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

By Iain Raeburn

During this textual content, the authors provide a latest remedy of the category of continuous-trace $C^*$-algebras as much as Morita equivalence. This encompasses a distinctive dialogue of Morita equivalence of $C^*$-algebras, a evaluate of the required sheaf cohomology, and an creation to fresh advancements within the sector. The e-book is out there to scholars who're starting study in operator algebras after a typical one-term direction in $C^*$-algebras. The authors have incorporated introductions to beneficial yet nonstandard history. hence they've got constructed the final thought of Morita equivalence from the Hilbert module, mentioned the spectrum and primitive perfect area of a $C^*$-algebra together with many examples, and provided the required proof on tensor items of $C^*$-algebras ranging from scratch. Motivational fabric and reviews designed to put the idea in a extra common context are integrated. The textual content is self-contained and will be compatible for a sophisticated graduate or an autonomous learn path.

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Since we can choose the same / throughout a neighbourhood of any given point t, and mf is continuous, the new function (j) is continuous. It is bounded by ||ra||. Given h G £2(T) of finite support, we can choose / such that / = 1 on supp/i, and then n((/))h = ji(mf)h = mfi(f)h = mh. Thus m — /x(0) is in fi{Cb{T)), as required. • In view of the above, it is natural to speculate on the multiplier algebra of Co(T,JC(H)). A naive guess might be Cb(T,B(H)), but for infinite-dimensional H this algebra is too small.

Then A n l = q(Xnl + Lan) —> q(T). Since CI is always closed, q(T) = XI for some A G C. Then (An - A)l + L a n -> T - Al G ker

We want to build a representation of A. The underlying vector space will be X 0 H^, and the inner product will be characterized by ( x 0 / i I y®k) := (7r((y , x) )h\k). (2,25) To see where this comes from, consider the case B = C. 25) is (y , x)c(h \k) = {x\ y)(h I k). 23) on the tensor product of Hilbert spaces. 64. 25). 25). 59. 25) is bilinear in (x, /i), and hence defines a linear map / y <8>/fc : XoWfr -> C. Then (y, k) i-> fy®fk is bilinear from X x Hn to (X©H*)", hence gives a linear map L on X 0 7 ^ , and we define (a \ (3) := L@(a).

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