bio | website | maths.gla.ac.uk/~gtornetta |
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location | Glasgow, Scotland | |
age | 27 | |
visits | member for | 3 years, 6 months |
seen | 3 hours ago | |
stats | profile views | 103 |
I'm currently involved in a PhD in Mathematics. My research area is that of Functional Analysis. Specifically I'm working on classification of C*-algebra by defining a bivariant version of the Cuntz semigroup, an invariant used to compare positive elements in a C*-algebra and hence infer information about its internal structure.
During my MSc I focused on Algebraic Quantum Field Theory and I have worked on DFR models for Quantum Space-time.
Jul 15 |
awarded | Commentator |
Jul 15 |
comment |
C*-algebras and bounded relations
Thanks for the reference. My question is on the topic, but asks some clarifications about that particular method of proof of said universality. |
Jul 15 |
comment |
Universal $C^*$-algebra with generators and relations
As an application of this in Quantum Mechanics, one can consider the position and momentum operators $x$ and $p$, which, according to the standard quantization, must satisfy $xp - px \subset 1$ (assuming natural units, where $\hbar = 1$). Then the above argument shows that $x$ and $p$ cannot be bounded operators. To deal with bounded operators one can do the Weil trick to take the exponentials $e^{i\xi p}$ and $e^{i\eta x}$ in order to get unitaries through Borel functional calculus. |
Jun 12 |
revised |
C*-algebras and bounded relations
added 6 characters in body |
Jun 12 |
revised |
Continuous linear functionals in strong operator and $\sigma$-strong topologies
added 11 characters in body |
Jun 12 |
answered | Continuous linear functionals in strong operator and $\sigma$-strong topologies |
Jun 12 |
asked | C*-algebras and bounded relations |
Apr 5 |
comment |
Projective limit construction of a semigroup
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! |
Apr 5 |
comment |
Projective limit construction of a semigroup
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? |
Apr 5 |
comment |
Projective limit construction of a semigroup
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. |
Apr 5 |
comment |
Projective limit construction of a semigroup
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! |
Apr 4 |
awarded | Scholar |
Apr 4 |
comment |
Projective limit construction of a semigroup
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! |
Apr 4 |
accepted | Projective limit construction of a semigroup |
Apr 4 |
revised |
Projective limit construction of a semigroup
edited tags |
Apr 3 |
asked | Projective limit construction of a semigroup |
Jan 13 |
answered | *-homomorphisms between matrix algebras |
Jun 28 |
awarded | Teacher |
Mar 16 |
awarded | Supporter |
Jul 27 |
awarded | Student |