By Kjeld Knudsen Jensen
The KK-theory of Kasparov is now nearly twelve years previous; its energy, software and significance were amply proven. Nonethe much less, it continues to be a forbiddingly tricky subject with which to paintings and examine. there are lots of purposes for this. For something, KK-theory spans numerous often disparate mathematical regimes. for an additional, the literature is scattered and hard to penetrate. a number of the significant papers require the reader to provide the main points of the arguments in accordance with just a tough define of proofs. eventually, the topic itself has come to include a few tough segments, each one of which calls for lengthy and in depth examine. is to house a few of these difficul Our aim in scripting this e-book ties and give the chance for the reader to "get all started" with the idea. we haven't tried to provide a complete treatise on all features of KK-theory; the topic turns out too very important to undergo this sort of remedy at this aspect. What appeared extra vital to us was once a well timed presen tation of the very easy components of the idea, the functoriality of the KK-groups, and the Kasparov product.
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Then JL rv ¢ 0 j1 0 JL where j1 : B [JIf JI] -+ is the imbedding into the lower right-hand corner. Note that j10JL(A)~[~ ~]~[JIf J:]. 3) as elements of M2(M(J)), we can define At E Hom (A, B) by At = ¢ 0 AdRt 0 j1 0JL, t E [0,1). This gives a homotopy showing that ¢ 0h 0 JL rv ¢ 0 j2 0 JL, where 12 is the imbedding of J into the upper left-hand corner. Note that . J2 Thus if we set JLo = 0 JL (A) [JBJ = [JBJ JB] BJ JB] BOO . 1/1 012 0 JL E Hom (A, [P%: P%]), the properties of 1/1 (cf. 17), ensure that JLo = [PBP PB] (A) [PBP Bp PB] BOO .
It is then clear that S E®C is the grading operator for a grading of E®C. Define by 4>®id(a® c) = 4>(a)®c = j(4)(a) ® c), a E A, c E C. 4 it follows that (E®C, >®id, F®id) is a Kasparov A ® C - B ® O-module. We denote it by rc(£). 7. Two Kasparov A - B-modules £1 = (El, >1, F1 ), (E2 , >2, F 2) are isomorphic when there is an isomorphism 'ljJ : El --+ E2 of Hilbert B- modules such that S E2 o'ljJ = 'ljJ 0 SEll F2 o'ljJ = 'ljJ 0 Fl and >2 (a) 0 'ljJ = 'ljJ 0 >1 (a), a EA. We write £1 ~ £2 in this case.
E9 Fn) is a Kasparov A - B-module which we denote by £1 E9 £2 E9 ... E9 £n and call the direct sum of £11 £2, ... 3. Pullback. ,p : C -+ A be a *-homomorphism of graded C* -algebras. ,p*(£). 4. Internal tensor product. ,p : B -+ C be a *-homomorphism of graded C* -algebras. 3. From the construction of E ®", C and the properties of S E it follows that there is a linear bijection SE®",C on E ®", C with the property that SE®",c(e ®", c) = SE(e) ®", f3c(c), e E E, c E C. ,p( < x, y »b > ). It follows that SE®",C is the grading operator for a grading of E ®", C.