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In page 280 of "C^* algebra" by Dixmier, in the context of group representation, it is written 'We never encounter the concept of algebraic irreducibility except in finite dimensional Hilbert space".

According to the fact that "For $C^*$ algebra representation both topological and algebraic irreducibility are the same" so why it is not the case for group representation? What does it mean "We never encounter algebraic irreducible representation"? From the view of irreducibility, what is the difference between algebra representation and group representation?

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    $\begingroup$ Algebraic irreducibility for the group is the same as algebraic irreduciblility for the complex group algebra which is much smaller than the C*-algebra if you ignore all topology. So it shouldn't be that surprising if there are algebraically irreducible reps if the C*-algebra which are not algebraically irreducible for the group. $\endgroup$
    – Benjamin Steinberg
    Commented Mar 25, 2022 at 12:08
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    $\begingroup$ The comments to math.stackexchange.com/questions/4384903/… give an example $\endgroup$
    – Benjamin Steinberg
    Commented Mar 25, 2022 at 14:35
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    $\begingroup$ There is no algebraically irreducible representation of any countable group $G$ on an infinite dimensional Hilbert space! Any vector $\xi$ lies in the (countably generated, hence) proper subspace $\text{span}\ G\xi$. On the other hand, there are surely many examples of topologically irreducible representations! $\endgroup$
    – Ruy
    Commented Mar 26, 2022 at 15:56
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    $\begingroup$ @Ali, I didn't quite understand your question. Could you please expand it. Do you mean $G\zeta$ finite for all $\zeta$ and all representations? $\endgroup$
    – Ruy
    Commented Mar 27, 2022 at 16:32
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    $\begingroup$ @Ruy: I think it's true that if a Banach space $X$ admits an algebraically cyclic continuous action of a second countable locally compact group, then it is finite-dimensional. $\endgroup$
    – Narutaka OZAWA
    Commented Mar 28, 2022 at 6:49

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