On the Nuclearity of Completely 1-Summing Mapping Spaces
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
ON THE NUCLEARITY OF COMPLETELY 1-SUMMING MAPPING SPACES∗
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Zhe DONG (
School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China E-mail : [email protected]
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Yafei ZHAO (
Department of Mathematics, Zhejiang International Studies University, Hangzhou 310012, China E-mail : zhaoyafei [email protected] Abstract In this paper, we investigate the λ-nuclearity in the system of completely 1summing mapping spaces (Π1 (·, ·), π1 ). In Section 2, we obtain that C is the unique operator space that is nuclear in the system (Π1 (·, ·), π1 ). We generalize some results in Section 2 to λ-nuclearity in Section 3. Key words
λ-nuclearity; completely 1-summing mapping space
2010 MR Subject Classification
1
46B07
Introduction
The theory of operator spaces is a natural non-commutative quantization of Banach space theory. For the convenience of the reader, we frist recall some basic notations and results in operator spaces, the details of which can be found in [5] and [14]. Given a Hilbert space H, we let B(H) denote the space of all bounded linear operators on H. For each natural number n ∈ N, there is a canonical norm k · kn on the n × n matrix space Mn (B(H)), given by identifying Mn (B(H)) with B(H n ). We call this family of norms {k · kn } an operator space matrix norm on B(H). An operator space V is a norm closed subspace of some B(H) equipped with the distinguished operator space matrix norm inherited from B(H). For any operator spaces V and W and an element u ∈ Mn (V ⊗ W ), we say that k u k∧ = inf{k α k · k v k · k w k · k β k: u = α(v ⊗ w)β}, where the infimum is taken over arbitrary decompositions with v ∈ Mp (V ), w ∈ Mq (W ), α ∈ Mn,p×q and β ∈ Mp×q,n , with p, q ∈ N arbitrary. This is indeed an operator space matrix norm and we call k · k∧ the operator space projective tensor product norm on V ⊗ W . We let V ⊗∧ W = (V ⊗ W, k · k∧ ), ∗ Received April 6, 2019; revised July 24, 2020. The project was partially supported by the National Natural Science Foundation of China (11871423).
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ACTA MATHEMATICA SCIENTIA
Vol.40 Ser.B
ˆ and we define the operator space projective tensor product V ⊗W to be the completion of the algebraic tensor product V ⊗∧ W . The operator space injective tensor product norm of an element u ∈ Mn (V ⊗ W ) is defined by k u k∨ = sup{k (f ⊗ g)n (u) k: f ∈ Mp (V ∗ ), g ∈ Mq (W ∗ ), k f k, k g k≤ 1}. We let V ⊗∨ W = (V ⊗ W, k · k∨ ), ˇ is called the operator space injective tensor product. For v = α ⊗ v0 ∈ and its completion V ⊗W Mm,r (V ) and w = β ⊗ w0 ∈ Mr,n (W ), the matrix inner product v ⊙ w ∈ Mm,n (V ⊗ W ) is defined by v ⊙ w = αβ ⊗ v0 ⊗ w0 , and for general v ∈ Mm,r (V ) and w ∈ Mr,n (W ), we let (v ⊙ w)i,j =
r X
vi,k ⊗ wk,j .
k=1
Given an element u ∈ Mn (V ⊗ W ), we state that k u kh = inf{k v k · k w k: u = v ⊙ w, v ∈ Mn,r (V ), w ∈ Mr,n (W ), r ∈ N}. We call k · kh the Haagerup tensor product norm on V ⊗ W . We let V ⊗h W = (V ⊗ W, k · kh ), h
and we define
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