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Regularity for operator algebras on a Hilbert space
| Content Provider | Semantic Scholar |
|---|---|
| Author | Froelich, John |
| Copyright Year | 1993 |
| Abstract | Four notions of reguilarity for operator algebras are introduced. An algebra A is called 1-regular if for any two linearly independent vectors x, y e H there is an a e A such that ax = 0 and ay #0. We show that the only weakly closed transitive 1-regular algebra is B(H), thus providing a natural generalization of the Rickart-Yood density theorem. We construct an example of a 1-regular algebra which contains no nonzero compact operators. This example is related to the "thin set" phenomena of classical harmonic analysis. Let A be a commutative semisimple Banach algebra with maximal ideal space F. The algebra A is called regular if for any closed K c F and any x 0 K there is a E A such that ai(K) = {,0} and a(x) = 1. Regular algebras occupy an important position in the commutative theory as they provide an appropriate context in which to study the relationships between elements of the algebra and their Gelfand transforms and especially general spectral synthesis questions. Operator algebras themselves have frequently and fruitfully been viewed as spaces of "noncommutative" functions, for example, the C*-algebras as noncommutative topology [2] and reflexive operator algebras as noncommutative harmonic analysis [1, 5, 8]. It seems likely therefore that appropriate notions of regularity can provide useful frameworks for the analysis of operator algebras. In this paper we introduce four notions of regularity and study their logical relations. We prove that the only transitive 1-regular algebra is B(H), thus providing a natural generalization of the Rickart-Yood transitivity theorem and establishing a surprising connection between these notions and the transitive algebra problem [9]. These notions should also prove useful in studying the interplay between the algebraic structure of an operator algebra and its spatial action. Throughout the paper we only consider strongly (= weakly) closed algebras of operators on a separable complex Hilbert space H. An interesting feature of many of the proofs is the decisive role of the Hilbert space geometry. Received by the editors April 11, 1992. 1991 Mathematics Subject Classification. Primary 47D25, 47A 1 5. ( |
| Starting Page | 1269 |
| Ending Page | 1277 |
| Page Count | 9 |
| File Format | PDF HTM / HTML |
| DOI | 10.1090/S0002-9939-1993-1181164-8 |
| Alternate Webpage(s) | http://www.ams.org/journals/proc/1993-119-04/S0002-9939-1993-1181164-8/S0002-9939-1993-1181164-8.pdf |
| Alternate Webpage(s) | https://doi.org/10.1090/S0002-9939-1993-1181164-8 |
| Volume Number | 119 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
