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Extra resources for Abstract linear algebra

Example text

W24-1t-1) dw for all Note that the integral converges because of the rapid t > 0. 1 exp(-w2 4- t- entire function for all 1 ) F(t,z) = (4Trt)-1/2 = near infinity, and that t e H. If z is in Bt, Jexp(_(w+4tz)241tl) exp(4772z2t) . z J* F(t,z) is an then dw exp(47r2z2t) 27 Thus for all z e T, and hence A E R and t > O. Using analyticity again this F(t,z) = exp(4Tr2z2t) for all = exp(-47r2a2t) formula holds for all t e H. From the definition of function for all for all Gtt(A) = F(t,iA ) Also t > O.

This is the intuitive idea behind the next three corollaries. 5 corresponds for Banach algebras with a countable bounded approximate identity bounded by 1 to the following trivial observation for unital algebras. e(T). 5 A be a Banach algebra. Then A Let approximate identity bounded by 0 from into L1OR+) Proof. 0(L7R+)). exists, then a countable bounded approximate identity 0 in 1 is easily transferred to a countable bounded L1(R+) approximate identity ('(fn)) in A. Conversely suppose that A has a countable bounded approximate identity bounded by 1.

Tb ? 0 An operator for all T b ? O. on a C Let Mn(B) * denote the C -algebra of B, and let onto M (Q). n In n x n matrices with entries from the C -algebra denote the identity operator from the C -algebra We shall think of M n (B) as M (f) 0 B. n An operator Mn(C) T on 43 a C -algebra is said to be completely positive if B, positive operator on for all M (B) * n T 0 I is a n One of the equivalent n E IN. formulations that a C -algebra is nuclear is that has a left bounded CL(B) approximate identity bounded by 1 consisting of completely positive continuous finite rank operators (see Lance [1973] and Choi and Effros [1978]).