All Questions
Tagged with oa.operator-algebras c-star-algebras
181 questions with no upvoted or accepted answers
3
votes
0
answers
203
views
Uniqueness of the reduced free product of unital completely positive maps
For $1\leq i\leq n$, let $\psi_i$ be a faithful state on the C$^*$-algebra $A_i$ and $\phi_i$ be a faithful state on the C$^*$-algebra $B_i$. Let $(A,\psi) = *_{i=1}^n (A_i,\psi_i)$ and $(B, \phi) = *...
- 3,984
3
votes
0
answers
146
views
Closed containment of open projections in C*-algebras
For a C*-algebra $A$ and open projections $p,q\in A^{**}$, consider the following statements.
$\overline{p}\leq q$
$p\leq q$ and there exists open $r\in A^{**}$ with $rp=0$ and $r\vee q=1$
$p\leq q$ ...
- 1,307
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
3
votes
0
answers
109
views
Two questions on topological and geometric structure of projections in a simple $C^{*}$ algebra
Let $A$ be a simple $C^{*}$ algebra. Assume that the space of projections has a connected component homeomorphic to the complex Grassmanian $G(k,n)$. Is it true to say that, for all $k'<k$, the ...
- 356
3
votes
0
answers
170
views
Closure of pseudodifferential operators of order 0 on compact manifolds
Let $M$ be a compact manifold. If we have a pseudodifferential operator $P$ of order $0$ on $M$, then $P$ is pseudolocal, i.e., every commutator $[f,P]$ with a continuous function $f \in C(M)$ is a ...
- 2,998
2
votes
0
answers
232
views
$\mathcal{B}(\mathcal{H})$ as the reduced $C^*$-algebra of a groupoid
Given an infinite dimensional Hilbert Space $\mathcal{H}$ what is the underlying locally compact Hausdorff etale groupoid $G$ such that $C_r^{\ast}(G)$ is $\ast$-isomorphic to $\mathcal{B}(\mathcal{H})...
- 15
2
votes
0
answers
158
views
Question about the ergodic mean
This is a repost from this MathStackExchange question, where unfortunately I was not able to resolve this question.
I've read a thesis where there is an example on ergodic mean, where however there is ...
- 33
2
votes
0
answers
202
views
The trigonometric $C^*$-algebra
The trigonometric $C^*$-algebra is the universal $C^*$-algebra generated by $\mathcal{G}=\{x,y,z\}$ subject to relations \begin{align}x^2=x=x^*, &\quad y^2=y=y^*\\ [x, z]=y, &\quad [y,z]=...
- 356
2
votes
0
answers
70
views
The $K_0$ mapping of an automorphism induced by a derivation
Let $\mathfrak{A}$ be a unital $C^*$-Algebra and let $\delta: \mathfrak{A} \rightarrow \mathfrak{A}$ be a linear map that is not constantly zero and satisfies, for every $A, B\in\mathfrak{A}$, $\delta(...
- 307
2
votes
0
answers
228
views
Irreducible group representation(algebraic and topological irreducibility)
In page 280 of "C^* algebra" by Dixmier, in the context of group representation, it is written 'We never encounter the concept of algebraic irreducibility except in finite dimensional ...
- 356
2
votes
0
answers
141
views
Is it true that $\varinjlim A_n^{**}=(\varinjlim A_n)^{**}$?
Let $(A_n, \phi_n)$ be an inductive system of $C^*$ algebras. It's well known double dual of $C^*$-algebra is again a $C^*$ algebra.
Is it true that $\varinjlim A_n^{**}=(\varinjlim A_n)^{**}$
Can ...
- 1,115
2
votes
0
answers
203
views
Quasidiagonal C*-algebras
Let $A$ be a nuclear $C^*$-algebra satisfying UCT condition. Then under what assumptions $A$ is quasidiagonal?
- 581
2
votes
0
answers
243
views
Reference request: definiitions of exact C* algebra and group C* algebra
I am writing my Ph.D. thesis and I would like to cite the specific papers where the concept of exact $C^*$ algebra and group $C^*$ algebra was defined.
In the book of Brown and Ozawa "$C^*$-...
- 115
2
votes
0
answers
119
views
Does this groupoid have a quasi-diagonal reduced $C^*$-algebra?
Let $H$ be a discrete group, and let $X$ be the one-point compactification of $\mathbb{N}$. Consider the étale groupoid $G = H \times \{\infty\} \sqcup \mathbb{N}$, whose unit space is $X$, and with ...
- 733
2
votes
0
answers
84
views
Are quasitrace extensions unique?
I'm trying to understand the basics of quasitraces on $C^*$-algebras. Using the terminology of Haagerup, given $n \geq 2$, an $n$-quasitrace $\tau$ on a $C^*$-algebra $A$ is a 1-quasitrace on $A$ ...
- 285
2
votes
0
answers
108
views
Special case of Elliott's Theorem
Let $A$ and $B$ be unital $AF$-algebra. By Elliott's theorem we know that if there an order isomorphism $\psi: K_0(A) \rightarrow K_0(B)$ with $\psi([1_{A}]) = [1_{B}]$, then there exists an ...
- 581
2
votes
0
answers
124
views
Representation of $C^{*} (S_{\infty})$
I was wondering what is the group $C^{*}$-algebra of infinite symmetric group?
Mainly, I was trying to calculate the k-theory of $C^{*}$-algebra of infinite symmetric group and I found K-Theory of $C^{...
- 581
2
votes
0
answers
305
views
Ideals of maximal tensor product of $C^{\ast}$-algebras
Let $A$ and $B$ be $C^{\ast}$-algebras. It is well known that maximal tensor product of simple $C^{\ast}$-algebras need not be simple. So basically the ideal structure of $A\otimes_{max}B$ does not ...
- 1,115
2
votes
0
answers
116
views
General construction of enveloping C*-algebra, left/right-regular representation, etc
In a number of contexts (e.g. groups, crossed products, groupoids, Fell bundles) there are similar constructions of enveloping C*-algebras and left/right-regular representations that incorporate ...
- 3,816
2
votes
0
answers
135
views
On crossed product of L^{P} spaces
Let $M$ be a von Neumann algebra with faithful normal state $\varphi$, and $G$ be a group action on $M$ preserving $\varphi$. Then is it true
\begin{align*}
L^{p}(M\rtimes G, \varphi^{M\rtimes G})\...
- 677
2
votes
0
answers
89
views
Computing norms of polynomials of operator in Hilbert space and generalized von Neumann inequality
Let $T$ be an operator $l^2({\mathbb{Z}_{\geq 0}}) \to l^2({\mathbb{Z}_{\geq 0}})$, $e_n \mapsto \sqrt{1 - q^{2(n+1)}}e_{n+1} $, where $0<q<1$. I want to compute $\|f(T,T^{*}) \|$ (operator ...
- 161
2
votes
0
answers
82
views
Atkinson-Mingo theorem
I've been read the book "$K$-theory and $C^*$-algebras" by Olsen. In the chapter on the generalized Fredholm index the following definitions are given:
Let $A$ be a $C^*$-algebra and $H_A$ the ...
- 41
2
votes
0
answers
99
views
Invertibility modulo the intersection of ideals in $C^*$-algebras
This is a crosspost from math.se because it did not get any attention whatsoever. I therefore assume that it fits here better.
Let $\mathcal{A}$ be a $C^*$-algebra and $A \in \mathcal{A}$. I am ...
- 171
2
votes
0
answers
151
views
A Banach or $C^*$ algebraic analogy of a classical fact in real analysis
Let $A$ be a commutative unital Banach algebra.The maximal ideal space of $A$ is denoted by $\hat A$.
Assume that $D:A \to A$ is a derivation. Fix an element $a\in A$.
Assume that for every $\phi\in \...
- 356
2
votes
0
answers
116
views
Closable operators on Hilbert modules
For $T:{\frak{Dom}}(T) \to H$ a densely defined operator, with $H$ a (separable) Hilbert space, we know that $T$ is closable if its adjoint $T^*$ has dense domain in $H$.
Does this extend to the (...
- 993
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
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
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
2
votes
0
answers
100
views
Amenability for Actions twited with 2-cocycles
Let $A \subset B(H)$ be a unital $C^\ast$-algebra and $\theta: G \rightarrow \mathrm{Aut}(A)$ an action and let $\omega: G \times G \rightarrow U(\mathcal{Z}(A))$ be a $2$-cocyle with respect to $\...
- 1,090
2
votes
0
answers
210
views
The functional equation $T(x\otimes y)=T(x)\otimes T(y)$ on certain $C^{*}$ algebras
Is there a name for the following property of a $C^{*}$ algebra $A$?
$$A \simeq A \otimes A\;\;\; \text{The spatial tensor product}$$
Example of this situation is $A=C(X)$ where $X$ is the ...
- 356
2
votes
0
answers
157
views
Primitive ideal space and unitary dual of a [SIN] group - when are they Hausdorff?
Recall that a locally compact group $G$ is said to be
an $[FC]^-$ group, if each conjugacy class in $G$ has a compact closure;
an $[SIN]$ group, if each neighborhood of the identity includes a ...
- 399
2
votes
0
answers
208
views
A functor on the category of rings, algebras or compact Hausdorff topological space
Assume that $R$ is a unital ring or a complex or real (Banach or $C^{*}$) algebra.
We define a relation $M$ on $R$ as follows: $$a\;M b \;\;\; \text{iff}\;\; a=xy,\;b=yx \;\; \text{for ...
- 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
2
votes
0
answers
209
views
Number of connected components of a $C^{*}$ algebra
Inspired by the concept in the following post
What are these compact sets called?
We introduce the following concept:
Let $A$ be a unital $C^{*}$ algebra. We consider the unitary equivalent ...
- 356
2
votes
0
answers
171
views
tensor product of the disc algebra with itself
Let $A=\mathcal{A}(\mathbb{D})$ be the disc algebra. Is there a cross norm on $A\otimes A$, which is compatible to the Banach algebra multiplication, such that the resulting Banach algebra has a $C^{*}...
- 356
2
votes
0
answers
200
views
Fredholmness and invertibility in a C* algebra generated convolution-type operators
Let $PC$ be the algebra of complex-valued, piecewise-continuous functions from $[-\infty,+\infty]$, $SO$ be the algebra of bounded, continuous, complex-valued functions on $\mathbb R$ which are slowly ...
- 21
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
1
vote
0
answers
86
views
Proof mistake of: $M_0A(G) = B(G)$ for a locally compact group
I am posting my question of mathstack exchange here. (see: My post on MSE)
Let $G$ be a locally compact group with Haar measure $\mu$, and $B(G),A(G),C_r^*(G),L(G)$ be its Fourier-Stieltjes algebra, ...
- 261
1
vote
1
answer
89
views
Continuous functions on HLS groupoids
I am reading a paper about property (T) for groupoids: Topological property (T) for groupoids. In section 4.4 they discuss the HLS groupoids which I describe define here.
Let $\Gamma$ be a discrete ...
- 261
1
vote
0
answers
79
views
Doubts on convergence of series of operators
Given an operator algebra of bounded operators $\mathcal{A}$ acting on a Hilbert space $\mathbb{H}$, I am interested in the algebra of tensor products $\mathcal{A}^{N} = \otimes_{k=1}^{N} \mathcal{A}...
- 33
1
vote
0
answers
137
views
Representation of states in $C^*$-algebras
Let $\mathfrak{A}$ be a $C^* $-algebra, let $\pi : \mathfrak{A} \to \mathcal{B}(H)$ be a representation of $\mathfrak{A}$ on the space of bounded linear operators on a Hilbert space $H$ and let $\...
- 139
1
vote
0
answers
87
views
irreducible subfactor inclusion and commutativity of induced projections
Let $N\subset M$ be an irreducible subfactor inclusion, i.e., $N'\cap M =\mathbb C1$, acting on a Hilbert space $H$.
Let $\Omega\in H$.
Does it follow that the projections onto $[N\Omega]$ and $[M'\...
- 759
1
vote
0
answers
106
views
A locally convex $C^*$ algebraic structure on the disk algebra
A locally convex $C^*$ algebra is a locally convex topological vector space $A$ whose topology is generated by a familly of complete $C^*$ semi norm and $A$ is a $*$ algebra. Moreover all algebra ...
- 356
1
vote
0
answers
178
views
A locally convex $C^*$ algebra without zero divisor
Let we have a locally convex $C^*$ algebra $A$. That is $A$ is a TVS equipped with an algebra and an involution structure such that all operations are continuous. Moreover the topology on $A$ ...
- 356
1
vote
0
answers
45
views
Examples of TRO $V $ and $C^{\ast} $-algebra $B $ for which $V\otimes^hB $ is a TRO
Let $V $ be a ternary ring of operator(TRO) and $B $ be a $\mathbb {C}^{\ast} $-algebra. Let $V \otimes^hB $ denotes the Haagerup tensor product of $V $ and $B $. Obviously if $V $ or $B $ is $\mathbb ...
- 1,115
1
vote
0
answers
113
views
Is it true that $\omega = \sum_{(k,l)\in I^2}\omega(p_k - p_l)$
Let $\{p_i\}_{i \in I}$ be a collection of projections in a $C^*$-algebra $A$ such that $\sum_{i \in I} p_i = 1$ in the strict topology (note here that $1$ is the unit of the multiplier algebra $M(A)$)...
- 175
1
vote
0
answers
129
views
Socle of an operator algebra
Let $H, K$ be Hilbert spaces.
Let $A\subseteq B(H)$ be a nonselfadjoint closed subalgebra such that the identity map is in $A$.
Let $C_A$ denote the $C^*$-algebra generated by $A$.
Q1: (this question ...
- 2,605
1
vote
0
answers
57
views
Is the universal representation an order isomorphism?
Let $A$ be a Banach *-algebra. By a *-representations of $A$, we mean a *-homomorphism $\pi:A\to B(H_\pi)$, where $B(H_\pi)$ is the space of all bounded linear maps on a Hilbert space $H_\pi$. Let $\...
- 4,058
1
vote
0
answers
139
views
Reduced twisted $C^*$-algebra and twisted crossed product
Let $G$ be a discrete group. Is it possible to represent $C^*_r(G, \sigma)$, the reduced twisted group $C^*$-algebra as a twisted crossed product?
- 581
1
vote
0
answers
283
views
Continuous fields of Hilbert spaces arising from representations of abelian C*-algebras
This is a followup to a previous question [1] on MO.
Let $X$ be a second-countable, locally compact, Hausdorff space, and let $\mathscr H =\{H_x\}_{x\in X}$, be a
measurable field of Hilbert spaces ...
- 483