0 Invariant Subspaces of Voiculescu ’ S Circular Operator
نویسنده
چکیده
The invariant subspace problem relative to a von Neumann algebra M ⊆ B(H) asks whether every operator T ∈ M has a proper, nontrivial invariant subspace H0 ⊆ H such that the orthogonal projection p onto H0 is an element of M; equivalently, it asks whether there is a projection p ∈ M, p / ∈ {0, 1}, such that Tp = pTp. Even when M is a II1–factor, this invariant subspace problem remains open. In this paper we show that the circular operator and each circular free Poisson operator (defined below) has a continuous family of invariant subspaces relative to the von Neumann algebra it generates. These operators arise naturally in free probability theory, (see the book [17]), and each generates the von Neumann algebra II1–factor L(F2) associated to the nonabelian free group on two generators. Given a von Neumann algebra M with normal faithful state φ, a circular operator is y = (x1 + ix2)/ √ 2 ∈ M, where x1 and x2 are centered semicircular elements having the same second moments and that are free with respect to φ. For specificity, we will always take cicular elements to have the normalization φ(y∗y) = 1, which is equivalent to φ(xi ) = 1. Voiculescu found [15] a matrix model for a circular element, showing that if X(n) is a random matrix whose entries are i.i.d. complex (0, 1/n)–Gaussian random variables then X(n) converges in ∗–moments as n → ∞ to a circular element, meaning that
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