Timeline for Notions in the literature capturing the "symmetric" or "homogeneous" flavour of $L_p$?

4 events

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Jun 9, 2019 at 14:10	comment	added	YCor		Irreducibility of the action of the group of isometries is tempting. However, it has the drawback of being only an isometry invariant, not isomorphism invariant. "Irreducibility of the (linear) isometry group for some equivalent norm" is an isomorphism invariant, but is a less handy notion; I don't know if it's satisfied by $\ell^p\oplus\ell^q$, $p\neq q$.
Jun 9, 2019 at 11:11	answer	added	Tomasz Kania		timeline score: 4
Jun 9, 2019 at 8:27	comment	added	Jochen Glueck		A (maybe only loosely related) remark: Within the class of all Banach function spaces, the "symmetric" nature of, for instance, $L^p$ and $\ell^p$ can in a way be described by the notion of a "rearrangement invariant space"; and within the class of all Banach lattices one could perhaps consider a property such as "the group of all isometric lattice isomorphisms acts ideal irreducibly on the given space" to capture the notion of symmetry (although this is admittedly only a vague idea). But I'd guess that you are perhaps more interested in the category of Banach spaces?
Jun 8, 2019 at 23:15	history	asked	Yemon Choi	CC BY-SA 4.0