Lattice of intermediate subalgebras

Abstract

Analogous to subfactor theory, employing Watatani's notions of index and $C^{\ast }$-basic construction of certain inclusions of $C^{\ast }$-algebras, (a) we develop a Fourier theory (consisting of Fourier transforms, rotation maps and shift operators) on the relative commutants of any inclusion of simple unital $C^{\ast }$-algebras with finite Watatani index, and (b) we introduce the notions of interior and exterior angles between intermediate $C^{\ast }$-subalgebras of any inclusion of unital $C^{\ast }$-algebras admitting a finite index conditional expectation. Then, on the lines of [2], we apply these concepts to obtain a bound for the cardinality of the lattice of intermediate $C^{\ast }$-subalgebras of any irreducible inclusion as in (a), and improve Longo's bound for the cardinality of intermediate subfactors of an inclusion of type $III$ factors with finite index. Moreover, we also show that for a fairly large class of inclusions of finite von Neumann algebras, the lattice of intermediate von Neumann subalgebras is always finite.

arXiv commentMinor changes, one reference added, no figures, 43 pages

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• Lie Ideals in Operator Algebras
• On Banach space projective tensor product of $C^{\ast }$-algebras
• On closed Lie ideals of certain tensor products of $C^{\ast }$-algebras
• On the Lie ideals of C*-algebras
• On closed Lie ideals of certain tensor products of C*-algebras II
• Closed ideals and Lie ideals of minimal tensor product of certain C*-algebras

Cited by in corpus (1)

• Center of Banach algebra valued Beurling algebras