Abstract:
Kadison-Singer algebras are a new class of non-self-adjoint reflexive operator algebras on a Hilbert space, which make connections between theories of self-adjoint algebras and non-self-adjoint algebras. Kadison-Singer algebras are closely associated with von Neumann algebras, and they are full of topological structures. Such class of operator algebras provide a new way to develop the theories of self-adjoint algebras. Pentagon subspace lattices and double triangle lattices are playing important roles in theories of lattices, and most of the focus was only put on double triangle lattices before. So we begin the study in pentagon subspace lattices to show more examples of Kadison-Singer algebras. Since any pentagon subspace lattice with gap-dimension 1 is reflexive, we construct a pentagon subspace lattice with gap-dimension 1 to ensure the reflexivity of the algebra. Let
H be a complex Hilbert space. Suppose
L=(0),
L, M, K, H is a pentagon subspace lattice satisfying that
M is the closed linear span of
L and
x0, where
x0 is a nonzero vector in
K┴ but not in
K+
L. We prove
AlgL is a semi-simple Kadison-Singer algebra. Besides, we know that nest algebras are Kadison-Singer algebras and nest algebras are invariant under similar transformation. However, any CSL algebra, not nest algebra, is not a Kadison-Singer algebra, and an algebra, which is similar to a Kadison-Singer, is not necessarily a Kadison-Singer algebra. So there are two natural questions:whether there exist a CSL algebra, not nest algebra, which can be similar to a Kadison-Singer algebra, and whether a CSL algebra exist but is not similar to any Kadison-Singer algebra.