In this paper we investigate some hereditary properties of amenability modulo an ideal of Banach algebras. We show that if $(e_alpha)_alpha$ is a bounded approximate identity modulo I of a Banach algebra A and X is a neo-unital modulo I, then $(e_alpha)_alpha$ is a bounded approximate identity for X. Moreover we show that amenability modulo an ideal of a Banach algebra A can be only considered ...

We show that for any abelian topological group G and arbitrary diffused submeasure μ, every continuous action of L0(μ,G) on a compact space has a fixed point. This generalizes earlier results of Herer and Christensen, Glasner, Furstenberg and Weiss, and Farah and Solecki. This also answers a question posed by Farah and Solecki. In particular, it implies that if H is of the form L0(μ,R), then H ...

We associate Popa systems (= standard invariants of subfactors, cf. [P3],[P4]) to the finite dimensional representations of compact quantum groups. We characterise the systems arising in this way: these are the ones which can be “represented” on finite dimensional Hilbert spaces. This is proved by an universal construction. We explicitely compute (in terms of some free products) the operation o...

We will discuss the motivating background leading to the consideration of the concept of amenability for von Neumann subfactors of nite Jones index and respectively for the combinatorial objects associated with them (standard graphs, standard lattices, paragroups, etc). Then we will review some of the results related to these notions of amenability, outlining the papers 20]{{25] and the lecture...

We present several examples of supersymmetric quantum mechanical systems with weak superalgebra $su(N|1)$. One them is the $su(N|1)$ oscillator. It has a singlet ground state, $N +1$ degenerate states at first excited level, etc. Starting from level $k = N+1$, system complete multiplets each involving $2^N$ states. Due to fact that supermultiplets are not for \leq N$, Witten index represents no...

We survey the recent investigations on (bounded, sequential) approximate Connes amenability and pseudo-Connes amenability for dual Banach algebras. We will discuss the core problems concerning these notions and address the signicance of any solutions to them to the development of the eld.