@terceira thqnk you for your comment.yes but in the question i search for globally smooth solutions

@YCor But I meant a proper inclusion. The motivation for question was the scalar case

I mean the usual category of all C^* algebra(including unital one

Thank you and +1 very much for your answer

@TomGoodwillie dear Prof. Goodwillie thank you very much foryour attention to my question. Is not possible two conjugate points p and q on a closed geodesics(closed leaf of the foliation) woul be jointe to eh other via amilly of geodesics which are not necessarilly leaves of the foliation? Why does foliation by geodesics imply no conugate points?

A space X is approximately sigma compact if it has a dense sigma compact subspace. For exqmple the long line is not approximately sigma compqct

@YemonChoi Yes algebras you mentioned are not $Z^*$ algebra. But a commutative algebra $C_0(X)$ is a $Z^*$ algebra iff $X$ is not approximately sigma compact