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Results tagged with c-star-algebras
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user 36688
A C*-algebra is a complex Banach algebra together with an isometric antilinear involution satisfying (a b)* = b* a* and the C*-identity ‖a* a‖ = ‖a‖². Related tags: [banach-algebras], [von-neumann-algebras], [operator-algebras], [spectral-theory].
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The groupoid $C^*$ algebra associated to a certain groupoid
Let $\mathbb{N}$ be the set of all natural numbers. We define a groupoid structure on $\mathbb{N}^{\mathbb{N}}$ as follows:
We put $G^1=\mathbb{N}^{\mathbb{N}},\;G^0 =\{(a_n)\in G^1\mid a_{2n-1} …
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3
votes
1
answer
450
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On equation $e^{xy-yx}=e^xe^ye^{-x}e^{-y}$ in $C^*$ algebras
Inspired by this MSE question we ask the following question:
Is there a noncommutative $C^*$-algebra $A$ for which the following identity holds for all $x,y \in A$?
$$e^{(xy-yx)}= e^xe^y e^{-x}e^{- …
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3
votes
1
answer
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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?)
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2
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1
answer
170
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Certain groupoid and its $C^{*}$ algebra
Let $X$ be a finite subset of real numbers. Let $G$ be the collection of all non empty subsets of $X$ and $G_{0}$ be the collection of all singleton subsets of $X$.
We define two maps $r,s:G \t …
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1
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0
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93
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A question on Stable rank 1
My apology in advance if my question is elementary
According to the initial definition of topological stable rank introduced by Marc Rieffel we have the following:
An algebra has tsr 1 if the space …
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4
votes
0
answers
202
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Connectivity of the group of invertible elements of $C(S^{2})\otimes A$
For what type of $C^{*}$ algebras $A$, the group of invertible elements of $C(S^{2}) \otimes A$ is a connected group?
All finite dimensional $A$ satisfy this property.
Is it true to say t …
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5
votes
1
answer
176
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(Noncommutative) Tietze $C^*$ algebras
A unital $C^*$ algebra $A$ is said a Tietze algebra if it satisfies the following:
For every ideal $I$ of $A$ and every unital morphism $\phi: C[0,1] \to A/I$ there is a unital morphism $\tilde{\phi} …
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2
votes
0
answers
171
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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^{* …
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2
votes
1
answer
181
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A Possible characterization of F.D or AF commutative $C^{*}$ algebras
By F.D or AF $C^{*}$ algebra,we mean finite dimensional or approximately finite dimensional $C^{*}$ algebra.
Let $A$ be a unital commutative $C^{*}$ algebra with the property that for every …
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1
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1
answer
377
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A question on K- theory of non commutative $C^\star$ algebra
Edit: According to the comment of Andre Henriques I revise the question:
What is an example of a noncommutative unital $C^\star$ algebra $A$, which is not Morita equivalent to a commutative alg …
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3
votes
0
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109
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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 s …
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1
vote
1
answer
171
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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 $C^{*}$ alg …
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answer
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Comparision of two $C^{*}$ algebras associated to a non vanishing vector field on a compact ...
Let $X$ be a non vanishing vector field on a compact manifold $M$ so we have a one dimensional foliation $F$ of $M$ with orbits of $X$.
This foliation defines a $C^{*}$ algebra $C^{*}(F)$. On the ot …
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1
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4
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365
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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 ar …
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4
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Classification of $C^*$ algebras whose all non scalar elements have disconnected spectrum
Regarding the non commutative case, I just realized that a non commutative $C^*$ algebra has a non scalar element $1+a$ with connected spectrum $\{1\}$ where $a$ is a non zero nilpotent eleme …
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