Mixing
Identifying tensor product of four Banach spaces, out of which two are finite dimensional.
$begingroup$
Does the following identity (bicontinuously) hold:
$$left(mathbb{M}_nbigotimesnolimits_epsilon Xright) bigotimesnolimits^gamma left( mathbb{M}_mbigotimesnolimits_epsilon Yright)cong mathbb{M}_{nm}bigotimesnolimits_epsilonleft(Xbigotimesnolimits^gamma Yright),$$
where $X,Y$ are Banach spaces, $bigotimes_epsilon$ and $bigotimes_gamma$ denote respectively the Banach space injective and the Banach space projective tensor tensor products respectively. I am asking about the bicontinuity of the natural map given by $(aotimes x)otimes (botimes y)mapsto (aotimes b)otimes (xotimes y)$.
I think, one way to do it would be by identifying $mathbb{M}_nbigotimesnolimits_epsilon X$ (bicontinuously) with $mathbb{M}_nbigotimesnolimits_gamma X$ (and similar identification for $mathbb{M}_nbigotimesnolimits_epsilon Y$), and then use associativity and commutativity of the projective tensor product to get the conclusion. Is there another way to do it?
Will it still hold if the projective tensor product is replaced by some other Banach space cross tensor product, for which we have no idea about its commutativity or associativity?
Any reference would be highly appreciated.
banach-spaces tensor-products
asked Jan 11 at 8:01
JansonJanson
507
$endgroup$
add a comment |
$begingroup$
Does the following identity (bicontinuously) hold:
$$left(mathbb{M}_nbigotimesnolimits_epsilon Xright) bigotimesnolimits^gamma left( mathbb{M}_mbigotimesnolimits_epsilon Yright)cong mathbb{M}_{nm}bigotimesnolimits_epsilonleft(Xbigotimesnolimits^gamma Yright),$$
where $X,Y$ are Banach spaces, $bigotimes_epsilon$ and $bigotimes_gamma$ denote respectively the Banach space injective and the Banach space projective tensor tensor products respectively. I am asking about the bicontinuity of the natural map given by $(aotimes x)otimes (botimes y)mapsto (aotimes b)otimes (xotimes y)$.
I think, one way to do it would be by identifying $mathbb{M}_nbigotimesnolimits_epsilon X$ (bicontinuously) with $mathbb{M}_nbigotimesnolimits_gamma X$ (and similar identification for $mathbb{M}_nbigotimesnolimits_epsilon Y$), and then use associativity and commutativity of the projective tensor product to get the conclusion. Is there another way to do it?
Will it still hold if the projective tensor product is replaced by some other Banach space cross tensor product, for which we have no idea about its commutativity or associativity?
Any reference would be highly appreciated.
banach-spaces tensor-products
asked Jan 11 at 8:01
JansonJanson
507
$endgroup$
add a comment |
$begingroup$
Does the following identity (bicontinuously) hold:
$$left(mathbb{M}_nbigotimesnolimits_epsilon Xright) bigotimesnolimits^gamma left( mathbb{M}_mbigotimesnolimits_epsilon Yright)cong mathbb{M}_{nm}bigotimesnolimits_epsilonleft(Xbigotimesnolimits^gamma Yright),$$
where $X,Y$ are Banach spaces, $bigotimes_epsilon$ and $bigotimes_gamma$ denote respectively the Banach space injective and the Banach space projective tensor tensor products respectively. I am asking about the bicontinuity of the natural map given by $(aotimes x)otimes (botimes y)mapsto (aotimes b)otimes (xotimes y)$.
I think, one way to do it would be by identifying $mathbb{M}_nbigotimesnolimits_epsilon X$ (bicontinuously) with $mathbb{M}_nbigotimesnolimits_gamma X$ (and similar identification for $mathbb{M}_nbigotimesnolimits_epsilon Y$), and then use associativity and commutativity of the projective tensor product to get the conclusion. Is there another way to do it?
Will it still hold if the projective tensor product is replaced by some other Banach space cross tensor product, for which we have no idea about its commutativity or associativity?
Any reference would be highly appreciated.
banach-spaces tensor-products
asked Jan 11 at 8:01
JansonJanson
507
$endgroup$
Does the following identity (bicontinuously) hold:
$$left(mathbb{M}_nbigotimesnolimits_epsilon Xright) bigotimesnolimits^gamma left( mathbb{M}_mbigotimesnolimits_epsilon Yright)cong mathbb{M}_{nm}bigotimesnolimits_epsilonleft(Xbigotimesnolimits^gamma Yright),$$
where $X,Y$ are Banach spaces, $bigotimes_epsilon$ and $bigotimes_gamma$ denote respectively the Banach space injective and the Banach space projective tensor tensor products respectively. I am asking about the bicontinuity of the natural map given by $(aotimes x)otimes (botimes y)mapsto (aotimes b)otimes (xotimes y)$.
I think, one way to do it would be by identifying $mathbb{M}_nbigotimesnolimits_epsilon X$ (bicontinuously) with $mathbb{M}_nbigotimesnolimits_gamma X$ (and similar identification for $mathbb{M}_nbigotimesnolimits_epsilon Y$), and then use associativity and commutativity of the projective tensor product to get the conclusion. Is there another way to do it?
Will it still hold if the projective tensor product is replaced by some other Banach space cross tensor product, for which we have no idea about its commutativity or associativity?
Any reference would be highly appreciated.
banach-spaces tensor-products
banach-spaces tensor-products
asked Jan 11 at 8:01
JansonJanson
507
asked Jan 11 at 8:01
JansonJanson
507
asked Jan 11 at 8:01
JansonJanson
507
asked Jan 11 at 8:01
JansonJanson
507
asked Jan 11 at 8:01
JansonJanson
507
507
add a comment |
add a comment |
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