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Results tagged with c-star-algebras
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user 12482
A C*-algebra is a complex Banach algebra together with an isometric antilinear involution satisfying (a b)* = b* a* and the C*-identity ‖a* a‖ = ‖a‖². Related tags: [banach-algebras], [von-neumann-algebras], [operator-algebras], [spectral-theory].
6
votes
Non degenerate representations for C*-algebras
In fact, for unital $C^*$-algebras non-degeneracy just means $\pi(1) = 1$. In the non-unital case there is even a sharper statement than your item (2): One can find for every $\phi$ and every $\epsilo …
- 8,098
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 …
- 8,098
7
votes
Accepted
Checking complete positivity of maps between C* algebras
Yes, an $n$ positive map is also $n-1$ positive. Hence you map is $n$-positive for all $n$, i.e. completely positive.
For a proof, you include $M_{n-1}(A)$ as upper left block into $M_n(A)$.
- 8,098
4
votes
Reference: Learning noncommutative geometry and C^* algebras
For $C^*$-algebras in general, there are many textbooks. Famous names are e.g. the two volume book by Kadison&Ringrose or the (by now somehow old but still very nice) book by Sakai.
I also enjoyed the …
- 8,098
4
votes
Accepted
Hermitian vector bundles and Hilbert $C^*$-modules
In addition to Nik Weaver's references, let me just sketch the proof which is in fact not very difficult:
A construction of Kaplansky (Rings of operators, Thm 26) shows that if $\mathcal{A}$ is a $*$ …
- 8,098