All Questions
Tagged with ra.rings-and-algebras c-star-algebras
27 questions
1
vote
0
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148
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Vanishing (infinite) tensor products
Since the advent of free probabilities and QFT, infinite tensor products of $R$-associative algebras with units has become more familiar to the working mathematician.
Starting from the (permuting) ...
- 3,738
6
votes
1
answer
322
views
Pairwise orthogonality for partitions of unity in a *-algebra
Let $\mathcal{A}$ be a $*$-algebra with unit $1_{\mathcal{A}}$. As in the $\mathrm{C}^*$-setting, a projection is an element $p\in\mathcal{A}$ such that $p=p^2=p^*$. A partition of unity is a finite ...
- 1,037
2
votes
2
answers
190
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Do positive-definite elements in finite-dimensional $*$-algebras over $\mathbb R$ always admit square roots?
Let $A$ be a finite-dimensional $*$-algebra over $\mathbb R$. We say that an element $x \in A$ is positive definite if $x$ admits an inverse and if $x = y y^*$ for some $y \in A$. Does every such $x$ ...
- 4,943
1
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1
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286
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A subalgebra of $B(H)$ which does not contain a commutator element
Is there a commutative subalgebra $A\subset B(H)$ containing the 1 dimensional scalars with the following property:
The algebra $A$ has trivial intersection with the set of commutator ...
- 356
3
votes
1
answer
388
views
Closed prime ideal in $C[0, 1]$
I know that maximal ideals of $C[0, 1]$ corresponds to singleton. Also, using Zorn's lemma one can construct a prime ideal in $C[0, 1]$ which is not maximal.
Is there any $\textbf{closed}$ prime ...
- 1,115
1
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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
2
votes
0
answers
72
views
Example of a ternary Lie ideal which is not a Lie ideal
Let $H$ and $K$ be Hilbert spaces and $V\subset B(H,K)$ be a ternary ring of operators i.e. $xy^*z \in V$ for all $x,y,z \in V$. Let $I$ be a closed subspace of $V$. $I$ is called a ternary Lie ideal ...
- 1,115
3
votes
1
answer
219
views
Extending $*$-morphisms to the multiplier algebras
I'm reading the following fragment in the paper "Notes on compact quantum groups":
While I'm familiar with the multiplier algebra (constructed via double centralizers) and its universal ...
user167952
7
votes
1
answer
479
views
Characterisation of finite dimensional C*-algebras?
$\DeclareMathOperator\Spec{Spec}$Let $A$ be a finite dimensional $*$-algebra over $\mathbb C$.
(Namely, an associate algebra equipped with an involution $*:A\to A$ satisfying $(ab)^*=b^*a^*$ and $(\...
- 43.2k
1
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0
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164
views
When a finite codimensional subalgebra contains a finite codimension ideal?
What is a classification of all algebras $A$ (purely algebraic algebras, Banach or $C^*$ algebras or Lie algebras) with the following property:
Every finite codimensional subalgebra $B$ of $A$ ...
- 356
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
199
views
Groups for which all projections of $C^*_{\text{red}}G$ belong to $\mathbb{C}G$
Revision: According to comment of Wojowu we give a complete revise for this post.
A group $G$ is a pr-group if all projections of $C^*_{\text{red}} G$ are contained in its dense subalgebra $\mathbb{...
- 356
2
votes
1
answer
151
views
Automorphism of algebras with certain initial conditions on given idempotents
The First question
Let $A$ be a Banach or a $C^*$ algebra. Assume that $e,f$ are two idempotents or prjections in $A$ which satisfy $ef=fe=0$. Assume that there are two automorphisms $\phi, \psi: A \...
- 356
4
votes
0
answers
333
views
Other kinds of equivalence relations on the set of idempotents of a Banach or $C^*$-algebra or a ring (Can we get a new kind of K-theory?)
The standard equivalent relations on idempotents of a $C^*$ algebra or a Banach algebra are Murray von Neumann, similarity and homotopy equivalent. In this post we consider two other kinds of ...
- 356
1
vote
1
answer
306
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Simple $C^*$ algebras whose all commutator elements have scalar square
Is there a simple $C^*$ algebra $A$, not isomorphic to $M_2(\mathbb{C})$, such that for every commutator element $x=ab-ba$, $x^2$ is an scalar element?
- 356
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
0
answers
185
views
ring structure of $KK_*(A,A)$ for a separable $C^*$-algebra $A$
Motivation:
For a topological space $X$ one can consider under certain circumstances the cohomology ring of suitable cohomology theories, for example:
1) The cohomology ring $H^*(X;R)=\oplus_{i\ge ...
- 1,188
7
votes
0
answers
191
views
Generalizing C*-algebraic results to purely algebraic situations
In the category of C*-algebras there are many purely algebraic properties which often make life easier, such as the fact that every C*-algebra is idempotent (by Cohen-Hewitt). When speaking of closed ...
- 2,263
2
votes
1
answer
171
views
algebraic version and polar decomposition
I have been thinking about polar decomposition of $C^*$-algebras. I could not find a proper reference where it says: Let $A$ be a $C^*$-algebra, and $b$ an invertible element of $A$ with modulus $\...
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: $$a\;M b \;\;\; \text{iff}\;\; a=xy,\;b=yx \;\; \text{for ...
- 356
14
votes
1
answer
616
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How "nondegenerate" are amalgamated free products of C*-algebras?
In the following, I assume all algebras are unital. Let $A$ and $B$ be C*-algebras that each contain (isomorphic copies of) a common C*-subalgebra $C$. Let $A *_C B$ denote the amalgamated free ...
- 5,407
6
votes
2
answers
2k
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Positive Elements of a $\ast$-Algebra
In a $C^*$-algebra ${\cal A}$, a positive element is a one of the form $aa^*$, for some $a \in {\cal A}$. It is known that the set of positive elements is a cone, and that for $a,b$ two non-zero ...
- 465
8
votes
1
answer
366
views
Does the following $ C^{*} $-algebraic result have a purely algebraic proof?
While studying the proof of Bott periodicity for operator $ K $-theory in this set of notes, I learned this fact:
Theorem. Let $ A $ and $ B $ be $ C^{*} $-algebras. Let $ f,g: A \to B $ be $ * $-...
- 1,396
5
votes
0
answers
270
views
A generalization of real characters on a group
Yesterday I understood that I can't live without this construction:
Let $G$ be a group, $A$ an associative algebra over $\mathbb R$ and $n\in{\mathbb N}$. We consider a sequence of maps $\varphi_k:...
- 7,426
0
votes
0
answers
129
views
A special Lie subalgebra
Motivated by comments of the following post A question on involutions on the Lie algebra of vector fields we ask the following question:
Let $L$ be a Lie algebra. We consider the Lie subalgebra $$...
- 356
1
vote
0
answers
134
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Non commutative analogy of compact-open topology
Let $R$ be a ring, define a topology on $AUT(R)$(Or End(R)) with the following subbase:
For every two 2-sided Ideal $I$ and $J$, a subbase element is $B(I,J)=\{f\in AUT(R) \mid f(I)+J=R\}$.
We can ...
- 356
64
votes
4
answers
8k
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What is the current status of the Kaplansky zero-divisor conjecture for group rings?
Let $K$ be a field and $G$ a group. The so called zero-divisor conjecture for group rings asserts that the group ring $K[G]$ is a domain if and only if $G$ is a torsion-free group.
A couple of good ...
- 1,065