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Operators on \(C^*\)-algebras induced by conditional expectations. (English) Zbl 0853.46052

Summary: This paper investigates the relationship between a unital \(C^*\)-algebra \(\mathcal A\) and a \(C^*\)-subalgebra \(\mathcal B\) which is the range of a conditional expectation operator on \(\mathcal A\) by studying a certain algebra \(\mathcal D\) of operators on \(\mathcal A\). The investigation of \(\mathcal D\) was suggested by previous work of A. Lambert and B. Weinstock in the case where the conditional expectation operators were the classical ones of probability theory.
The commutant of \(\mathcal D\), the radical \(\text{Rad } {\mathcal D}\), the quotient \({\mathcal D}/\text{Rad } {\mathcal D}\), the spectra of elements of \(\mathcal D\) and the lattice of invariant subspaces for \(\mathcal D\) are studied, as well as the questions of when \(\mathcal D\) is closed in the norm and strong operator topologies.

MSC:

46L05 General theory of \(C^*\)-algebras
46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
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[1] K. Davidson, Nest algebras , Pitman Research Notes 191, Longman, 1988. · Zbl 0669.47024
[2] J. Dixmier, Les Algèbres d’Operateurs dans l’Espace Hilbertien , Second ed., Gauthier-Villars, Paris, 1969. · Zbl 0175.43801
[3] R.V. Kadison and J.R. Ringrose, Fundamentals of the theory of operator algebras I, II, Academic Press, Orlando, 1983, 1986. · Zbl 0518.46046
[4] I. Kaplansky, The structure of certain operator algebras , Trans. Amer. Math. Soc. 70 (1951), 219-255. JSTOR: · Zbl 0042.34901 · doi:10.2307/1990368
[5] A. Lambert and B.M. Weinstock, A class of operator algebras induced by probabilistic conditional expectations , Michigan Math. J. 40 (1993), 359-376. · Zbl 0820.46056 · doi:10.1307/mmj/1029004757
[6] C.E. Rickart, Spectral permanence for certain Banach algebras , Proc. Amer. Math. Soc. 4 (1953), 191-196. JSTOR: · Zbl 0051.09106 · doi:10.2307/2031790
[7] S. Stratila, Modular theory in operator algebras , Abacus Press, Tunbridge Wells, 1981. · Zbl 0504.46043
[8] S. Stratila and L. Zsido, Lectures on Von Neumann algebras , Abacus Press, Tunbridge Wells, 1979.
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