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Signature plays an important role in geometry and topology. In the space withsingularity, Goresky and MacPherson extend the signatures to orientedpseudomanifolds with only even codimensional stratums by using generalizedPoincare duality of intersection homology. After that Siegel extended thesignature on Witt spaces. Higson and Xie study the $C^*$- higher signature onWitt space. Followed by the combinatorial framework developed by Higson andRoe, this paper construct the $C^*$-signature on non Witt space withnoncommutative geometric methods. In conical singular case, we compareanalytical signature of smooth stratified non Witt space by Albin, Leichtnam,Mazzeo and Piazza.

C*-Algebraic higher Signature on Non-Witt space