All Questions
27 questions with no upvoted or accepted answers
27
votes
0
answers
1k
views
Unital $C^{*}$ algebras whose all elements have path connected spectrum
A unital $C^{*}$ algebra is called a "Path connected algebra" if the spectrum of all its elements is a path connected subset of $\mathbb{C}$.
What is an example of a non commutative ...
- 356
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 ...
7
votes
0
answers
85
views
Analogue of Friedrichs extension for Hilbert $C^*$-modules
Suppose one has a densely defined symmetric operator $T:\mathcal{M}\rightarrow\mathcal{M}$, where $\mathcal{M}$ is a Hilbert $A$-module for a $C^*$-algebra $A$. Suppose that $T$ is non-negative, so ...
- 1,903
6
votes
0
answers
243
views
For what kind of $C^*$ algebra $A$ every normal element $y\in A$ has a normal lift for every given surjective $C^*$ morphism $\phi:B\to A$
Is there a terminology for the following property of $C^*$ algebra $A$:
For every $C^*$ algebra $B$ and surjective $C^*$ morphism $\phi: B\to A$, every normal element $y\in A$ admits a normal ...
- 356
4
votes
0
answers
242
views
On the Dunford-Pettis property and multiplier algebras
I am not an expert in operator algebras, so if the answer to this question might be trivial, that might be one reason for that:
Let $\mathcal{A}$ be a $C^\ast$-algebra. Then $\mathcal{A}^{\ast \ast}$ ...
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 ${\rm C}^{\star}(X)$, ...
4
votes
0
answers
146
views
A question on extension of $Z^{*}$ algebras
A $Z^{*}$ algebra is a $C^{*}$ algebra which all elements are(two sided or equivalently one sided) zero divisor.
Are there two $Z^{*}$ algebras $A,B$ such that for every short exact sequence of ...
- 356
3
votes
0
answers
179
views
Stinespring's theorem: can we choose the dilation to be an isometry?
Let $A$ be a $C^*$-algebra and $\varphi: A \to B(H)$ be a completely positive contractive map. Stinespring's theorem says that there exists a $*$-representation $\pi: A \to B(H')$ and a bounded ...
- 175
3
votes
0
answers
69
views
Trying to understand morphisms in category of ternary $C^*$-rings
Recall that a ternary $C^*$-ring is a complex Banach space $X$, equipped with a associative ternary product $[.,.,.]:X^3 \to X$ which is linear in outer variables and conjugate linear in middle ...
- 1,115
3
votes
0
answers
97
views
Is the set of points in the irreducible decompositions of this C$^{*}$ -algebra's representations closed?
Suppose $X$ and $Y$ are compact Hausdorff 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
0
answers
178
views
A point concerning absolute value of functionals
Let $M$ be a von Neumann sub-algebra in $B(H)$. Let $\phi$ be a normal functional on $M$. Assume $\psi$ is a normal functional on $B(H)$ with $\psi_{|_M}=\phi$ (note that $\phi$ and $\psi$ may have ...
- 4,058
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
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
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
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
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
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
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
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
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
137
views
A continuous choice of invertible elements
Let $A$ be a simple unital $C^{*}$ algebra with invertible elements $G(A)$. Assume that $A^{*}$ is its dual space, which is equipped with the weak star topology.
Is there a continuous map $\alpha:A^...
- 356
0
votes
0
answers
71
views
Necessary conditions for $K_0(I_x\bigotimes A)$ to be the trivial group
Let $A$ be a unital $C^*$-Algebra with non-trivial $K_0$ group. Define $CA = \{f\in C\big([0, 1], A\big)\,\vert\,f(0) = 0\}$. It can be shown that $CA$ is homotopic equivalent to the set $\{0\}$ and ...
- 307
0
votes
0
answers
91
views
Proof of the isomorphism of the Toeplitz algebra and the algebra generated by the element and the relation
Please tell me where can I see the proof of this well-known fact?
enter image description here
0
votes
0
answers
88
views
Is A an amenable $C^{*}$-algebra?
Let $A$ be $C^{*}$-algebra. Suppose that, for any $\epsilon > 0$ and finite subset $F \subset A$, there are an amenable $C^{*}$-subalgebra $B \subset A$, contractive completely positive linear maps ...
- 581