TITLE: Most natural equivalence between $C^*$-algebras in NCG
QUESTION [7 upvotes]: I have listen or read that, in the context of noncommutative geometry,  Morita equivalence is a more natural equivalence for $C^*$-algebras than $*$-isomorphism. 
Can someone explain this sentence or know some text that could be helpful? 
Does anybody know some comparisons of different $C^*$-algebras categories? 
NOTE: I asked this same question en SE last week, it's still unanswered.

REPLY [11 votes]: Here I list some facts that may be useful for building your intuition:
1. Two commutative Morita equivalent $C^*$-algebra are in fact $*$-isomorphic.
2 If $A$ is $C^*$-algebra and you take $B=M_n(A)$ then $A$ and $B$ are Morita equivalent.
3 Many invariants for $C^*$-algebras such as $K$-theory or Hochschild or cyclic homology are the same for Morita equivalent $C^*$-algebras.
4 Two Morita equivalent $C^*$-algebras have the same representation theory so from this point of view they should represent the same "noncommutative space".