All Questions
33 questions
5
votes
0
answers
136
views
C^*-algebra theory with all the Koszul signs
I was wondering if someone knows of a reference in which $\mathbb{Z}_2$-graded $C^*$-algebra theory is developed using the sign convention $(ab)^* = (-1)^{|a||b|}b^* a^*$. I would be most enthusiastic ...
- 357
1
vote
1
answer
367
views
finitely generated C*-algebra as $C(X)$
In the question ($C(X)$ as finitely generated $C^*$-algebra), the answer show that spectrum of an abelian unital finitely generated C*-algebra is homeomorphic to compact subset of $\mathbb{C}^{n}$. I ...
- 523
0
votes
1
answer
163
views
Regarding socle of a C* algebra
I wanted to know if the socle of a complex C*-algebra is essential?
Can anyone suggest a text where the socle is studied in detail. I tried reading it from the book by Bernard Aupetit, A Primer in ...
- 149
2
votes
1
answer
316
views
Decomposition of Hilbert spaces via groups and algebras representations
Let $\mathcal{H}$ be a complex finite dimensional Hilbert space and let $\mathcal{A}\subseteq \mathcal{B}(\mathcal{H})$. I am looking to understand the different decompositions of $\mathcal{H}$ ...
- 23
3
votes
0
answers
196
views
Cuntz semigroups of basic C*-algebras
I am doing some research related to Cuntz semigroups, and I am trying to find concrete examples in simple cases. In one paper that I found, it says the following (p.103):
"[...] $A_i$ is ...
- 285
9
votes
1
answer
445
views
Reference request: Brown Ozawa and strong completely positive approximation property?
The notion of a $C^*$-algebra being nuclear has many equivalent characterisations. These are considered in the excellent, modern textbook $C^*$-Algebras and Finite-Dimensional Approximations by Brown ...
- 18.7k
3
votes
1
answer
119
views
Universal representations of quotient C*-algebras
Suppose that $\mathfrak{J}$ is a closed ideal of a C*-algebra $\mathfrak{A}$. Let $(\pi_u, H_u)$ be the universal representation of $\mathfrak{A}$. Is there a way to use these data to describe the ...
- 73
0
votes
1
answer
124
views
Definition of center of ternary ring of operators
Let $H$ and $K$ be Hilbert spaces and $B(H,K)$ denotes the space of bounded operators from $H$ to $K$. Recall that a ternary ring of operators (TRO) $V$ is a closed subspace of $B(H,K)$ which is ...
- 1,115
5
votes
1
answer
178
views
Hopf C-star algebra/comodules using a Fubini tensor product rather than the minimal tensor product?
Throughout, $\otimes$ denotes the minimal tensor product of $\newcommand{\Cst}{{\rm C}^{\ast}}\Cst$-algebras. and $\odot$ denotes the tensor product of (underlying) vector spaces.
Given two $\Cst$-...
- 25.8k
1
vote
1
answer
87
views
Projection (or idempotent) graph of a $C^*$ algebra(or a ring)
In the literature, are there some research around a directed graph associated to a $C^*$ algebra or a ring $A$ whose vertices are projections or idempotents of $A$ and $e$ is connected to $f$ iff $ef=...
- 356
4
votes
1
answer
341
views
On differential equation $Z'=Z^2-Z$ on a $C^*$ algebra
Let $A$ be a Banach or a $C^*$ algebra. We consider the differential equation $$(*)\;\;\;\;Z'=Z^2-Z$$ on $A$.
Obviously the singularities of this systems are just the idempotents of the ...
- 356
7
votes
1
answer
394
views
Inverse limit in the category of $C^{\ast}$-algebras or operator spaces
Does the inverse limits (projective limits) exist in the category of $C^{\ast}$-algebras or operator spaces?
I tried to search but could not find a proper reference. Any reference or comments about ...
- 1,115
2
votes
1
answer
189
views
Need a reference of a fact given in B. Blackadar's Operator Algebras
I am reading Blackadar's book on Operator algebras. In $\Pi 9.6.5$ Blackader says that
Maximal Tensor products commute with arbitrary limits.
In the same book the proof of this fact is not given....
- 1,115
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
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
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_*...
3
votes
1
answer
261
views
CBAP for the full group $C^*$-algebra
Let $G$ be a weakly amenable group, in the sense that it has a net of finitely supported functions $\varphi:G\to \mathbb{C}$ which converge point wise to 1 and their cb norm is bounded uniformly by ...
- 99
5
votes
1
answer
242
views
Spectral decomposition of a C$^*$algebra with respect to an action of a compact abelian group
Let $G$ be a compact abelian group (finite dimensional, but not finite) and $A$ be a $C^*$-algebra. Consider an action $\alpha: G\to Aut(A)$. In analogy with the case of finite abelian group, I ...
- 743
10
votes
2
answers
1k
views
Kazhdan's property (T) vs. residual finiteness
I have asked this question already on mathstackexchange but got no answer (see https://math.stackexchange.com/questions/1795795/kazhdans-property-t-vs-residual-finiteness) and it was suggested that I ...
- 721
4
votes
0
answers
135
views
References for a lemma about compact operators on a Hilbert module
I am looking for a reference for the following result:
If $A$ and $B$ are C* algebras, $H$ is a right Hilbert $A$-modules, $\phi :A \rightarrow B$ is a morphism, and assume that there is a map $\eta :...
- 42.4k
3
votes
0
answers
255
views
Graded structures for simple $C^{*}$ algebras without nontrivial idempotent
Edit(A confession): I just realized that the question is trivial: Since one can easily prove that the convex hull of the spectrum of every nontrivial homogeneous element of a $\mathbb{Z}_{n}$-graded $...
- 356
1
vote
1
answer
171
views
A $C^{*}$ algebra associated to a graded $C^{*}$ algebra
A $C^{*}$ algebra $A$ is graded by $\mathbb{Z}_{n}$ iff it can be acted by $\mathbb{Z}_{n}$. So we associate the $C^{*}$ algebra $A\rtimes \mathbb{Z}_{n}$ to a $\mathbb{Z}_{n}$-graded $C^{*}$ ...
- 356
2
votes
1
answer
320
views
Totally non hereditary $C^{*}$-subalgebras
Assume that $B$ is a $C^{*}$ subalgebra of $A$. We say $B$ is totally non hereditary subalgebra of $A$ if not only $B$ is not a hereditary subalgebra but also it is not isomorphic to any ...
- 356
2
votes
0
answers
234
views
The kernel of $C^{*}(G)\to C_{r}^{*}(G)$
Let $G$ be a locally compact group. Put $I(G)=\ker: C^{*}(G) \to C_{r}^{*} (G)$, the kernel of the canonical morphism.
What type of $C^{*}$ algebras can not be isomorphic to $I(G)$, for some ...
- 356
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
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
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
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
13
votes
1
answer
1k
views
Does the hyperfinite II_1 factor admit two irreducible representations that are not unitarily equivalent?
Regarding the hyperfinite $II_{1}$ factor $R$ as $C^{*}$-algebra, is it known whether any two irreducible representations of $R$ are unitarily equivalent? If it is known that there exists a pair of ...
- 7,047
8
votes
1
answer
904
views
The Haar state on compact quantum groups $A_u(Q)$ and $A_o(Q)$
Let $Q\in GL_n(\mathbb{C})$. The free unitary quantum group is the universal $C^*$-algebra $A_u(Q)$ with generators $u_{ij},1\leq i,j\leq n$ and relations making $u=(u_{ij})$ as well as $Q\bar{u}Q^{-1}...
- 1,702
7
votes
0
answers
573
views
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}}
\newcommand{\Cst}{\operatorname{...
- 25.8k
19
votes
2
answers
1k
views
Commutators in the reduced C*-algebra of the free group
Is it known whether any element of trace 0 in the reduced $C^*$-algebra of a non-abelian free group, is a limit of sums of (additive) commutators?
- 11.1k
2
votes
1
answer
653
views
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
$\mathcal{A}:=${$a\...
- 213