ORIGINS IN SINGLE OPERATOR THEORY 5 Consider the subspace c0 of /°° consisting of all sequences á satisfying Y\mn_+^an = 0. c0 is a closed selfadjoint ideal in /°°, and so we may form the quotient C*-algebra l°°/c0 and the canonical projection Notice that the equivalence dass of a given sequence á depends only on the projection of á to /°°/c0. It is not very hard to show that the norm of ä is given by the asymptotic expression (1.9) 11*11- lim IIK, *„+!,... }||. Ð~* 00 Let us now consider invariants for the equivalence relation ~ . One such invariant is the number p||, since á ~ b implies p | | = ||ft||. Another invariant is the essential spectrum, defined for an element á e l°° as the spectrum of ä in l°°/cQ. It is an interesting exercise to prove the following description of the essential spectrum of a sequence as the set of all its Cluster points: PROPOSITION 1.10. For every a e / 0 0 , 00 sp(o)= Ð '{ak:kn} . The following theorem is due to von Neumann, and asserts that the essential spectrum is a complete invariant for the equivalence relation ~ . THEOREM 1.11. á - b if and only if, sp(*) = sp(ft). Á proof of the interesting implication can be found in [19, pp. 81-82]. We move now from commutative asymptotics to noncommutative asymptotics. In place of l°° we consider the space i? of all infinite matrices Á of complex numbers^4 = (aiJ)i l 9 for which the norm sup is finite, the supremum extended over all sequences æ, ç of complex numbers such that £z = çß = 0 for all but a finite number of indices / and which satisfy ÓÉ€,É 2 é, E k l 2 i - JSf is a C*-algebra relative to the usual matrix Operations, and the usual *-operation in which A* means the conjugate transpose of A. Of course, «5? is isometrically *-isomorphic to the algebra^(^) of all bounded Operators on an infinite-dimensional separable Hubert space Jf, but for the moment, we wish to view JSf as a noncommutative analogue of the C*-algebra of sequences /°°. Let Jfbe the norm-closure in 3? of all matrices Á = (a^) for which a{j = 0 for all but a finite number of pairs /, j . Xis a closed two-sided ideal in JSf, occupying
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