Regularity of operators on essential extensions of the compacts

A semiregular operator on a Hilbert C^*-module, or equivalently, on the C^*-algebra of `compact' operators on it, is a closable densely defined operator whose adjoint is also densely defined. It is shown that for operators on extensions of compacts by unital or abelian C^*-algebras, semiregularity leads to regularity. Two examples coming from quantum groups are discussed.Comment: LaTeX2e, 13 pages, no figures, to appear in the Proceedings of the AM

A counter example on idempotent states on compact quantum groups

A simple example is given to illustrate that an idempotent state may not be the haar state of any subgroup in the case of compact quantum groups

Torus equivariant spectral triples for odd dimensional quantum spheres coming from C∗C^*-extensions

The torus group $(S^1)^{\ell+1}$ has a canonical action on the odd dimensional sphere $S_q^{2\ell+1}$. We take the natural Hilbert space representation where this action is implemented and characterize all odd spectral triples acting on that space and equivariant with respect to that action. This characterization gives a construction of an optimum family of equivariant spectral triples having nontrivial $K$-homology class thus generalizing our earlier results for $SU_q(2)$. We also relate the triple we construct with the $C^*$-extension 0\longrightarrow \clk\otimes C(S^1)\longrightarrow C(S_q^{2\ell+3}) \longrightarrow C(S_q^{2\ell+1}) \longrightarrow 0. Comment: LaTeX2e, 12 page