Limit Algebras: An Introduction to Subalgebras(Pitman by S C Power

By S C Power

Written by means of one of many key researchers during this box, this quantity develops the speculation of non-self adjoint restrict algebras from scratch.

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Extra info for Limit Algebras: An Introduction to Subalgebras(Pitman Research Notes in Mathematics Series, 278)

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P2] Show that bimodules for regular canonical masas are singly generated. 3. s. ) 28 5 Topological Equivalence Relations It will be shown how AF C*-algebras are classified by certain topological equivalence re­ lations. The methods and results here are essential for the analysis of non-self-adjoint subalgebras of AF C*-algebras in terms of topological binary relations. Let 5 , {Bk}^ j C' be the usual quartet of AF C*-algebra, subalgebra chain, matrix unit system and regular canonical masa C. ({ef^}).

10 ). In exactly the same way if Uk = riV2 .. (lim(Mn^,pjt)) with the analogous equivalence relation on the Cantor space [ri] x H x . . ({efj }) by specifying that the sets form a base of open sets. Each El^^^ is either disjoint from E^j or is a subset 31 of when I > k. From this it follows that the base consists of clopen sets. Identifying the diagonal A = {{x,x) : x G M{C)} with M{C) we see that the relative topology on A is the Gelfand topology. In fact it is straightforward to show that the sets E^j are compact, and so R{{^ij}) is a locally compact Hausdorff space with totally disconnected topology.

Furthermore it is assumed that for all k < £ the norms of the compositions i 0 ... o (f)k are bounded by an absolute constant. Define the seminorm || || on Aoo by ll<^it,oo(a)|| = limsup \\i o ... o (^it(a)ll e for a E Ak. Then the quotient of Aoo by the ideal of elements with zero seminorm becomes a normed algebra and the completion is a Banach algebra called the Banach algebra direct limit of the system. Write A = lim(A^, (f>k) for this limit algebra. If the given direct system is in fact an injective direct system of C*-algebras, with star injections, then the limit is a C*-algebra which can be viewed as the closed union of the subalgebras Ajt, which in this case are isometric copies of the Ak.

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