1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

3.1. Algebraic tensor products 61

Note that the vector space structures on X 0 Y and Xx Y are completely

different. For example, in XxY we have (x1, y1)+(x2, Y2) = (x1 +x2, YI +y2)

while there is no way to "simplify" x1 0 YI+ x2 0 Y2 (in general) and

(x1 + x2) 0 (y1 + Y2) = x1 0 YI+ x1 0 Y2 + x2 0 Y1 + x2 0 Y2·

In many proofs it will suffice to work only with elementary tensors, but this
is because they form a spanning set for X 0 Y - one must not forget that
X 0 Y contains a lot more than just the elementary tensors.
The other crucial fact about tensor products is their universal property;
they are designed to turn bilinear maps X x Y ---+ Z into linear maps X 0 Y ---+
Z. Moreover, X0Y is the unique vector space, up to isomorphism, with this
property. Before making this precise, first note that the natural mapping

XxY---+X0Y, (x,y)i---+x@y

is not linear - it is bilinear (since this map factors through Cc(X x Y) and
the subspace K is specifically designed for bilinearity).^2

Proposition 3.1.3 (Universality). For any vector space Z and any bilinear

map O" : X x Y ---+ Z, there exists a unique linear map 0--: X 0 Y ---+ Z such

that

commutes {i.e., 0-(x 0 y) = O"((x, y)) for all x EX, y E Y ).

Though important, we prefer to leave it to the reader to formulate and

prove uniqueness of X 0 Y. As usual, the rough version states that any

other vector space enjoying the same universal property as X 0 Y must be

isomorphic to X 0 Y.

The following mappings on tensor products will be indispensable.

Proposition 3.1.4 (Tensor product maps). If <p: W---+ Y and 1/J: X ---+ Z

are linear maps, then there is a unique linear map

r.p01/J:W0X---+Y0Z

such that <p 01/J(w 0 x) = r.p(w) 01/J(x) for all w E W, x EX.

Proposition 3.1.5 (Product maps). If C is an algebra and <p: X ---+ C,

1jJ: Y ---+ C are linear maps, then there is a unique linear map

r.px1/J:X0Y---+C

such that <p x 1/J(x 0 y) = r.p(x)1/J(y) for all x EX, y E Y.

2Note also that the map Xx Y--+ X 0 Y is (usually) not onto. For example, you can't hit

all of the vectors of the form x1 0 Yl + x2 0 Y2 (unless X or Y is one dimensional).