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 useful?
Does anybody know some comparisons of different $C^*$-algebras categories?
I post here the answer I received for the same question in mathoverflow:
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".
And a helpful comment:
One more crucial fact: the crossed product $C_0(X)\rtimes_r G$ of the $C^*$-algebra of a locally compact Hausdorff space by a free and proper action of a locally compact topological group is, in general, only Morita equivalent to the $C^*$-algebra of the quotient space $C_0(X/G)$. Indeed, $C_0(X)\rtimes_r G$ is necessarily noncommutative the moment $G$ is non-Abelian.