Smooth structures, actions of the Lie algebra \({\mathfrak su}(2)\) and Haar measures on non-commutative three dimensional spheres (Q1201471)

In previous papers Tomiyama and the author have deformed the ordinary 3- sphere \(S^ 3\) into non-commutative \(C^*\)-algebra -- their non- commutative 3-sphere \(S_ \theta^ 3\) (\(\theta\) is a real-valued continuous function on \([0,1]\)) is the biggest \(C^*\)-algebra generated by two normal operators \(M\), \(N\) with relations \[ M^*M+ N^*N=1, \qquad MN=\exp (2\pi i\Theta(M^*M)NM, \] where \(\Theta(M^*M)\) is the selfadjoint operator obtained by the functional calculus of \(M^*M\) by the function \(\Theta\). In this paper ``smooth'' 3-sphere is defined as a dense \(*\)-subalgebra of \(S_ \Theta^ 3\) for a ``smooth'' deformation function \(\Theta\). Equivalently (Theorem 3.11) it can be defined as an algebra of all smooth cross sections in some fibered space. The action of su(2) on \(S_ \Theta^ 3\) with a twisted Leibniz's rule is defined (Theorem 5.1). Finally, the existence of the Haar measure on \(S_ \Theta^ 3\) is shown.