^{\ast}$-algebra $A$ induces the identity map on $K_{0}(A)$ because unitary equivalence implies Murray-von Neumann equivalence. What is known ...
- 169
0
votes
1
answer
243
views
Unitary with full spectrum
I have a unitary element $u\in C(\mathbb{T},M_{n}(\mathbb{C}))$ such that $Spec(u)=\mathbb{T}$. Does there exist a unitary $v\in C(\mathbb{T},\mathbb{C})$ such that $Spec(uv)\subsetneqq\mathbb{T}$?
- 169
25
votes
2
answers
1k
views
Can nuclearity be determined by tensoring with a single C*-algebra?
A C*-algebra is nuclear if the algebraic tensor product $A\odot B$ ($B$ is any other C*-algebra) admits a unique C*-norm. This definition requires testing the condition for nuclearity with `all' C*-...
- 505
4
votes
0
answers
374
views
Hans Saar's thesis
I would love to have a look on some results which are claimed by some people to be in Saar's thesis:
H. Saar, Kompakte, vollständig beschränkte Abbildungen mit Werten in einer nuklearen C${}^\ast$-...
- 505
11
votes
2
answers
635
views
Quasinilpotent elements of group C-star algebras
If $G$ is a discrete torsion-free group, can its (reduced or full) group C-star algebra contain non-zero quasinilpotent elements? I've seen various examples in the group von Neumann algebra setting (...
- 25.8k
10
votes
2
answers
803
views
General recipe for building C*-algebras out of combinatorial object
I want to ask what should be a nice way to build C*-algebras out of objects like groups, inverse-semigroups, semigroups, ringgs or graphs. I know there are well known construction of C*-algebras out ...
- 493
20
votes
3
answers
3k
views
Realizing universal $C^*$-algebras as concrete $C^*$-algebras
How do I in general realize a universal C*-algebra generated by some generators and relation as concrete C*-algebras? For example, I know that universal C*-algebra generated by a single unitary is $C(\...
- 493
4
votes
1
answer
464
views
Kadison-Singer problem in exotic Hilbert spaces
The Kadison-Singer problem is considered in relation to the separable Hilbert space:
KS: Does every pure state on the diagonal (atomic) masa of $B(\ell_2)$ has a unique extension to $B(\ell_2)$?
...
- 387
11
votes
2
answers
576
views
Seeing topological (geom.) properties of the space via corresponding C^*-algebra
Compact Hausdorff spaces bijectively correspond to C^*-algebras with identity. One needs to consider the algebra of continuous functions C(X) to go in one direction and spectrum to go in the other. (...
- 24.8k
2
votes
1
answer
323
views
Ideal spanned by matrix units isomorphic to compact operators
Hello,
Assume we have $(n+1)$ isometries $S_1,...,S_{n+1}$ in the separable Hilbert space $H$ with the properties that $\sum_{i=1}^{n+1}S_iS_i^*=I, S_i^*S_j=0$ (i.e. $S_i$ are the generators of the ...
- 23
2
votes
1
answer
221
views
Non-perfect type one C^*-algebra, and a lemma in Fourier analysis
I would like to know if the following is true :
Let $\mathcal{H}$ be the complex Hilbert space $L^2([0,1])$ for the Lebesgue measure.
Let $q$ be the orthogonal projection on the subspace of $\mathcal{...
- 42.4k
1
vote
1
answer
611
views
Commutant of a von Neumann algebra as the linear span of unitaries.
I'm reading chapter 4 of Gerard Murphy's C*-algebras book and am confused by a statement in his proof of theorem 4.1.10. In his proof, he says, "$A'$ is the linear span of its unitaries" (where $A'$ ...
12
votes
2
answers
479
views
C*-algebras with no nontrivial endomorphisms
Pick a C*-algebra $A$ and call a (*-)endomorphism $\alpha:A\to A$ nontrivial if it is injective and $\alpha(A)\neq A$.
Question: Do there exist infinite dimensional C*-algebras with no nontrivial ...
- 1,411
10
votes
0
answers
508
views
Tensorial decomposition of $B(H)$
Let $H$ be an infinite-dimensional Hilbert space and let $\mathcal{B}(H)$ be the (C*/W*-)algebra of bounded operators on it. Actually, you may forget about the involution in $\mathcal{B}(H)$ because I ...
- 101
16
votes
4
answers
1k
views
Von Neumann algebra associated to the infinite Cuntz algebra
The Cuntz algebra $\mathcal{O}_{\infty}$ is the universal $C^*$-algebra generated by countably infinitely many isometries $s_i$ satisfying the relations $s_i^*s_j = \delta_{ij}$ (there is no condition ...
- 7,637
10
votes
1
answer
492
views
Which W*-algebras are the duals of C*-coalgebras?
A Banach algebra (assumed associative and unital) is precisely a monoid object in the monoidal category of Banach spaces, short linear maps, and the projective tensor product. A Banach coalgebra is ...
- 2,754
4
votes
2
answers
957
views
Do semi-continuous functions generate bounded Borel measurable functions as a $C^*$-algebra?
This question is related to Question 2 of my previous posting.
Question. Let $\mu$ be a Radon measure on a compact Hausdorff space $\Omega$ and $L^{\infty}(\Omega,\mu)$ the set of essentially bounded ...
- 619
4
votes
0
answers
282
views
Extensions of completely positive mappings
I would like to ask the following two questions.
Let $1_{\mathcal{H}}\in \mathcal{A}\subset\mathcal{B}\subset\overline{\mathcal{A}}^{SOT}\subset\mathbb{B}(\mathcal{H})$ be a sequence of $C^{\ast}$-...
- 619
8
votes
2
answers
961
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 $S^m$ ...
- 619