1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

64 3. Tensor Products

To see that cp 8 'lj; is injective is a simple exercise using what we know

about bases and linear independence.

Proposition 3.1.13 (Exact sequences). If 0--+ X __!:..,. Y ~ Z--+ 0 is a short

exact sequence and W is a vector space 1 then the sequence

0-+XGW iGidw YGW'iT~ Z8W-+ 0

is also exact. In particular 1 if X c Y is a subspace and W is arbitrary 1 we

have a canonical isomorphism

Y8 W ~ (Y/X) 8 W.5

XGW

Proof. Proposition 3.1.12 implies X 8 W {,~ Y 8 Wis injective while it is

trivial that Y 8 W 7rGidw Z 8 W is surjective. Hence on~ only needs to check

exactness in the middle. Since X8W C ker(?r8idw ), the linear map ?T8idw

induces a surjection 7r: (Y 8 W)/(X 8 W)--+ Z 8 W. To prove injectivity

of 1f, we construct a left inverse. Define (} : Z x W --+ (Y 8 W) / (X 8 W) by

(J((z, w)) = y ® w + (X 8 W),

where y E Y is any element such that ?r(y) = z. It is clear that (J((z, w))

is independent of the choice of the lift y and that (} is bilinear. Hence

by universality, there is a linear map 0-: Z 8 W --+ (Y 8 W) / (X 8 W)

such that 0-(z ® w) = (J((z, w)) for every z E Z and w E W. Evidently

& o 1f = id(YGW)/(XGW)' since this is clear for elementary tensors, so we're

done.

D

Though the results above have been formulated for general vector spaces,

we are primarily interested in C* -algebras. Let's put an involution on the

tensor product.

Proposition 3.1.14 (Involution). If A and B are C* -algebras 1 then A 8 B

carries a unique involution such that (a® b)* =a*® b* for all elementary

tensors.

Of course, the involution on all of A 8 B is defined by
Lai ® bi r-r L at ® bt
i i
and to prove that this is well-defined, it suffices to show that if :Ei ai®bi = 0,
then :Ei at ® bt = 0 as well. Expanding the ai 's out in terms of a basis for
A and playing around with the tensor calculus and linear independence will
show this to be true.

(^5) A C -analogue of this isomorphism can fail due to the existence o_f nonexact C -algebras.