Topological group actions by group automorphisms and Banach representations. (English) Zbl 07812392

Megrelishvili, Michael

Forum Math. 36, No. 2, 327-338 (2024).

Let a group \(G\) act on a topological space \(X.\) A representation of the \(G\) -space \(X\) on a Banach space \(V\) (and its dual \(V^{\ast }\)) is a continuous map \(\alpha :X\rightarrow V^{\ast }\) that is weak-star continuous bounded \(G \)-equivariant, with respect to some linear representation \(h:G\rightarrow Iso(V)\), where \(Iso(V)\) is the linear isometric group. A typical example is that \(X\) is compact and \(V=C(X,\mathbb{R}),\) the Banach space of continuous real functions on \(X.\) It is a less known fact (see [J. Adámek (ed.) and S. MacLane (ed.), Categorical topology and its relation to analysis, algebra and combinatorics. International conference CAT-TOP, Prague, Czechoslovakia, August 22-27, 1988. London: World Scientific. 220–237 (1989; Zbl 0849.18001); M. G. Megrelishvili, N. Z. J. Math. 25, No. 1, 59–72 (1996; Zbl 0848.22003)]) that the \(G\) -space \(G\) (conjugation action) is Banach representable (i.e. \(\alpha \) is topological embedding) for every topological group \(G\). In the article under review, the author considers Banach representability when \(V\) is additionally Hilbert, reflexive, Asplund, or Rosenthal.
Some examples and facts are discussed. For example, the conjugation action of \(\mathrm{SL}_{2}(\mathbb{R})\) on itself is not representable on reflexive Banach spaces; The conjugation action of \(\mathrm{SL}_{n}(\mathbb{R}),n\geq 4,\) on itself, the linear action of \(\mathrm{GL}_{n}(\mathbb{R}),n\geq 2,\) on \(\mathbb{R}^{n},\) are not representable on Asplund Banach spaces; On the other hand, the latter action is representable on a Rosenthal Banach space.

Reviewer: Shengkui Ye (Suzhou)

MSC:

20F29

Representations of groups as automorphism groups of algebraic systems

Keywords:

representation on Banach spaces; equivariant compactification

Citations:

Zbl 0849.18001; Zbl 0848.22003

References:

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