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Algebras of operators on Hilbert space, $C^*-$algebras, von Neumann algebras, non-commutative geometry

1 vote

Element Analytic, C*-dynamical system

In this case, a (improper) Riemann integral will suffice. The reason is that the function you want to integrate is continuous: you need (strong) continuity of $\alpha$ as a basic assumption throuhgout …
Stefan Waldmann's user avatar
3 votes

center of the algebra of bounded operators

The argument of BS works also in the case where $X$ is a Hausdorff locally convex space since the topological dual still separates points (by Hahn-Banach). This is enough to show that the (continuous) …
Stefan Waldmann's user avatar
4 votes

$*$-representation $\pi:A\odot B\to B(H_1\otimes H_2)$ such that $\pi \neq \pi_1\otimes \pi_2$

Suppose for simplicity that $A$ and $B$ are unital. If now $\pi = \pi_1 \otimes \pi_2$ then the operators $\pi(a \otimes 1)$ commute with all the operators of the form $id \otimes B$ with $B \in B(H_2 …
Stefan Waldmann's user avatar
28 votes
Accepted

Quantum mechanics formalism and C*-algebras

In addition to what has already been said I would like to add some more comments. I completely understand your suspicion that the passage from unbounded operators to bounded ones is at least tricky. F …
Stefan Waldmann's user avatar
8 votes
Accepted

Stinespring's dilation without $C^{\ast}$-algebras

The surprising fact is that the GNSStinespringKasparov theory is in fact completely algebraic, at least to a very large extend: the following results have been obtained by a PhD of mine but are, unfor …
Stefan Waldmann's user avatar
6 votes
Accepted

Formal series convergence in deformation quantization and $C^*$-condition

OK, let me give a try on this question. There are several problems hidden underneath which one has to address. First, for physical reasons a formal deformation is not sufficient. $\hbar$ is a constan …
Stefan Waldmann's user avatar