A theorem of Brown and Halmos [6] characterizes when the product of two scalar Toeplitz operators is again a Toeplitz operator. Although the theorem and its proof are quite simple, it has served as an inspiration for a variety of related and often significantly more difficult results on various function spaces in one or several variables. Motivated by its extension to Toeplitz operators with matrix-valued symbols [16], we characterize when the product of two Toeplitz operators with operator-valued symbols is again a Toeplitz operator. We also address the related question of when the product of a Toeplitz operator and a Hankel operator with operator-valued symbols is a Hankel operator. In addition, a partial answer is given to the problem of determining when the product of two Hankel operators with operator-valued symbols is a Hankel operator.
Let p ∈ (1, ∞ ) and let {n_k} be a strictly increasing sequence of positive integers satisfying n_k|n_k+1 . In this paper, we introduce an L^p -Bunce–Deddens algebra 𝔅^p({n_k}) . We prove that the K_0 and K_1 groups of 𝔅^p({n_k}) are independent of the choice of p. For a fixed p, two L^p -Bunce–Deddens algebras 𝔅^p({n_k}),𝔅^p({m_k}) are isometrically isomorphic if and only if the associated supernatural numbers δ _{n_k} and δ _{m_k} agree. Furthermore, for p, q ∈ (1, ∞ ) , 𝔅^p({n_k}) and 𝔅^q({n_k}) are (anti)isometrically isomorphic if and only if p = q or 1/p + 1/q = 1 . Finally, we show that every L^p -Bunce–Deddens algebra is unital, separable, simple, amenable, p-nuclear, and has stable rank one and a unique normalized trace.
The J-unitary group representations on Pontryagin Π _k -spaces combine in a special way the general finite-dimensional group representations with unitary finite- or infinite-dimensional group representations. The theory of such representations is closely linked to the structure of 1-cocycles of a special type—neutral cocycles. Thus, for dealing with J-unitary representations, one needs, apart from describing 1-cocycles, to be able to determine which of them are neutral. In this work we discuss a general approach to this process for Engel and, in particular, for nilpotent connected groups. For some important classes of nilpotent groups this approach leads to a precise description of neutral cocycles in terms of families of operators on finite-dimensional spaces.
We study ℓ _∞ -valued and ℓ _1 -valued noncommutative symmetric spaces. We define related Hardy spaces of noncommutative martingales, show duality and interpolation properties of these spaces. Using these results, we give a direct proof of algebraic atomic decompositions and Davis-type decompositions within the context of Hardy spaces affiliated with separable noncommutative symmetric spaces, which constitute an interpolation of the (L_p, L_q) pair for 1< p ≤ q < 2 . We also derive a version of Doob maximal inequality associated with symmetric spaces of measurable operators.
