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4 exercise(s) shown, 0 hidden
BBot
Apr 22'25
[math]
\newcommand{\mathds}{\mathbb}[/math]
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Prove that the finite dimensional [math]C^*[/math]-algebras are exactly the direct sums of matrix algebras
[[math]]
A=M_{N_1}(\mathbb C)\oplus\ldots\oplus M_{N_k}(\mathbb C)
[[/math]]
by decomposing first the unit into a sum of central minimal projections.
BBot
Apr 22'25
[math]
\newcommand{\mathds}{\mathbb}[/math]
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Given a matrix [math]M\in M_N(\mathbb C)[/math] having norm [math]||M||\leq1[/math], prove that
[[math]]
P=\lim_{n\to\infty}\sum_{k=1}^nM^k
[[/math]]
exists, and equals the projection onto the [math]1[/math]-eigenspace of [math]M[/math].
BBot
Apr 22'25
[math]
\newcommand{\mathds}{\mathbb}[/math]
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Work out the details of the abstract Weingarten integration formula in the group dual case, where [math]A=C^*(\Gamma)[/math] with [math]\Gamma= \lt g_1,\ldots,g_N \gt [/math].
BBot
Apr 22'25
[math]
\newcommand{\mathds}{\mathbb}[/math]
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Work out in detail the representation theory for the basic operations, namely products, dual free products, quotients, projective versions.