Abstract:
Suppose that for $i = 1,2$, $A_i$ is a unital subalgebra of a $C^*$-algebra $C_i$ and suppose $A_i$ generates $C_i$ as a $C^*$-algebra. Blecher, Muhly and Paulsen asked: If $C_1$ is strongly Morita equivalent to $C_2$ in the sense of $C^*$-algebra theory, and if $A_1$ is $CB$-Morita equivalent to $A_2$ in the category of operator algebras, must $A_1$ and $A_2$ be strongly Morita equivalent in the sense of their AMS Memoir, "Categories of Operator Modules: Morita Equivalence and Projective Modules"?  My purpose is to exhibit an example which answers this question in the negative.