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In Banach space theory, a super-property is a property of a Banach space that is preserved under ultrapowers. (Super-properties are also characterized--more commonly?--via finite representability).

What is the name for a property of a Banach space preserved under ultraproducts.

For example, a Banach space $B$ is super-reflexive iff it is reflexive and all its ultrapowers are reflexive. However, super-reflexitivity is not preserved under ultraproducts. Indeed, the spaces $L^p$ for $1<p<\infty$ are super-reflexive, but if $\mathcal{U}$ is a non-principle ultrafilter on $\mathbb{N}$, then the ultraproduct $(\prod_{n=2}^\infty L^n)/\mathcal{U}$ is not reflexive.

Every super-reflexive space is isomorphic (in the Banach space sense) to a $q$-uniformly convex space for $2\leq q <\infty$. Such a space is said to have martingale cotype $q$. The property of having martingale cotype $q$ is preserved under ultraproducts (for a fixed $q$).

Update 1 (2015-09-27): As Bill Johnson pointed out, having $q$-martingale cotype is a bad example, since it is not actually preserved under ultraproducts.

For a better example, being a uniformly convex Banach space with modulus $\delta(\varepsilon)$ is preserved under ultraproducts. [Proof: Let $\{B_i\}_{i\in I}$ be uniformly convex spaces with common modulus $\delta(\varepsilon)$. Let $x=(x_i)_\mathcal{U}$ and $y=(y_i)_\mathcal{U}$ be elements in the interior of the unit ball of the ultraproduct $B = \prod_{i\in I} B_i/\mathcal{U}$ with $\|x-y\|_B>\varepsilon$. Then $\mathcal{U}$-a.s. $\|x_i\|_{B_i}, \|y_i\|_{B_i} \leq 1$ and $\|x_i - y_i\|_{B_i} \geq \varepsilon$. By uniform convexity, $\|(x + y)/2\|_B = \lim_\mathcal{U} \|(x_i + y_i)/2\|_{B_i} \leq 1-\delta(\varepsilon)$.]

For another example, a space $B$ is of martingale cotype $q$ iff it is isomorphic to a Banach space with modulus of uniform convexity $\delta(\varepsilon) = C \varepsilon^q$. In other words, there are constants $C_1$ and $C_2$ and a new norm $\| \|_0$ such that $(1/C_1) \|x\|_0 \leq \|x\|_B \leq C_1 \|x\|_0$ and $(B,\|\|_0)$ is uniformly convex with modulus $\delta(\varepsilon) = C_2 \varepsilon^q$. Call such a $B$ a space of martingale cotype $(C_1,C_2,q)$. Unlike martingale cotype $q$, martingale cotype $(C_1,C_2,q)$ is preserved by ultraproducts. [The proof is similar to the previous one.]

Update 2 (2015-09-27): The reason I am interested in this is because such properties have a lot of uniformity associated with them. For example, Avigad and I gave a variational norm on the mean ergodic theorem which holds for every Banach space of martingale cotype $(C_1,C_2,q)$. Our inequality only depends on $C_1$, $C_2$ and $q$.

In Banach space theory, a super-property is a property of a Banach space that is preserved under ultrapowers. (Update (2015-09-28): The property must also be closed under isometric embeddings.) (Super-properties are also characterized--more commonly?--via finite representability).

What is the name for a property of a Banach space preserved under ultraproducts.

Update 3 (2015-09-28): Given that a super-property must also be closed under isometric embeddings (Thanks Bill Johnson!), I should update my question to the following version.

What is the name for a property of a Banach space preserved under ultraproducts and isometric embeddings.

My examples below satisfy this additional requirement.

For example, a Banach space $B$ is super-reflexive iff it is reflexive and all its ultrapowers are reflexive. However, super-reflexitivity is not preserved under ultraproducts. Indeed, the spaces $L^p$ for $1<p<\infty$ are super-reflexive, but if $\mathcal{U}$ is a non-principle ultrafilter on $\mathbb{N}$, then the ultraproduct $(\prod_{n=2}^\infty L^n)/\mathcal{U}$ is not reflexive.

Every super-reflexive space is isomorphic (in the Banach space sense) to a $q$-uniformly convex space for $2\leq q <\infty$. Such a space is said to have martingale cotype $q$. The property of having martingale cotype $q$ is preserved under ultraproducts (for a fixed $q$).

Update 1 (2015-09-27): As Bill Johnson pointed out, having $q$-martingale cotype is a bad example, since it is not actually preserved under ultraproducts.

For a better example, being a uniformly convex Banach space with modulus $\delta(\varepsilon)$ is preserved under ultraproducts. [Proof: Let $\{B_i\}_{i\in I}$ be uniformly convex spaces with common modulus $\delta(\varepsilon)$. Let $x=(x_i)_\mathcal{U}$ and $y=(y_i)_\mathcal{U}$ be elements in the interior of the unit ball of the ultraproduct $B = \prod_{i\in I} B_i/\mathcal{U}$ with $\|x-y\|_B>\varepsilon$. Then $\mathcal{U}$-a.s. $\|x_i\|_{B_i}, \|y_i\|_{B_i} \leq 1$ and $\|x_i - y_i\|_{B_i} \geq \varepsilon$. By uniform convexity, $\|(x + y)/2\|_B = \lim_\mathcal{U} \|(x_i + y_i)/2\|_{B_i} \leq 1-\delta(\varepsilon)$.]

For another example, a space $B$ is of martingale cotype $q$ iff it is isomorphic to a Banach space with modulus of uniform convexity $\delta(\varepsilon) = C \varepsilon^q$. In other words, there are constants $C_1$ and $C_2$ and a new norm $\| \|_0$ such that $(1/C_1) \|x\|_0 \leq \|x\|_B \leq C_1 \|x\|_0$ and $(B,\|\|_0)$ is uniformly convex with modulus $\delta(\varepsilon) = C_2 \varepsilon^q$. Call such a $B$ a space of martingale cotype $(C_1,C_2,q)$. Unlike martingale cotype $q$, martingale cotype $(C_1,C_2,q)$ is preserved by ultraproducts. [The proof is similar to the previous one.]

Update 2 (2015-09-27): The reason I am interested in this is because such properties have a lot of uniformity associated with them. For example, Avigad and I gave a variational norm on the mean ergodic theorem which holds for every Banach space of martingale cotype $(C_1,C_2,q)$. Our inequality only depends on $C_1$, $C_2$ and $q$.

In Banach space theory, a super-property is a property of a Banach space that is preserved under ultrapowers. (Super-properties are also characterized--more commonly?--via finite representability).

What is the name for a property of a Banach space preserved under ultraproducts.

For example, a Banach space $B$ is super-reflexive iff it is reflexive and all its ultrapowers are reflexive. However, super-reflexitivity is not preserved under ultraproducts. Indeed, the spaces $L^p$ for $1<p<\infty$ are super-reflexive, but if $\mathcal{U}$ is a non-principle ultrafilter on $\mathbb{N}$, then the ultraproduct $(\prod_{n=2}^\infty L^n)/\mathcal{U}$ is not reflexive.

Every super-reflexive space is isomorphic (in the Banach space sense) to a $q$-uniformly convex space for $2\leq q <\infty$. Such a space is said to have martingale cotype $q$. The property of having martingale cotype $q$ is preserved under ultraproducts (for a fixed $q$).

Update 1 (2015-09-27): As Bill Johnson pointed out, having $q$-martingale cotype is a bad example, since it is not actually preserved under ultraproducts.

For a better example, being a uniformly convex Banach space with modulus $\delta(\varepsilon)$ is preserved under ultraproducts. [Proof: Let $\{B_i\}_{i\in I}$ be uniformly convex spaces with common modulus $\delta(\varepsilon)$. Let $x=(x_i)_\mathcal{U}$ and $y=(y_i)_\mathcal{U}$ be elements in the interior of the unit ball of the ultraproduct $B = \prod_{i\in I} B_i/\mathcal{U}$ with $\|x-y\|_B>\varepsilon$. Then $\mathcal{U}$-a.s. $\|x_i\|_{B_i}, \|y_i\|_{B_i} \leq 1$ and $\|x_i - y_i\|_{B_i} \geq \varepsilon$. By uniform convexity, $\|(x + y)/2\|_B = \lim_\mathcal{U} \|(x_i + y_i)/2\|_{B_i} \leq 1-\delta(\varepsilon)$.]

For another example, a space $B$ is of martingale cotype $q$ iff it is isomorphic to a Banach space with modulus of uniform convexity $\delta(\varepsilon) = C \varepsilon^q$. In other words, there is a constant $C$ and a new norm $\| \|_0$ such that $(1/C) \|x\|_0 \leq \|x\|_B \leq C \|x\|_0$ and $(B,\|\|_0)$ is uniformly convex with modulus $\delta(\varepsilon) = C \varepsilon^q$. Call such a $B$ a space of martingale cotype $(C,q)$. Unlike martingale cotype $q$, martingale cotype $(C,q)$ is preserved by ultraproducts. [The proof is similar to the previous one.]

Update 2 (2015-09-27): The reason I am interested in this is because such properties have a lot of uniformity associated with them. For example, Avigad and I gave a variational norm on the mean ergodic theorem which holds for every Banach space of martingale cotype $(C,q)$. Our inequality only depends on $C$ and $q$.

In Banach space theory, a super-property is a property of a Banach space that is preserved under ultrapowers. (Super-properties are also characterized--more commonly?--via finite representability).

What is the name for a property of a Banach space preserved under ultraproducts.

For example, a Banach space $B$ is super-reflexive iff it is reflexive and all its ultrapowers are reflexive. However, super-reflexitivity is not preserved under ultraproducts. Indeed, the spaces $L^p$ for $1<p<\infty$ are super-reflexive, but if $\mathcal{U}$ is a non-principle ultrafilter on $\mathbb{N}$, then the ultraproduct $(\prod_{n=2}^\infty L^n)/\mathcal{U}$ is not reflexive.

Every super-reflexive space is isomorphic (in the Banach space sense) to a $q$-uniformly convex space for $2\leq q <\infty$. Such a space is said to have martingale cotype $q$. The property of having martingale cotype $q$ is preserved under ultraproducts (for a fixed $q$).

Update 1 (2015-09-27): As Bill Johnson pointed out, having $q$-martingale cotype is a bad example, since it is not actually preserved under ultraproducts.

For a better example, being a uniformly convex Banach space with modulus $\delta(\varepsilon)$ is preserved under ultraproducts. [Proof: Let $\{B_i\}_{i\in I}$ be uniformly convex spaces with common modulus $\delta(\varepsilon)$. Let $x=(x_i)_\mathcal{U}$ and $y=(y_i)_\mathcal{U}$ be elements in the interior of the unit ball of the ultraproduct $B = \prod_{i\in I} B_i/\mathcal{U}$ with $\|x-y\|_B>\varepsilon$. Then $\mathcal{U}$-a.s. $\|x_i\|_{B_i}, \|y_i\|_{B_i} \leq 1$ and $\|x_i - y_i\|_{B_i} \geq \varepsilon$. By uniform convexity, $\|(x + y)/2\|_B = \lim_\mathcal{U} \|(x_i + y_i)/2\|_{B_i} \leq 1-\delta(\varepsilon)$.]

For another example, a space $B$ is of martingale cotype $q$ iff it is isomorphic to a Banach space with modulus of uniform convexity $\delta(\varepsilon) = C \varepsilon^q$. In other words, there are constants $C_1$ and $C_2$ and a new norm $\| \|_0$ such that $(1/C_1) \|x\|_0 \leq \|x\|_B \leq C_1 \|x\|_0$ and $(B,\|\|_0)$ is uniformly convex with modulus $\delta(\varepsilon) = C_2 \varepsilon^q$. Call such a $B$ a space of martingale cotype $(C_1,C_2,q)$. Unlike martingale cotype $q$, martingale cotype $(C_1,C_2,q)$ is preserved by ultraproducts. [The proof is similar to the previous one.]

Update 2 (2015-09-27): The reason I am interested in this is because such properties have a lot of uniformity associated with them. For example, Avigad and I gave a variational norm on the mean ergodic theorem which holds for every Banach space of martingale cotype $(C_1,C_2,q)$. Our inequality only depends on $C_1$, $C_2$ and $q$.

In Banach space theory, a super-property is a property of a Banach space that is preserved under ultrapowers. (Super-properties are also characterized--more commonly?--via finite representability).

What is the name for a property of a Banach space preserved under ultraproducts.

For example, a Banach space $B$ is super-reflexive iff it is reflexive and all its ultrapowers are reflexive. However, super-reflexitivity is not preserved under ultraproducts. Indeed, the spaces $L^p$ for $1<p<\infty$ are super-reflexive, but if $\mathcal{U}$ is a non-principle ultrafilter on $\mathbb{N}$, then the ultraproduct $(\prod_{n=2}^\infty L^n)/\mathcal{U}$ is not reflexive.

Every super-reflexive space is isomorphic (in the Banach space sense) to a $q$-uniformly convex space for $2\leq q <\infty$. Such a space is said to have martingale cotype $q$. The property of having martingale cotype $q$ is preserved under ultraproducts (for a fixed $q$).

In Banach space theory, a super-property is a property of a Banach space that is preserved under ultrapowers. (Super-properties are also characterized--more commonly?--via finite representability).

What is the name for a property of a Banach space preserved under ultraproducts.

For example, a Banach space $B$ is super-reflexive iff it is reflexive and all its ultrapowers are reflexive. However, super-reflexitivity is not preserved under ultraproducts. Indeed, the spaces $L^p$ for $1<p<\infty$ are super-reflexive, but if $\mathcal{U}$ is a non-principle ultrafilter on $\mathbb{N}$, then the ultraproduct $(\prod_{n=2}^\infty L^n)/\mathcal{U}$ is not reflexive.

Every super-reflexive space is isomorphic (in the Banach space sense) to a $q$-uniformly convex space for $2\leq q <\infty$. Such a space is said to have martingale cotype $q$. The property of having martingale cotype $q$ is preserved under ultraproducts (for a fixed $q$).

Update 1 (2015-09-27): As Bill Johnson pointed out, having $q$-martingale cotype is a bad example, since it is not actually preserved under ultraproducts.

For a better example, being a uniformly convex Banach space with modulus $\delta(\varepsilon)$ is preserved under ultraproducts. [Proof: Let $\{B_i\}_{i\in I}$ be uniformly convex spaces with common modulus $\delta(\varepsilon)$. Let $x=(x_i)_\mathcal{U}$ and $y=(y_i)_\mathcal{U}$ be elements in the interior of the unit ball of the ultraproduct $B = \prod_{i\in I} B_i/\mathcal{U}$ with $\|x-y\|_B>\varepsilon$. Then $\mathcal{U}$-a.s. $\|x_i\|_{B_i}, \|y_i\|_{B_i} \leq 1$ and $\|x_i - y_i\|_{B_i} \geq \varepsilon$. By uniform convexity, $\|(x + y)/2\|_B = \lim_\mathcal{U} \|(x_i + y_i)/2\|_{B_i} \leq 1-\delta(\varepsilon)$.]

For another example, a space $B$ is of martingale cotype $q$ iff it is isomorphic to a Banach space with modulus of uniform convexity $\delta(\varepsilon) = C \varepsilon^q$. In other words, there is a constant $C$ and a new norm $\| \|_0$ such that $(1/C) \|x\|_0 \leq \|x\|_B \leq C \|x\|_0$ and $(B,\|\|_0)$ is uniformly convex with modulus $\delta(\varepsilon) = C \varepsilon^q$. Call such a $B$ a space of martingale cotype $(C,q)$. Unlike martingale cotype $q$, martingale cotype $(C,q)$ is preserved by ultraproducts. [The proof is similar to the previous one.]

Update 2 (2015-09-27): The reason I am interested in this is because such properties have a lot of uniformity associated with them. For example, Avigad and I gave a variational norm on the mean ergodic theorem which holds for every Banach space of martingale cotype $(C,q)$. Our inequality only depends on $C$ and $q$.