1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

68 3. Tensor Products

is an orthonormal basis. Moreover, for each x E 1{ ® K there is a unique

set of vectors { ki} C K such that

x = LVi ®ki.^6

i

If S E IIB(H) and T E IIB(K), then we can consider the algebraic tensor

product mapping S 8 T: 1{ 8 K----+ 1{ 8 K. Naturally, this map is bounded

and has the expected norm.

Proposition 3.2.3 (Tensor product operators). If SE IIB(H) andT E IIB(K),

. then there is a unique linear operator S ®TE IIB(H ® K) such that

S ®T(v ®w) = Sv ®Tw

for all v E 1i, w EK. Moreover, JJS ® TJI = JJSJJ JITJJ.

Proof. Let us first consider the case that S = 17-l. For any vector x E 1i8K

we can find an orthonormal set { ei} c 1{ and unique vectors { ki} C K such

that x = I:i ei ® ki.^7 Now we compute

JllH 8 T(x)Jl^2 =II Lei® T(ki)ll^2

i

= L llT(ki)ll^2

i

::::; JJTJJ^2 JJxJJ^2.

It follows that 17-l 8 T has a unique extension to a bounded linear operator

on 1{ ® K, which we denote by 17-l ® T, and its norm is bounded by JITJJ.

Evidently the same holds for S 8 l!C. Hence we may define S ® T to be

(S ® lK)(lH ® T) and we have the inequality

11s0r11::::; IJSllllTIJ.

One should also check that

s ® TIH01C = s 8 T,

but this is easy.

To show the opposite inequality JIS ® TJJ ;::: JJSJJ JJTJJ, first note that the

norm on 1{ ® K is a cross norm. In other words, for all h E 1{, k E K we

have JJh ® kJJ = JJhJJ JJkJJ. This is obvious but has the nice consequence that

(^6) 0f course, this is no longer a finite sum and convergence is in the norm topology.
(^7) rs this sum finite or infinite? Are we appealing to Proposition 3.2.2 or Corollary 3.1.10?
Actually we can neither appeal to Proposition 3.2.2, since 7-l 0 lC is not complete, nor to Corollary
3.1.10, since an orthonormal basis is not an algebraic basis. What are we to do? How about start-
ing with an orthonormal set which spans a prescribed finite-dimensional subspace, then extending
it to an algebraic basis and invoking Corollary 3.1.10. Or, if you prefer, appeal to Corollary 3.1.10
first, and then apply the Gram-Schmidt procedure to arrange orthogonality. Either way, we must
pay attention.