I have recently realised that the Paleo-Hebrew (and Phoenician) graph for the Hebrew letter ט (Teth) is $\otimes$. This made me wonder if there is any relation between the choice of the symbol and the choice of the name (as far as I am aware, tensor product is meant to be a stretched product, coming from latin tendere although I confess I can't understand why the tensor product of two spaces, or modules, or elements, should be thought as being "stretched"), since ט is the letter used in modern and ancient Hebrew to denote the sound T in words borrowed from other languages.
I have checked the Archives Bourbaki, in particular the second redaction (n°034) of Algèbre. Chapitre II, algèbre linéaire and the corresponding discussion which seem to me the first occurrences of the name (still called "produit tensoriel (ou kroneckerien)", see ibid. p. 198) and of the symbol, but found no hints. The tensor product is presented as a special case of the bilinear product of two modules, which is denoted by $\odot$, itself not a graph that I am aware of in the Paleo-Hebrew alphabet: the circle in $\otimes$ could simply come from the one in $\odot$, but it might also be related to the first letter T of tensor.
Is there any reference I can look at for more information?
Best Answer
According to John Aldrich's list of "Earliest Uses of Symbols for Matrices and Vectors", the notation $\times$ for direct product (as well as the name itself) goes back to Wedderburn's 1934 Lectures on Matrices (page 74).
Some further search gave a much earlier source, Hurwitz's 1894 Zur Invariantentheorie
I still have to track down the step from $\times$ to $\otimes$.
Incidentally, On the history of the Kronecker product argues that it should more appropriately be called the Zehfuss product.
In any case, since tensor product is $\otimes$ and tensor sum is $\oplus$, it seems obvious that the $\times$ and the $+$ refer to the arithmetical operations of multiplication and addition, not to a letter.