A Hilbert module defined in "L^2-invariants: theory and applications to geometry and K-theory", Springer-Verlag, 2002, by W. Lück: A Hilbert $\mathcal N(G)$-module $V$ is a Hilbert space $V$ together with a linear isometric $G$-action such that there exists a Hilbert space $H$ and an iso...

Let $A = C(X)$ be a commutative $C^*$-algebra. An example of a full finitely generated Hilbert $A$-bimodule E such that the crossed product $C(X) \rtimes_E \mathbb{Z}$, as defined by Abadie, Eilers and Exel is simple is given by $E =C(X)$ viewed as a bimodule over itself with mulitplication on th...

Rubik's cube and its generalizations attracts certain attention of mathematical community. It is somehow "noteworthy" that it has been proved that diameter of the Rubik's cube group is 20, i.e. cubik can be turned into initial position at worst at 20 moves. It rises certain interesting question...

I and my friend are thinking about a smooth analog of Rubik's cube. One idea is the following: Consider the 2-dimensional sphere $S^{2}$. We choose three parameters: $(L, H, \theta)$. Here $L$ is a ray that passes through the origin, $H$ is a plane that intersects with $L$, the sphere, and ortho...