An Essay in the Time of Corona
One hundred and twenty two days ago, I was dining with a friend in a empty restaurant: a piece of grilled rump and chips with cream cheese on pita bread. So simple a meal, and yet so far way now. The world was already at the dawn of a new age of perplexi
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1 Is There? One hundred and twenty two days ago, I was dining with a friend in a empty restaurant: a piece of grilled rump and chips with cream cheese on pita bread. So simple a meal, and yet so far way now. The world was already at the dawn of a new age of perplexity. After taking my friend to her place, I headed to mine into lockdown. This was on a Sunday, March 15 of the virulent year of 2020; it was just four months before but it seems four long decades ago. This is a COVID 19 pandemic scenario. At that time I was thinking on revisiting tensor products, a subject on which I had written some papers a dozen years before in connection with transferring properties from a pair of Hilbert-space operators to their tensor product. It was quite a fashionable subject at the time. So perhaps this Corona social distance enforcement and consequent home imprisonment might be a chance to give it a try. As an aftermath came the next few lines addressed to a wide audience. There is a conventional protocol to build up a tensor product of Hilbert spaces where a reasonable crossnorm comes nicely and naturally from the factors’ inner products. There is, however, an intriguing point: why is it, and where does it come from? As George Pólya [3] taught us: is there an easier question to ask? So, are there other ways to construct those tensor product spaces? Are they somehow equivalent, thus boiling down to the same thing? Yes, indeed. Here is the yellow brick road.
Mathematics Subjects Classification (2010): 46M05, 47A80, 15A63; Keywords Algebraic tensor product
C. Kubrusly (*) Mathematics Institute, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. Wonders (ed.) Math in the Time of Corona, Mathematics Online First Collections, https://doi.org/10.1007/16618_2020_10
C. Kubrusly
2 Is This a Possible Start?
These are all rooted on firm grounds, no abstract nonsense required: X, Y , Z and T are linear spaces (all over the same field), and a pair ðT , θÞ is a tensor product of X and Y if (a) θ is a bilinear map whose range spans T , and (b) for every bilinear map ϕ into any Z there is a linear transformation Φ for which the above diagram commutes. These are the axioms of tensor product whose definition can be rewritten as “a tensor product space T of X and Y has the universal property with respect to a bilinear map θ on the Cartesian product X Y ” , and so θ factors every bilinear map ϕ through T thus “linearising ϕ by Φ”. Enough is enough. It is a rather clean start and it seems impressive to me how much follows from these axioms. Basically all properties of concrete tensor products (yes, they do exist) follow from such an abstract formulation. First of all if ðT , θÞ and ðT 0 , θ0 Þ are tensor products of X and Y , then they are essentially (i.e., up to isomorphism) the same, and so it is usual to write X Y for “the” tensor product space T of X and Y : Also, the linear space b½X Y , Z of all bilinear maps ϕ from the Cartesian product X
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