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Oct 3, 2014 at 19:05 comment added Ben Grossmann My mistake. I think you're right about all that.
Oct 3, 2014 at 18:51 comment added arne Oh, I see now that my question wasn't well defined. I actually ment for the operation to be linear, meaning $\sum_n A_n \otimes B_n$ goes to $\sum_n A_n B_n$. This should uniquely define a linear map, I hope. But I don't think the one you gave is the one I'm after. I have edited my question to emphasize that the map should be linear.
Oct 3, 2014 at 18:31 history answered Ben Grossmann CC BY-SA 3.0