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When is c 0 ( τ ) complemented in tensor products of ℓ p ( I ) ?
Author(s) -
Morelli Cortes Vinícius,
Medina Galego Elói,
Samuel Christian
Publication year - 2019
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201700348
Subject(s) - tensor product of hilbert spaces , tensor product , injective function , mathematics , cardinality (data modeling) , banach space , tensor (intrinsic definition) , sequence (biology) , generalization , product (mathematics) , space (punctuation) , set (abstract data type) , pure mathematics , cofinality , tensor product of modules , cardinal number (linguistics) , discrete mathematics , combinatorics , tensor contraction , mathematical analysis , countable set , computer science , geometry , uncountable set , linguistics , philosophy , genetics , biology , data mining , programming language , operating system
Let X be a Banach space, let I be an infinite set, let τ be an infinite cardinal and let p ∈ [ 1 , ∞ ) . In contrast to a classical c 0 result due independently to Cembranos and Freniche, we prove that if the cofinality of τ is greater than the cardinality of I , then the injective tensor product ℓp ( I )⊗ ̂ ε X contains a complemented copy ofc 0 ( τ )if and only if X does. This result is optimal for every regular cardinal τ. On the other hand, we provide a generalization of a c 0 result of Oja by proving that if τ is an infinite cardinal, then the projective tensor product ℓp ( I )⊗ ̂ π X contains a complemented copy ofc 0 ( τ )if and only if X does. These results are obtained via useful descriptions of tensor products as convenient generalized sequence spaces.