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Results tagged with oa.operator-algebras
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user 75
Algebras of operators on Hilbert space, $C^*-$algebras, von Neumann algebras, non-commutative geometry
11
votes
Accepted
Who gave the generalized Stone-Weierstrass Theorem?
This (for real-valued rather than complex-valued functions) was in Stone's original paper that proved the Stone-Weierstrass theorem, as Theorem 84. The statement is a bit funny, since he defines the …
- 31.2k
6
votes
Accepted
Classification of $C^*$ algebras whose all non scalar elements have disconnected spectrum
In the commutative case $A=C(K)$, this is equivalent to asking for which compact Hausdorff spaces there are no continuous surjections $K\to[0,1]$, since the spectrum of $f\in C(K)$ is its image. A co …
- 31.2k
5
votes
Accepted
A version of the spectral theorem for group actions
This is true for locally compact abelian $G$; you ought to be able to find it in any text on abstract harmonic analysis (a reference I have at hand is Theorem 4.44 in Folland's A Course in Abstract Ha …
- 31.2k
6
votes
Accepted
Number of II${}_1$ factors
Your argument is correct. An alternate and more "intrinsic" argument is to look at the predual, which is a separable Banach space. There are only continuum many separable Banach spaces, since they a …
- 31.2k
7
votes
Accepted
Transfinite induction, a theorem of Pedersen, and chains of subalgebras of $B(H)$
The answer to Question 2 is yes; such a chain can be found using only diagonal operators with respect to some fixed basis $(e_n)_{n\in{\mathbb N}}$ for $H$. As a starting point, you can find an $\ome …
- 31.2k
8
votes
Accepted
Schur's Lemma for Hilbert spaces
To elaborate on my comment, let us suppose that $G$ is closed under taking adjoints (in particular, this holds if $G$ is unitary). Then it is easy to see $D$ is also closed under adjoints, so for any …
- 31.2k