Morita Equivalence and Continuous-Trace C*-algebras by Iain Raeburn

By Iain Raeburn

During this textual content, the authors supply a contemporary remedy of the type of continuous-trace $C^*$-algebras as much as Morita equivalence. This contains a certain dialogue of Morita equivalence of $C^*$-algebras, a assessment of the required sheaf cohomology, and an advent to contemporary advancements within the quarter. The publication is on the market to scholars who're starting learn in operator algebras after a regular one-term path in $C^*$-algebras. The authors have incorporated introductions to important yet nonstandard history. therefore they've got built the overall concept of Morita equivalence from the Hilbert module, mentioned the spectrum and primitive excellent house of a $C^*$-algebra together with many examples, and offered the required evidence on tensor items of $C^*$-algebras ranging from scratch. Motivational fabric and reviews designed to put the idea in a extra normal context are integrated. The textual content is self-contained and will be appropriate for a complicated graduate or an self reliant examine direction.

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Suppose that X is a right Hilbert A-module, and that T is a linear operator from X to X. Then T is a positive element of C(X) if and only if (T(x) , x) > 0 for all x G X. Hilbert C*-Modules 20 Proof. If T > 0 in £(X), then T = S*S for some S £ £(X) and (T(x) , X)A = (S(x) , S(X))A > 0. Now assume (T(x) , x) > 0 for all x G X. 16) and (T(z) , 2;) = (z , T(z)) for all 2 G X, it follows that (T(x) , y) = (x , T(y)) for all x, y G X. Thus T is adjointable with T* = T. Now the functional calculus allows us to write T = S - R with S, i?

57. a) and / G C 0 (T,/C). We first show that Lm(f) G Co(T, K). Since 11-» m(t)h is continuous for each ft, a standard compactness argument shows that for every compact set K in H, t — i > m(t)h is uniformly continuous for ft G if. Thus £ — i > m(t)S is continuous from T into /C for each S e JC. Fix s e T and e > 0. Choose a neighbourhood (7 of 5 such that ||m(t)/(s) - m(s)f(s)\\ < e/2 and \\f(t) - f(s)\\ < 6/(2117711100) for all t G U. Then t G C/ implies \\m(t)f(t)-m(s)f(s)\\ < e. Since ||m(*)/(*)|| < ||m|UI/WII, this proves that L m ( / ) G Co(T,/C).

So the standard procedure is to construct such an algebra M(A), and to prove that this concretely defined object is a maximal unitization, but to remember how individual elements were constructed. 2]). Since we are interested in the interaction between multipliers and Hilbert modules, it is more helpful for us to define M(A) to be the C*-algebra £(AA), SO that individual multipliers are adjointable operators on A A- This elegant approach comes from [94]. For those who have seen the traditional definition of M(A), the next Theorem says that M(A) = C{AA)] for others, it will motivate our definition of M(A).

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