Local Multipliers of C*-Algebras by Prof. Dr. Pere Ara, Dr. rer. nat. habil. Martin Mathieu

By Prof. Dr. Pere Ara, Dr. rer. nat. habil. Martin Mathieu (auth.)

Many difficulties in operator idea result in the distinction ofoperator equa­ tions, both at once or through a few reformulation. normally, how­ ever, the underlying house is simply too 'small' to include recommendations of those equa­ tions and therefore it should be 'enlarged' in a roundabout way. The Berberian-Quigley growth of a Banach house, which permits one to transform approximate into actual eigenvectors, serves as a classical instance. within the conception of operator algebras, a C*-algebra A that seems to be small during this feel culture­ best friend is enlarged to its (universal) enveloping von Neumann algebra A". This works good on account that von Neumann algebras are in lots of respects richer and, from the Banach area standpoint, A" is not anything except the second one twin house of A. one of the various fruitful purposes of this precept is the well known Kadison-Sakai theorem making sure that each derivation eight on a C*-algebra A turns into internal in A", although eight is probably not internal in A. The transition from A to A" in spite of the fact that isn't an algebraic one (and can't be because it is widely known that the valuables of being a von Neumann algebra can't be defined basically algebraically). for this reason, ifthe C*-algebra A is small in an algebraic experience, say uncomplicated, it can be irrelevant to maneuver directly to A". In any such scenario, A is sometimes enlarged through its multiplier algebra M(A).

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Putting (x'Y)A = x*y for x,y E E, E can be considered as a right Hilbert A-module. The closed linear span of ( E, E )A(= E* E) is the closed ideal generated by E, and will be denoted by I. On the other hand, K(EA) ~ D through the map (}x,y I-t xy* because EE* = En E* = D. Note that E is a full Hilbert D-I-bimodule and so D and I are strongly Morita equivalent C*-algebras. The isomorphism from K(DE) to I is given by (}~,y I-t x*y for x, y E E, where (}~,y is the map on the left Hilbert D-module E defined by Z(}~,y = D(Z,X)Y (Z E E).

Then, in particular, exp(iT) is a surjective isometry on A. The following result due to Kadison [156; Theorem 7] reveals that isometries on C*-algebras have rather special properties. 44 Kadison's Theorem. Let S: A -+ A be a unital surjective isometry on a unital C*-algebra A. Then S is hermitian-preserving and S(x 2 ) = S(x)2 for all x E A. 8. A detailed discussion of numerical ranges can be found in [58], [59; Section 10], and [130]. 2 Analytic Tools 45 MISCELLANEA. This final subsection compiles various results that are scattered throughout the literature and for which we felt it appropriate to include a proof.

TYPE I C*-ALGEBRAS AND COMPACT ELEMENTS. When a C*-algebra A is represented, via 7f, on a Hilbert space H, the represented C*-algebra 7f(A) ~ B(H) mayor may not contain a non-zero compact operator on H. 4]. In the case that A is simple, this leaves only two possibilities: either A is isomorphic to K(H) or 7f(A) never contains any non-zero compact operator for any irreducible representation 7f of A. In the latter case, A is said to be antiliminal. If the ideal structure of the C*-algebra A is non-trivial, an intricate hierarchy of structural properties arises.

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