Vahid Shirbisheh
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Registered User
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I am a mathematician primarily interested in C*-algebras, functional analysis, harmonic analysis and related areas.
Some of my previous and current works have some overlaps with one or more other areas such as K-theory, KK-theory, cyclic cohomology, Galois theory, group theory, number theory, deformation quantization.
Recently, I've written a lecture notes on C*-algebras, which is available at arXiv:1211.3404.
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Apr 27 |
asked | Torsion version of HNN extensions. |
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Apr 3 |
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Books on advanced galois theory I think inverse Galois problems and Galois embedding problems could be interesting subjects to continue. Just google these phrases to find reading materials that suits you. You can download my book on Galois embedding problem at researchgate: (researchgate.net/publication/…). |
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Mar 31 |
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Are there any nontrivial ring homomorphisms $M_{n+1}(R)\rightarrow M_n(R)$? When $R=\mathbb{C}$ (and probably most fields), it is easy to see that every homomorphism from $M_n(R)$ into another algebra is either an injection or the zero homomorphism, because $M_n(R)$ (in this case) is a simple algebra. For an easy proof, see Proposition 4.3.23 of arXiv:1211:3404. Certainly, by modifying the proof, you can relax some of the conditions on $R$. |
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Mar 30 |
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Quasinilpotent elements of group C-star algebras For a proof for the above statement see, Page 110 (Proposition II.6.4.14) of Blackadar's book "Operator algebras". |
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Mar 24 |
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What information about a locally compact group $G$ is encoded in $C_r^\ast(G)$ which is not in $L^1(G)$? @Martin: That's exactly my point. I'd like to know what makes mathematicians to consider $C_r^\ast(G)$ instead of $L^1(G)$. When $G$ is abelian, $C_r^\ast(G)$ appears in the Pontryagin duality, $C_r^\ast(G)$ is a $C^\ast$-algebra and therefore easier to work with. What else can be said about the benefits of $C_r^\ast(G)$. For example, I am tempted to say the computation of $K$-theory of $C_r^\ast(G)$ is easier, but I am not sure about it. |
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Mar 24 |
asked | What information about a locally compact group $G$ is encoded in $C_r^\ast(G)$ which is not in $L^1(G)$? |
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Mar 18 |
awarded | ● Fanatic |
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Mar 17 |
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Can we generalize the result of Urysohn’s lemma to countable collection of pairwise disjoint closed subsets of a normal space..? No, take $A_n=\{1/n\}\subset \mathbb{R}$. |
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Mar 14 |
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Orthonormal basis for $L^2(G/H)$. Thanks for references. The original problem is to show the Hecke algebra $\mathcal{H}(G,H)$ has a left regular representation on $L^2(H\backslash G)$. When $H\backslash G$ is discrete, there is a proof based on an orthonormal basis which I wanted to generalize to locally compact case. However, when $H$ is cocompact, one can use standard techniques like the Fubini theorem to show that the convolution product defines a representation. |
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Mar 14 |
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Orthonormal basis for $L^2(G/H)$. Your suggestion looks promising (in special cases), but my original problem has an easier solution when $H\backslash G$ is compact. Anyway, I would like to learn more about the technique you suggested. Could you give some references? |
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Mar 14 |
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Orthonormal basis for $L^2(G/H)$. I am interested in groups or homogeneous spaces, because I am looking for a suitable way to define a convolution like product and define a regular representation for certain algebras on $L^2(H\backslash G)$. I have been thinking about a similar construction as you described in your answer. But it does seem have several problems: 1. There is no general recipe to partition a general group as the union of cubes in $\mathbb{R}^n$. The Gram-Schmidt orthogonalisation process does not give us an explicit orthonormal basis to work with, but it proves the existence of an orthonormal basis. |
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Jan 26 |
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How similar/different are dense subgroups of a compact group. A very simple example of two completely different dense subgroups of $\mathbb{T}$, the circle in $\mathbb{C}$ can be considered by letting $H_1$ be the subgroup of all roots of unity and $H_2$ be the cyclic infinite subgroup generated by $e^{2\pi i \lambda}$ where $\lambda$ is an irrational number. |
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Jan 25 |
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Stabilization in Banach algebras @Yemon: You are welcome. Following Vincent Lafforgue's works, Walter Paravicini has studied Morita equivalence of Banach algebras too. |
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Jan 21 |
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Orthonormal basis for $L^2(G/H)$. Thanks Alain for sharing your insight. |
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Jan 21 |
asked | Orthonormal basis for $L^2(G/H)$. |
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Jan 20 |
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Why all irreducible representations of compact groups are finite-dimensional ? [EDIT: Subtleties: AC,etc] I suggest the book "Principles of Harmonic Analysis" by Deitmar and Echterhoff. Chapter 7 of this books answers your first question, see Thm. 7.2.4. |
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Jan 17 |
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Stabilization in Banach algebras @Yemon: I list some instances that clarify the importance of stabilization in $C^*$-algebras: 1. Both K-theory and KK-theory are stable functors meaning $K(A)\simeq K(A\otimes K(H))$. 2. Two separable $C^*$-algebras $A$ and $B$ are Morita equivalent if and only if they are stably isomorphic, i.e. $A\otimes K(H)\simeq B\otimes K(H)$. 3. Tensoring by $K(L^2(G))$ also appears in some theorems too, for example see Takai-Takesaki duality. So, it is nice to have a similar notion in Banach algebras, for instance proving item 2 for Banach algebras would be a good start. |
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Jan 16 |
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Stabilization in Banach algebras @Yemon: I explained my ideas in the above. |
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Jan 15 |
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Some career start-up (postdoc) questions @kreck: I think being a bad teacher is just being a bad teacher and an inappropriate behavior is just a bad behavior. I call someone jerk (in an office or institute) if he (or she) abuses his power or his position to suppress others, in other words a bully. Honestly, yes I have seen a few of such people among postdocs as well as senior faculties and all of them somehow got the job. Unfortunately the current system is unable to address issues. My point is such vague accusations can open the door for prejudices and discrimination in academia. So we must be a little more careful. |
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Jan 15 |
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Stabilization in Banach algebras @Alain: It is a good point. I guess choosing the type of the tensor product should be part of the stabilization process too. Do you have any suggestion? |
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Jan 15 |
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Stabilization in Banach algebras Thanks Ulrich. I guess one can choose is $K(H)$ for stabilization of Banach algebras too. But I am not sure if this gives rise to similar theorems about Morita equivalence and $K$-theory of Banach algebras. So, it seems there is no obvious choose "known yet" and it is open for more investigations. |
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Jan 15 |
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Some career start-up (postdoc) questions @Julia: Could you tell us what kind of jerky actions a postdoc (and young faculties until they get tenure) can commit to deserve your expression "certain jerks"? I am asking this question because postdocs have little or no power in academia so they cannot do anything bad even if they are really evil. |
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Jan 15 |
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Stabilization in Banach algebras After being able to define (an appropriate notion for) the stabilization of a Banach algebra, say $A$, I'd like to see if it is Morita equivalent with $A$. Of course, the equality of $K$-groups is the next. And so on. Stabilization of $C^\ast$-algebras is a elementary notion, so I thought, maybe there is a similar notion for Banach algebras too. That's why I asked this question. |
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Jan 14 |
asked | Stabilization in Banach algebras |
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Jan 11 |
answered | Why is $\mathcal{M}(\text{SL}_2(\mathbb{Z}))$ spanned by $E_4$ and $E_6$? |
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Jan 9 |
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Calkin Algebra and the embedding @Nik: Could you give some reference about your answer in (1)? I am interested in to read more in this subject. |
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Jan 9 |
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Induced representations for profinite groups Thank you Marc for sharing your insight. However what you described is more about irreducible representations of profinite groups. My question was about the possible applications of induced representations from closed (or open) subgroups of profinite groups which they can be reducible. If I understood correctly your answer implies that if $H$ is (open) subgroup of a profinite group $G$ and its index in $G$ is not finite, then $ind_H^G 1$ is not irreducible, which is interesting too. |
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Jan 9 |
asked | Induced representations for profinite groups |
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Jan 7 |
awarded | ● Enthusiast |
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Jan 7 |
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Seeing topological (geom.) properties of the space via corresponding C^*-algebra @Yemon: You are right. But I assumed $X$ is a locally compact and Hausdorff topological space. And I was trying to show that the correspondence between open subsets and closed ideals does not work in noncommutative $C^*$-algebras, because there are $C^*$-algebras without any non-trivial closed two sided ideal which are not isomorphic to $\mathbb{C}$. Therefore, the concept of open subsets, which is the core of any local structure, does not exists for noncommutative spaces. |
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Jan 6 |
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Seeing topological (geom.) properties of the space via corresponding C^*-algebra @Yemon Choi: Assuming hausdorffness of $X$ having few open sets means having few points. For the noncommutative case I used the word "naive" to say it is not a suitable point of view. |
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Jan 6 |
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Seeing topological (geom.) properties of the space via corresponding C^*-algebra It is complex Radon measures. There is a bijective correspondence between open subsets of $X$ and closed two sided ideals of $C_0(X)$. But we cannot consider closed two sided ideals as a substitute for open sets in noncommutative $C^*$-algebras, because there are simple $C^*$-algebras, for example $M_n(\mathbb{C}$ for every $n\in \mathbb{N}$. Then all simple $C^*$-algebras should be considered as a noncommutative space with one point which is obviously naive. |
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Jan 6 |
answered | Seeing topological (geom.) properties of the space via corresponding C^*-algebra |
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Jan 6 |
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Should science authors discourage / boycott the recent push for author IDs @Criag: Honestly, I don't see any disadvantages for myself as an author. But people have different opinions and I'd like to see what others think about this issue. |
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Jan 6 |
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Should science authors discourage / boycott the recent push for author IDs You mean we should have the right to pretend that the past never existed? |
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Dec 28 |
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Old books still used Also I would add his book "Functional Analysis". |
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Dec 24 |
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Morita equivalence for *-algebras Yes, that's true. Of course there is also a notion of Morita equivalence for Banach algebras which is developed mostly by Walter Paravicini. But I do not know whether he considers involutive Banach algebras too. |
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Dec 24 |
answered | Morita equivalence for *-algebras |
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Dec 22 |
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On the definition of ‘smooth vectors’ in Rieffel’s “Deformation Quantization for Actions of $ \mathbb{R}^{d} $”. By definition, $a\in A^\infty$ if $f_a$ a differentiable function from $\mathbb{R}^n$ into $A$. So, the answer is "yes" regarding the definition of $A^\infty$. |
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Dec 22 |
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Automatic continuity of the inverse map $Y$ also should be compact in the above! |
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Dec 22 |
asked | Extending length functions regarding certain group extensions. |
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Dec 19 |
awarded | ● Nice Answer |
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Dec 18 |
revised |
Publishing a bad paper? deleted 4 characters in body |
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Dec 18 |
answered | Publishing a bad paper? |
