All Questions
Tagged with oa.operator-algebras rt.representation-theory
70 questions
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Unitary representations of Fuchsian and Kleinian groups
Let $\Gamma$ be a discrete group that is either Fuchsian ($\Gamma \subseteq \text{PSL}(2,\mathbb R)$) or Kleinian ($\Gamma \subseteq \text{PSL}(2,\mathbb{C})$).
I have a unitary representationL
$$
\...
- 357
3
votes
0
answers
141
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Location of nontrivial Gleason parts on the topological boundary of the polydisc - Was it described anywhere else besides in Bekken's PhD Thesis?
My PhD advisor and me need the exact description of the location of the non-trivial Gleason parts on the topological boundary of the polydisc $\mathbb{D}^n$.
It was described in Otto B. Bekken's PhD ...
- 63
0
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0
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57
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Decomposition of a contractive representation into an orthogonal sum for the $n$-dimensional case. Has this been done yet?
I know that it has been done for the two-dimensional case. Marek Kosiek showed it in his work "Decomposition of operator representations of the algebra $R(K_1 \times K_2)$" and "...
- 63
4
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0
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168
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Representations of $C\left(SO_q(n)\right)$
A complete classification of irreducible representations of the $C^*$-algebra $C(G_q)$, where $G_q$ is the $q$-deformation of a classical simply connected semisimple compact Lie group, was provided by ...
- 73
5
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2
answers
1k
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Are umbral moonshine and umbral calculus connected?
In a 2013 article, Cheng, Duncan and Harvey introduce the concept of umbral moonshine as a generalization of monstrous moonshine. The terminology they use, starting with the title, is common in umbral ...
- 10.5k
0
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0
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131
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Can a non-separable C$^*$ algebra have separable GNS Hilbert space
Suppose we have a $C^*$ algebra $\mathfrak{U}$ that is non-separable. Consider a state $ω$ of $\mathfrak{U}$ and the GNS representation $(H_ω,π_ω,Ω)$. Is it possible for $H_ω$ to be separable, and if ...
- 151
5
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1
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165
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Is norm-continuous representation factored through a Lie quotient group?
I asked this 11 days ago at MSE, but there was no answer, I hope people here could help.
Let $G$ be a locally compact group, and $X$ a Hilbert space. A unitary representation $\varphi:G\to B(X)$ is ...
- 7,426
2
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1
answer
94
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Unitary dual of universal cover
The universal covering group $G$ of $\mathrm{SL}_2({\mathbb R})$ has infinite center. Is there an irreducible unitary representation $\pi$ of $G$, whose central character is injective? Or does every $\...
- 63.9k
1
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0
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137
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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
6
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1
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623
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Is the conditional expectation faithful?
Let $G$ be a locally compact group and let $H$ be an open subgroup in $G$.
Then the full group $C^*$-algebra of $H$, $C^*(H)$, is a subalgebra of $C^*(G)$ and there is a conditional expectation $$E\...
- 55
8
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0
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411
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Semigroups of matrices closed under conjugate transposition
An involution semigroup or $\star$-semigroup is a unary semigroup $\langle S,{\cdot}\,,{}^\star\rangle$ that satisfies the equations $$ (x^\star)^\star = x \quad \text{and} \quad (xy)^\star = y^\star ...
- 563
0
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0
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59
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Are banach space representations of commutative $C^*$ algebras decomposable?
It is well known that, if $\pi:A\to \mathbb B(\mathcal H)$ is a $^*$-representation of a type I $C^*$-algebra, then $\pi$ is unitarily equivalent to a direct integral of irreducible representations.
...
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2
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208
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Generalisation of the equivalence between $C^*(H)$ and $C_0(G/H) \rtimes G$; induction of group actions on C*-algebras
There is a well known Morita equivalence between the group C*-algebra $C^*(H)$ and $C_0(G/H) \rtimes G$, where $H$ is a subgroup of $G$. The corresponding equivalence of representations is an ...
- 12.9k
2
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0
answers
228
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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
9
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1
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434
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Questions on the group $\mathrm{GL}(H)$
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\U{U}$Let $H$ be an infinite dimensional complex Hilbert space. Consider the group $\GL(H)$ of bounded invertible operators on $H$.
Question 1. I've ...
- 18.7k
5
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1
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354
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Density of matrix coefficients of unitary representations of a locally compact group
Let $G$ be a locally compact group, $C_0(G)$ the $C^*$-algebra of continuous functions on $G$ that vanish at infinity, $C_b(G)$ the $C^*$-algebra of bounded continuous functions on $G$. We know that $...
- 25.8k
3
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0
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205
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Status of RFD groups and $C^*$-algebras
Motivated by this question and its great answers, I become very curious to know what do we know about RFD (residually finite dimensional) groups and $C^*$-algebras, e.g. do we know how these ...
- 1,586
2
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0
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147
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About the algebraic structure of the $G$-equivariant $KK$-theory
Let $ G $ be a second countable locally compact group.
Let $ A $ and $ B $ be two $G$-$C^*$-algebras.
Let $ KK^G (A, B) $ be the $G$-equivariant $KK$-theory of the pair $ (A, B) $.
Could you tell me ...
- 35.4k
1
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0
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283
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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
2
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1
answer
316
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Decomposition of Hilbert spaces via groups and algebras representations
Let $\mathcal{H}$ be a complex finite dimensional Hilbert space and let $\mathcal{A}\subseteq \mathcal{B}(\mathcal{H})$. I am looking to understand the different decompositions of $\mathcal{H}$ ...
- 12.9k
18
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1
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458
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For what topological groups $G$ can we take $EG \rightarrow BG$ to be of the form $S^{\infty} \rightarrow BG$?
Title. For what topological groups $G$ can we take $EG \rightarrow BG$ to be of the form $S^{\infty} \rightarrow BG$?
If $G$ is a subgroup of either $S^0,S^1,S^3$ or $S^7$ this induces a free action ...
- 27.5k
8
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0
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157
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Relation between the Laplace eigenvalue of Maass forms and the Virasoro algebra
I attended today a talk of Dorian Goldfeld. In the talk, he mentioned that for a Maass cusp form $\phi$ of $\mathrm{SL}(n, \mathbb{Z})$, with Langlands parameter $\alpha = \left( \alpha_1, \dots, \...
- 3,065
21
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2
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944
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Which p-adic algebraic groups are type I?
It was proved by Jacques Dixmier (Sur les représentations unitaires des groupes de Lie algébriques, Annales de l'institut Fourier, 7 (1957), p. 315-328, doi: 10.5802/aif.73, MR 20 #5820, Zbl 0080....
- 412
8
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0
answers
251
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When does a semisimple $\mathbb{C}$-algebra come from a group?
Let $\mathcal{A}$ be a semisimple $\mathbb{C}$-algebra. By the Artin-Wedderburn theorem, it is isomorphic to a direct product of matrix algebras:
$$ \mathcal{A} = \prod_{i=1}^m M_{n_i}(\mathbb{C})$$
...
- 297
7
votes
1
answer
201
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Unitary representation is strictly continuous
Let $G$ be a compact group and $u: G \to B(H)$ be a strongly continuous unitary representation on the Hilbert space $H$. Then is $u: G \to B(H)$ strictly continuous?
That is, give $B(H)$ the topology ...
- 18.7k
6
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2
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874
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Is the set of all ICC amenable groups countable?
Is the set of all ICC amenable groups countable?
If "yes", then in general, the classes of all countable ICC groups that give rise to the same von Neumann algebra (factor) -- are these ...
- 63.9k
0
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1
answer
325
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Group algebras and group automorphisms
Say, we have a countable ICC group $G$, a Hilbert space $H$ with a basis indexed by the group elements, the group algebra generated by the left regular representation of $G$ on this Hilbert space, and ...
- 456
2
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0
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116
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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 ...
- 63.9k
1
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1
answer
178
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Representation of algebras as bounded nilpotent operators
Let $A$ be a real/complex algebra (just a real/complex vector space with a multiplication; none PI's are required). Let $\mathcal{H}$ be a real/complex Hilbert space. Let $\operatorname{B}(\mathcal{H})...
- 42.8k
3
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0
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270
views
How to compute a simultaneous block-diagonalization?
Let $n$ be a positive integer and consider of finite set $S \subset M_n(\mathbb{C})$ such that $S^* = S$ (i.e. if $a \in S$ then $a^* \in S$). The algebra generated by $S$ is a finite dimensional $*$-...
1
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1
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448
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irreducible representation of a $C^*$ algebra
Suppose we have a $C^*$ algebra $A=\{(x_n)\in \prod M_n(\Bbb C),lim_n tr_n(x_n^*x_n)=0\}$.
If $B$ is any nonzero $C^*$-sub algebra of $A$,does there exist a finite dimensional irreducible ...
- 451
6
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2
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487
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Induction theorems for finite-dimensional complex representations of infinite groups
Let $G$ be a group, usually infinite. I am interested in finite-dimensional complex unitary representations of $G$, i.e. group homomorphisms $G \rightarrow U_n(\mathbb{C})$. The category of these ...
- 12.9k
1
vote
0
answers
229
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Tensor product decomposition of commuting representations
If $\mathscr{X}$ is a Hilbert space, we denote by $\mathrm{GL}(\mathscr{X})$ the group of all bounded operators on $\mathscr{X}$ with bounded inverses. Let $\mathbb{F}_2$ be the free group on two ...
- 1,769
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0
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290
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A robust version of Schur's lemma?
Does a robust version of Schur's lemma exist? Specifically, I was wondering about something like this:
Let $B$ be a bounded operator over a vector space $V$, with underlying field $\mathbb{C}$ and ...
- 91
3
votes
1
answer
1k
views
injective tensor norms for real tensors
If $A$ is an element of $\mathbb{R}^n \otimes\mathbb{R}^n \otimes\mathbb{R}^n$, then define its injective tensor norm to be
$$\|A\|_{\rm inj} := \max_{x,y,z\in \mathbb{R}^n, \|x\|=\|y\|=\|z\|=1}
|\...
- 316
9
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1
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521
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Which group algebras in analysis are "true group algebras"?
Let $G$ be a group, $A$ a unital associative algebra over ${\mathbb C}$, and let us call a representation of $G$ in $A$ an arbitrary map $\pi:G\to A$ such that
$$
\pi(1)=1,\qquad \pi(a\cdot b)=\pi(a)\...
- 7,426
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0
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230
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How to decompose the left regular representation of a real reductive group?
In Dixmier ($C^*$-algebras), the Plancherel theorem states (I will not mention the right regular representation even though the theorem does talk about it):
Let $G$ be unimodular, $\lambda$ be the ...
- 4,713
0
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1
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144
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Dualizing the trivial action on a $C^*$-algebra
Let $G$ be a finite abelian group (cosidered as a discrete topological group), $A$ a unital separable $C^*$-algebra. Let $T\colon G\to \operatorname{Aut}(A)$, $T_g(a)=a$ for all $g\in G$ the trivial ...
- 830
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1
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542
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Strengthening of Connes' embedding conjecture
If $A$ and $B$ are $C^*$ algebras I will write $A \overline{\otimes} B$ for the maximal tensor product and $A \underline{\otimes} B$ for the minimal tensor product. If $G$ is a countable discrete ...
- 5,005
3
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0
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141
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Existence of a unique cyclic and separating vector in a *-representation
I'm interested in knowing the requirements for a $*$-representation, $\pi_{\omega}$, of a C*-algebra, $\mathbb{C}(\mathcal{G})$, (or equivalently the requirements for the unitary representation, $U_{\...
- 31
3
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0
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269
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Finite dimensional representation of tensor product
Let $A$ and $B$ be $C^*$ algebras, and let $\pi:A \odot B \to B(H)$ be a $*$-representation of the algebraic tensor product on a finite dimensional Hilbert space $H$. Let $x \in A \odot B$. Since $H$ ...
- 1,769
11
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2
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537
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Groups without property (T) but all finite quotients are expanders
What is an example of a group $G$ which
1- is finitely generated by $S$,
2- does not have property (T),
3- admits infinitely many finite quotients which do not factor through an homomorphism $G \...
- 638
3
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0
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237
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Orthogonality relations for unitary representations of infinite (finitely generated) groups
Let $G$ be a group, and consider the matrix elements of finite dimensional irreducible unitary representations of $G$ over $\mathbb{C}$ as functions $f:G\to \mathbb{C}$. If $G$ is finite, any two ...
- 63.9k
3
votes
0
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61
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Isometry from a representation to the representation tensored with itself
Suppose, the group $ G=S(2^{\infty})$ has a unitary representation $ \pi $ on a separable infinite dimensional Hilbert space $ H $.
(The group $ S(2^{\infty}) $ is the direct limit of the following ...
- 817
2
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0
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157
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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 ...
- 25.8k
8
votes
1
answer
422
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The positive cone of the standard representation of a Von Neumann algebra
Let $A$ be a von Neumann algebra, let $L^2(A)$ be the underlying Hilbert space of the standard form of $A$, and $P \subset L^2(A)$ the canonical positive cone (see for example this paper by Haagerup). ...
- 42.4k
3
votes
1
answer
506
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Is there a link between $H_2(G,\mathbb{Z})$, the Schur Multiplier of a group, and the "other" Schur multipliers of a group?
The name for the the following 2 mathematical objects:
$$H_2(G,\mathbb{Z})$$ and
$$\{K:G\times G\longrightarrow\mathbb{C}\ |\ \forall T\in B(l^2(G))\text{we have that}~S:G\times G\longrightarrow\...
- 424
3
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0
answers
168
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The left regular representation of the Jacobi groups over local fields of characteristic >2 is type I?
Let $K$ be a non-archimedean local field of characteristic $>2$. Consider the Jacobi group $G=H_{2n+1}(K)\rtimes Sp_{2n}(K)$, which is the semidirect product of the Heisenberg group $H_{2n+1}(K)$ ...
- 1,702
3
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0
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144
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Deformation and Representations
Let $\widetilde{U_q(sl_n)}$ denote a deformation of the algebra $U_q(sl_n)$. In particular, $\widetilde{U_q(sl_n)}$ is defined by the same generators and relations and $*$-operations as $U_q(sl_n)$ ...
- 11.2k
2
votes
1
answer
291
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Mysterious central projections in the full group $C^*$-algebra
Let me quote the following theorem about the structure of $C^*(G)$ for property $T$ group (the reference is Higson and Roe "Analitycal K-homology"):
Let $G$ be a property $T$ (discrete) group....
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