User phoenix87 - MathOverflow

Projective limit construction of a semigroup

2013-04-03T16:05:16Z

Let $\tilde{\mathbb N}$ be the Abelian semigroup (under addition) given by $\mathbb N\cup\{0,\infty\}$, and let $S_n$ be the Abelian monoid $\tilde{\mathbb N}^{2^n}$ under point-wise addition. Introduce the transition maps $\{\phi_n:S_{n+1}\to S_n\}_{n\in\mathbb N}$ defined by

$$\phi_{n-1}(n_1,n_2,n_3,n_4,\ldots,n_{2^n-1},n_{2^n}) := (n_1+n_2,n_3+n_4,\ldots,n_{2^n-1}+n_{2^n}),$$

i.e. summing all consecutive pairs of integers.

Now consider the sequence $(S_n,\phi_n)_{n\in\mathbb{N}}$. Does the projective limit

$$S=\lim_{\longleftarrow}S_n\subset\prod_{k\in\mathbb N}S_n$$

have a name as an Abelian semigroup, namely is such $S$ an object in a well-studied family of semigroups so that it can be actually identified among them?

Answer by Phoenix87 for *-homomorphisms between matrix algebras

2013-01-13T21:17:43Z

I do understand this is an old question but, considering that: 1. you might still be interested in a simpler (and surely less elegant) proof; 2. I had the same problem, so this might be helpful for others in the future I'll tell you how I solved this problem

Take your *-homomorphism $\lambda:M_n\to B$, where $B$ is any other C*-algebra. Set $$F_{ij}:=\lambda(E_{ij}),\qquad\forall i,j=1,\ldots,n,$$ where the $E_{ij}$ is the canonical basis of the underlying vector space of $M_n$ (then the $F_{ij}$ satisfy the very same algebra, i.e. $F_{ij}F_{mn}=\delta_{jm}F_{in}$), and uppose that the kernel of $\lambda$ is non-trivial. Then there exists $a\in M_n\smallsetminus {0}$ s.t. $\lambda(a)=0$. This is a constraint between all the $F_{ij}$, namely there are coefficients $\alpha_{ij}$ such that $$\sum \alpha_{ij}F_{ij}=0.$$ Note that if just one of the $F_{ij}$ for some $(i,j)$ is 0, then $\lambda$ is 0, because $F_{kk}$ are all Murray-von Neumann equivalent projections, and the "off-diagonal" elements $F_{ij}$, $i\neq j$ are partial isometries linking them. Therefore sandwich the above linear combination between $F_{kk}$ and $F_{mm}$ to obtain $$0=\sum_{ij}\alpha_{ij}F_{kk}F_{ij}F_{mm} = \alpha_{km}F_{km},$$ which implies $$\alpha_{km}=0$$ for arbitrary $(k,m)$. Hence all the $F_{ij}$ are linearly independent, meaning that, as a vector space, $B$ must have enough space to accommodate at least a copy of $M_n$. But taking into account once again that all the projections $F_{kk}$ are Murray-von Neumann equivalent, we can conclude that there must exists a subalgebra $C$ of $B$ such that $M_n\otimes C\subset B$ and a projection $P\in C$ such that, up to unitary equivalence $$\lambda(E_{ij})=E_{ij}\otimes P\in M_n\otimes C.$$ If the rank of $P$ is $k$, then $B$ should be large enough to accommodate $k$ copies of $M_n$, i.e. the underlying vector space of B must have dimension greater than $kn$ (if it is not exactly $kn$ the you'll have some 0-padding).

Hope this helped like it did for me!

Is the F.T of $\operatorname{Span}(\mathscr S(\mathbb R^2)\otimes\mathscr D(D))$, $D\subset\mathbb R^2$ dense in $L^2(\mathbb R^4)$?

2011-06-27T07:47:29Z

Let $K$ be the real vector space generated by elements $f$ in $\mathscr S(\mathbb R^2,\mathbb R)\otimes\mathscr D(D, \mathbb R)$, where $D$ is any bounded subset of $\mathbb R^2$. Let $\hat K$ be the vector space generated by the Fourier transform of each $f\in K$, i.e. $\hat K = \{\hat f\ |\ f\in K\}$. Is $\hat K+i\hat K$ dense in $L^2(\mathbb R^4)$?

Answer by Phoenix87 for Is the F.T of $\operatorname{Span}(\mathscr S(\mathbb R^2)\otimes\mathscr D(D))$, $D\subset\mathbb R^2$ dense in $L^2(\mathbb R^4)$?

2011-06-29T10:20:18Z

I'm sorry if this is not the appropriate place to ask such questions. I'll be more careful next time. By the way my question is strongly related to an actual problem in QFT (actually QFT on QST). I'm not sure about M. Bischoff conclusion on orthogonality pointed out in one of the comments to the question.

There is a result from Araki that state that the real Hilbert subspace of the one-particle Hilbert space $K(O)=\{\hat f\big\vert_{\Omega_m^+}\ |\ f\in\mathscr D(O,\mathbb R)\}$, where $\Omega_m^+$ is the hyperboloid of mass $m$ in the future light-cone and $O\subset\mathbb R^4$ is a non-empty simply connected bounded open subset of $\mathbb R^4$, is standard, i.e. $\overline{K(O)+iK(O)}$ is dense and $K\cap iK=\{0\}$. The vector space $\hat K$ I defined in the question ought to contain such a vector space $K(O)$ associated with a region $O\subset\mathbb R^2\times D$, where $D$ satisfies some suitable regularity condition (e.g. regular boudary, simply connectedness,...) and therefore $K(O)\subset \hat K$, that would imply $\hat K$ standard, according to the result from Araki.

Comment by Phoenix87

2013-04-05T22:32:45Z

Ok so the difference is just in the set $X$, and the restrictions on the universal property account for the different nature of such set $X$. Anyway that is exactly what I was expecting, for this former case actually comes from $F(C_0(\mathbb N),\mathcal K)$, where $C_0(\mathbb N)$ denotes the C*-algebra of continuous functions on $\mathcal N$ (which isn't compact) vanishing at $\infty$. So thanks a lot for your help with this matter!

Comment by Phoenix87

2013-04-05T21:10:27Z

Yes I actually meant 0 by "dropping" in this particular example. So the limit would not be a pro-ACom object, but I guess this non compact $X$ is just $\mathbb N$ in this other particular example?

Comment by Phoenix87

2013-04-05T18:59:19Z

Dear kar, thank you very much for the suggested reading and for your further explanation, it helped a lot and I think I can more clearly see the link with the Cantor set. I actually have a similar construction with $\mathbb N$ viewed as the limit of $\{1,\ldots,n\}$ with transition maps that act, on the generators, as $x_k\mapsto x_k$ for any $k=1,\ldots,n$ and $x_{n+1}$ is dropped. Hence this should lead to $\tilde{\mathbb N}^{\mathbb N}$ in the limit on the free pro-ACom semigroups.

Comment by Phoenix87

2013-04-05T15:48:27Z

I was trying to locate some references for such category of semigroups, in particular some sources about how to construct transition maps between the semigroups in the projective limit given the topological space (the Cantor set in the example in question)? Could you please refer me to some literature? That would be great. Cheers!

Comment by Phoenix87

2013-04-04T23:28:32Z

The object $S$ in question arises from considering the C*-algebra C(X) of continuous functions on the Cantor set as first argument in a bivariant functor $F(C(X), \mathcal K)$, where $\mathcal K$ is the C*-algebra of compact operators. So the answer you provided makes a lot of sense in this context. Thank you very much!

Comment by Phoenix87

2012-06-13T17:59:18Z

Ok I'll probably go ask this question to a kid from a kindergarten nearby then. Thanks.

Comment by Phoenix87

2012-06-13T17:00:09Z

Could you please provide an example of a vector subspace which is not closed nor open? Thank you!

Comment by Phoenix87

2011-06-28T07:02:58Z

$\mathscr D(D)$ is the space of smooth function with compact support contained in a fixed bounded subset $D\subset\mathbb R^2$. Of course M. Bischoff is right, my mistake. Now I've fixed the question.