Interpolation of Protoquantum Spaces
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terpolation of Protoquantum Spaces O. Yu. Aristov and N. V. Volosova Received July 3, 2019; in final form, November 8, 2019; accepted November 22, 2019
Abstract. We consider the complex interpolation method in the context of protoquantum spaces. We apply it to construct an Lp -protoquantization of an arbitrary normed space for 1 < p < ∞. Key words: complex interpolation, protoquantum space, Lp -space. DOI: 10.1134/S0016266320020070
The purpose of this paper is to show that any normed space has an Lp -protoquantization for each 1 < p < ∞. We use the Lions–Calder´on complex interpolation method in the context of protoquantum, or matricially normed, spaces. (The notion of a matricially normed space was introduced by Effros and Ruan in [1]. We follow the equivalent coordinate-free approach to these objects proposed by Helemskii in [2] and use the term protoquantum space, which better suits this point of view.) For the important class of operator protoquantum spaces, the theory of complex interpolation was developed by Pisier in [3]. Here we suggest a generalization of this theory to arbitrary protoquantum spaces and use it to construct Lp -protoquantizations. The main result of this paper (Corollary 7) was conjectured by the anonymous referee of the paper [4]. The authors are grateful to A. Ya. Helemskii for this information. Protoquantum spaces. First, we recall the necessary definitions from [2] and [4]. By B we denote the Banach algebra of bounded operators on a separable infinite-dimensional Hilbert space and by F, the two-sided ideal in B consisting of finite-dimensional operators. We set F P := {P aP | a ∈ F}, where P ∈ B is a projection (i.e., a self-adjoint idempotent). We denote the tensor product F ⊗E, where E is a linear space, by FE and elementary tensors in it by ax : = a⊗x. The space FE is a bimodule over B with respect to outer multiplication given by b·ax·c := (bac)x. Next, E is called a protoquantum space if FE is endowed with a norm such that a·u·b a u b for all u ∈ FE and a, b ∈ B (it is called a protoquantum norm on E). The embedding E → FE : x → px, where p is a one-dimensional projection, allows us to regard E as a normed subspace of the space FE. The corresponding norm on E does not depend on the choice of p: if q is another one-dimensional projection, then qx = S · px · S ∗ S px S ∗ = px, where S is a partial projection with initial projection p and final projection q, and, similarly, px = S ∗ · qx · S qx. If E is given directly as a normed space, then its protoquantization is any protoquantum space with underlying space E such that the restriction of the protoquantum norm to E coincides with the norm of E. A protoquantum space whose underlying space is complete is called a Banach protoquantum space. A projection P ∈ B is a support of an element u ∈ FE if P · u · P = u, and projections P and Q are said to be orthogonal if P Q = 0. We note the following simple property. Lemma 1. Let E be a protoquantum space. Then P · u · P + Q · u · Q u for any u ∈ FE and any
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