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Apr
18
awarded  Yearling
Apr
16
comment Morita equivalence base equivalence relation for discrete groups
@Shakiba: Indeed it seems that you can replace "countable" by "discrete", if you insist on dealing with huge groups. The reason is that Cech cohomology works well for arbitrary compact spaces, not only metrizable ones. Of course in the above argument you must then work with arbitrary directed sets, not only $\mathbb{N}$.
Apr
15
answered Morita equivalence base equivalence relation for discrete groups
Apr
15
comment Morita equivalence base equivalence relation for discrete groups
@user89334: Oops I meant the 3-element group! :-)
Apr
14
comment Morita equivalence base equivalence relation for discrete groups
If two C*-algebras are Morita equivalent, they have equivalent representation theories and homeomorphic duals. For finite groups: $G \sim H$ if and only if $|\hat{G}|=|\hat{H}|$. For example the two-element group is equivalent in your sense to the symmetric group $Sym(3)$. May I ask whether you have non-trivial examples among torsion-free groups?
Apr
14
comment Morita equivalence base equivalence relation for discrete groups
@Sebastian: Observe: $G\mapsto C^*(G)$ is indeed functorial, but $G\mapsto C^*_r(G)$ is not: the latter is functorial only for group monomorphisms; and whenever $C^*_r(G)$ is simple, i.e. frequently, you have a counter-example to functoriality in general, by looking at the map $G\rightarrow \{1\}$.
Mar
27
awarded  Necromancer
Jan
29
comment $K$-Theory of finite dimensional Banach algebras
@Fermando: Because, for non-unital algebras $K_0(A)$ is defined as $\ker[K_0(\tilde{A})\rightarrow K_0(k)=\mathbb{Z}]$, where $\tilde{A}$ is the unitization of $A$, so you don't deal with f.g. projective modules, but with differences of modules having the same dimension.
Jan
29
comment $K$-Theory of finite dimensional Banach algebras
@Fernando: I think you are implicitly assuming that your algebra has a unit.
Jan
25
comment Spectrum of the Laplace-Beltrami operator on a domain of finite volume in the hyperbolic space $H^n$
@Vladimir: The link provided by Marcel contains a number of useful references. See also: en.wikipedia.org/wiki/Laplace–Beltrami_operator
Jan
25
comment Spectrum of the Laplace-Beltrami operator on a domain of finite volume in the hyperbolic space $H^n$
You should check Chapters XII, XIII and XIV of Serge Lang's book ``$SL_2(\mathbb{R})$'' (where SL doesn't stand for Serge Lang): you will see that, for $\Gamma=SL_2(\mathbb{Z})$, the spectrum of the Laplace operator has both a continuous and a discrete part. Everything is very explicit. On the other hand, it is a classical fact that, on a compact Riemannian manifold, the Laplace operator has compact resolvent, hence has discrete spectrum.
Jan
25
revised Construct discrete series of SL(2,R) as kernel of twisted Dirac operators
added 13 characters in body
Jan
25
answered Construct discrete series of SL(2,R) as kernel of twisted Dirac operators
Jan
24
comment Construct discrete series of SL(2,R) as kernel of twisted Dirac operators
This way you get only "half of" the discrete series of $SL_2(\mathbb{R})$, namely the holomorphic discrete series, corresponding to characters $z\mapsto z^n$ of $K$ with $n>0$. You miss the anti-holomorphic discrete series, corresponding to $n<0$.
Jan
14
comment The positive cone of the standard representation of a Von Neumann algebra
@Yemon: indeed, and weakly continuous representations as well.
Jan
8
comment Is there a link between $H_2(G,\mathbb{Z})$, the Schur Multiplier of a group, and the “other” Schur multipliers of a group?
I fully support Yemon's answer and comment! Sometimes it happens in mathematics that two completely unrelated objects have similar names...
Jan
3
comment The left regular representation of the Jacobi groups over local fields of characteristic >2 is type I?
Liminality of $H_3(K)$ ($K$ non-archimedean) follows from the explicit description of the unitary irrep's, as given e.g. in section 1.2 of V. Lafforgue's paper: vlafforg.perso.math.cnrs.fr/haagerup-rem.pdf I expect the proof to be essentially the same for $H_{2n+1}(K)$.
Dec
23
comment Which finite groups can be characterized by their automorphism groups?
Carmichael's conjecture states that the equation $\varphi(x)=n$ (where $n$ is given) never has a unique solution. This amounts to saying that the finite cyclic group $\mathbb{Z}/n\mathbb{Z}$ is never characterized by the order of its automorphism group.
Oct
24
comment Infinite k-connected planar graphs
Recently, there has been extensive work on planar Cayley graphs by Agelos Georgakopoulos, see his website: homepages.warwick.ac.uk/~maslar/pubsFrames.html
Oct
9
comment Is the sum of spectral projections a projection?
@Frank: It is rather the holomorphic functional calculus that is well-behaved... This holds in any Banach algebra!