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I don't know how fruitful it proved, but my impression of Pisier's counterexample to a conjecture of Grothendieck (a Banach space on whose tensor square all reasonable cross-norms coincide) is that it was unexpected at the time. Perhaps Bill Johnson can correct me on this if I am mistaken? It certainly qualifies as ingenious, in my book.

The article is

G. Pisier, Counterexamples to a conjecture of Grothendieck. Acta Math. 151 (1983), no. 3-4, 181--208.

Since the MathReview explains things quite well, and people may not have easy access, here are some selected parts of the review:

A. Grothendieck posed the following problem ... if $X$ and $Y$ are Banach spaces such that $X\check\otimes Y$ and $X\hat\otimes Y$ coincide, then is one of them necessarily finite-dimensional? Grothendieck conjectured a positive answer [Bol. Soc. Mat. Sao Paulo 8 (1953), 1--79; MR0094682 (20 #1194) ]. The author solves this problem in the negative by giving an example of a separable infinite-dimensional Banach space $X$ such that $X\check\otimes X= X\hat\otimes X$.

... any (separable) Banach space $E$ of cotype 2 can be imbedded isometrically into a (separable) Banach space $X$ such that (a) $X$ and $X^*$ are both of cotype 2 and they "verify Grothendieck's theorem'' (every operator into a Hilbert space is absolutely summing), and (b) $X\check\otimes X=X\hat\otimes X$.

... Any infinite-dimensional space $X$ satisfying (a) cannot be isomorphic to a Hilbert space, although $X$ and $X^*$ are of cotype 2...

... the natural map $X^* \hat\otimes X\rightarrow X^*\check \otimes X$ is not injective. However, it is shown that for any Banach space $X$ satisfying (a) this map is surjective. In other words: every operator in $X$ which is a uniform limit of finite-rank operators is nuclear.

I don't know how fruitful it proved, but my impression of Pisier's counterexample to a conjecture of Grothendieck (a Banach space on whose tensor square all reasonable cross-norms coincide) is that it was unexpected at the time. Perhaps Bill Johnson can correct me on this if I am mistaken? It certainly qualifies as ingenious, in my book.

The article is

G. Pisier, Counterexamples to a conjecture of Grothendieck. Acta Math. 151 (1983), no. 3-4, 181--208.

Since the MathReview explains things quite well, and people may not have easy access, here are some selected parts of the review:

A. Grothendieck posed the following problem ... if $X$ and $Y$ are Banach spaces such that $X\check\otimes Y$ and $X\hat\otimes Y$ coincide, then is one of them necessarily finite-dimensional? Grothendieck conjectured a positive answer [Bol. Soc. Mat. Sao Paulo 8 (1953), 1--79; MR0094682 (20 #1194) ]. The author solves this problem in the negative by giving an example of a separable infinite-dimensional Banach space $X$ such that $X\check\otimes X= X\hat\otimes X$.

... any (separable) Banach space $E$ of cotype 2 can be imbedded isometrically into a (separable) Banach space $X$ such that (a) $X$ and $X^*$ are both of cotype 2 and they "verify Grothendieck's theorem'' (every operator into a Hilbert space is absolutely summing), and (b) $X\check\otimes X=X\hat\otimes X$.

... Any infinite-dimensional space $X$ satisfying (a) cannot be isomorphic to a Hilbert space, although $X$ and $X^*$ are of cotype 2...

... the natural map $X^* \hat\otimes X\rightarrow X^*\check \otimes X$ is not injective. However, it is shown that for any Banach space $X$ satisfying (a) this map is surjective. In other words: every operator in $X$ which is a uniform limit of finite-rank operators is nuclear.

I don't know how fruitful it proved, but my impression of Pisier's counterexample to a conjecture of Grothendieck (a Banach space on whose tensor square all reasonable cross-norms coincide) is that it was unexpected at the time. Perhaps Bill Johnson can correct me on this if I am mistaken? It certainly qualifies as ingenious, in my book.

The article is

G. Pisier, Counterexamples to a conjecture of Grothendieck. Acta Math. 151 (1983), no. 3-4, 181--208.

Since the MathReview explains things quite well, and people may not have easy access, here are some selected parts of the review:

A. Grothendieck posed the following problem ... if $X$ and $Y$ are Banach spaces such that $X\check\otimes Y$ and $X\hat\otimes Y$ coincide, then is one of them necessarily finite-dimensional? Grothendieck conjectured a positive answer [Bol. Soc. Mat. Sao Paulo 8 (1953), 1--79; MR0094682 (20 #1194) ]. The author solves this problem in the negative by giving an example of a separable infinite-dimensional Banach space $X$ such that $X\check\otimes X= X\hat\otimes X$.

 

... any (separable) Banach space $E$ of cotype 2 can be imbedded isometrically into a (separable) Banach space $X$ such that (a) $X$ and $X^*$ are both of cotype 2 and they "verify Grothendieck's theorem'' (every operator into a Hilbert space is absolutely summing), and (b) $X\check\otimes X=X\hat\otimes X$.

 

... Any infinite-dimensional space $X$ satisfying (a) cannot be isomorphic to a Hilbert space, although $X$ and $X^*$ are of cotype 2...

 

... the natural map $X^* \hat\otimes X\rightarrow X^*\check \otimes X$ is not injective. However, it is shown that for any Banach space $X$ satisfying (a) this map is surjective. In other words: every operator in $X$ which is a uniform limit of finite-rank operators is nuclear.

I don't know how fruitful it proved, but my impression of Pisier's counterexample to a conjecture of Grothendieck (a Banach space on whose tensor square all reasonable cross-norms coincide) is that it was unexpected at the time. Perhaps Bill Johnson can correct me on this if I am mistaken? It certainly qualifies as ingenious, in my book.

The article is

G. Pisier, Counterexamples to a conjecture of Grothendieck. Acta Math. 151 (1983), no. 3-4, 181--208.

Since the MathReview explains things quite well, and people may not have easy access, here are some selected parts of the review:

A. Grothendieck posed the following problem ... if $X$ and $Y$ are Banach spaces such that $X\check\otimes Y$ and $X\hat\otimes Y$ coincide, then is one of them necessarily finite-dimensional? Grothendieck conjectured a positive answer [Bol. Soc. Mat. Sao Paulo 8 (1953), 1--79; MR0094682 (20 #1194) ]. The author solves this problem in the negative by giving an example of a separable infinite-dimensional Banach space $X$ such that $X\check\otimes X= X\hat\otimes X$.

... any (separable) Banach space $E$ of cotype 2 can be imbedded isometrically into a (separable) Banach space $X$ such that (a) $X$ and $X^*$ are both of cotype 2 and they "verify Grothendieck's theorem'' (every operator into a Hilbert space is absolutely summing), and (b) $X\check\otimes X=X\hat\otimes X$.

... Any infinite-dimensional space $X$ satisfying (a) cannot be isomorphic to a Hilbert space, although $X$ and $X^*$ are of cotype 2...

... the natural map $X^* \hat\otimes X\rightarrow X^*\check \otimes X$ is not injective. However, it is shown that for any Banach space $X$ satisfying (a) this map is surjective. In other words: every operator in $X$ which is a uniform limit of finite-rank operators is nuclear.

I don't know how fruitful it proved, but my impression of Pisier's counterexample to a conjecture of Grothendieck (a Banach space on whose tensor square all reasonable cross-norms coincide) is that it was unexpected at the time. Perhaps Bill Johnson can correct me on this if I am mistaken? It certainly qualifies as ingenious, in my book.

(out of office right now, will add details later)

I don't know how fruitful it proved, but my impression of Pisier's counterexample to a conjecture of Grothendieck (a Banach space on whose tensor square all reasonable cross-norms coincide) is that it was unexpected at the time. Perhaps Bill Johnson can correct me on this if I am mistaken? It certainly qualifies as ingenious, in my book.

The article is

G. Pisier, Counterexamples to a conjecture of Grothendieck. Acta Math. 151 (1983), no. 3-4, 181--208.

Since the MathReview explains things quite well, and people may not have easy access, here are some selected parts of the review:

A. Grothendieck posed the following problem ... if $X$ and $Y$ are Banach spaces such that $X\check\otimes Y$ and $X\hat\otimes Y$ coincide, then is one of them necessarily finite-dimensional? Grothendieck conjectured a positive answer [Bol. Soc. Mat. Sao Paulo 8 (1953), 1--79; MR0094682 (20 #1194) ]. The author solves this problem in the negative by giving an example of a separable infinite-dimensional Banach space $X$ such that $X\check\otimes X= X\hat\otimes X$.

... any (separable) Banach space $E$ of cotype 2 can be imbedded isometrically into a (separable) Banach space $X$ such that (a) $X$ and $X^*$ are both of cotype 2 and they "verify Grothendieck's theorem'' (every operator into a Hilbert space is absolutely summing), and (b) $X\check\otimes X=X\hat\otimes X$.

... Any infinite-dimensional space $X$ satisfying (a) cannot be isomorphic to a Hilbert space, although $X$ and $X^*$ are of cotype 2...

... the natural map $X^* \hat\otimes X\rightarrow X^*\check \otimes X$ is not injective. However, it is shown that for any Banach space $X$ satisfying (a) this map is surjective. In other words: every operator in $X$ which is a uniform limit of finite-rank operators is nuclear.