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](https://math.stackexchange.com/questions/1722380/most-natural-equivalence-between-c-algebras) en SE last week, it's still unanswered. 



Most natural equivalence between $C^*$-algebras in NCG
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".


