Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [ultrapowers]

The tag has no usage guidance.

0
votes
0answers
85 views

Why is $\widetilde{W}$ closed?

let's consider $\mathscr{U}$ a free ultrafilter on the natural numbers and consider its corresponding ultrapower \begin{align*} \widetilde{X} = (\ell ^{\infty}(X _{i})/\operatorname{ker}\mathcal{N},\...
16
votes
1answer
502 views

Is there a pair of non-isomorphic structures each of which is isomorphic to an ultrapower of the other?

Does there exist a pair of non-isomorphic structures $\mathfrak{A}$ and $\mathfrak{B}$ as well as sets $I$ and $J$ and ultrafilters $\mathcal{U}$ on $I$ and $\mathcal{F}$ on $J$ such that $\mathfrak{A}...
4
votes
1answer
138 views

If $j:V\prec M$ has critical point $κ$ and for any $X\in M$ with $|X|=μ$, $|X|^M=μ$, what properties does $κ$ have?

More specifically, if $j:V\prec M$ has critical point $κ$ and for any $X\in M$ with $|X|=μ$, $|X|^M=μ$, does $κ$ necessarily have some form of $μ$-compactness? Is it related to strong compactness in ...
5
votes
1answer
201 views

Model theory of Banach algebras

Let us consider the (metric) theory of Banach algebras. I have a sentence encoding the (possible) openness of multiplication in a given Banach algebra: $$(\forall x) (\forall y) (\forall \varepsilon &...
2
votes
1answer
88 views

A Question about an irreducible ultra-power II,

Let $E$ be an irreducible Banach $A$-module, for a Banach algebra $A$. One can easily show that for an ultra filter $\mathcal U$, $(E)_\mathcal U$ is a Banach $(A)_\mathcal U$-module. Is it possible ...
0
votes
0answers
64 views

A question about an irreducible ultra-power

Let $A$ be a Banach algebra and $E$ be an irreducible Banach $A$-module. Is there a countably incomplete ultra filter $\mathcal U$ on $\mathbb N$, the set of natural numbers, such that the ultra power ...
1
vote
1answer
188 views

About reflexivity of ultrapower

It is obvious that for a Banach space $E$, $E$ is reflexive iff $\ell^2(E)$ is reflexive. Let $\mathcal U$ be an ultrafilter. Is the reflexivity of $(E)_\mathcal U$ equivalent to refelxivity of $(\ell^...
0
votes
1answer
191 views

What strengthenings of measurability do the Mostowski collapses of ultrapowers possess?

What strengthenings of measurability does the Mostowski collapse of the ultrapowers possess? Ok, I already posted this question, but a couple of notational errors and assumptions were made in the ...
3
votes
1answer
155 views

A question on ultraproducts of $L_{p}(\mu)$-spaces

Let $I$ be a set, $\mathcal{U}$ be an ultrafilter on $I$ and $1\leq p<\infty$. Let $X_{i}=L_{p}(\mu_{i})(i\in I)$, where $\mu_{i}$ is a probability measure for each $i\in I$. Relying on standard ...
7
votes
1answer
220 views

What is the Turing degree associated with an ultrafilter $U$?

I asked Turing degree of a turing machine with access to an (arbitrary) nonstandard integer, not thinking about the possiblity that this could depend on the model used. The question was not formulated ...
-1
votes
1answer
202 views

On ultraproducts of topological spaces

Intuitively, I understand the construction of the hyperreals by ultraproducts as a process of turning the limit operation into an algebraic object. More precisely, to check the existence of the limit $...
3
votes
1answer
105 views

Biduals of Banach algebras

For a Banach algebra $A$ the bidual $A^{**}$ may be given two natural products called the Arens products. By local reflexivity, there is an ultrafilter $U$ so that $A^{**}$ embeds into the ultrapower $...
2
votes
0answers
103 views

Ultrapowers of $c_0(\ell_1)$ and $\ell_1(c_0)$

I would like to know if there exist an explicit decription of ultrapowers of $c_0(\ell_1)$ and $\ell_1(c_0)$. The best option would be -- "they are complemented subspaces of $C(K, L_1(\mu))$ and $L_1(\...
5
votes
0answers
122 views

Banach spaces complemented in their ultrapowers

By the principle of local reflexivity, the second dual $X^{**}$ of a Banach space $X$ is complemented in some ultrapower $X^U$ of $X$. Even when $X$ is separable, the index set of $U$ cannot be ...
5
votes
2answers
431 views

Statements that Could be Forced by Ultrapowers

Ultrapower of a structure is a very flexible mathematical creature in comparison with the ground structure and its ordinary products. Depending on the nature of ground structure and the good ...
2
votes
0answers
103 views

Ultraproducts and subobjects of projectives

Usually the question whether the class of projective algebras in a given variety is closed under taking subalgebras seems to be quite hard. In varieties with well understood dual geometry (e. g. ...
1
vote
0answers
70 views

Reduced products of (abelian and triangulated) categories: references?

For a filter $U$ on a set $X$ and for a family of categories $C_x$ indexed by $X$ I would like to consider the (corresponding categorical version of) reduced product of $C_x$ (for $x\in X$) with ...
3
votes
0answers
132 views

Which compact topological spaces are homeomorphic to their ultrapower?

It is well known that for any compact metric space $(X, d)$, and any ultrafilter $\mu$ there is a map $i_\mu:\prod_\mu (X, d) \to (X_d)$ in the category of metric spaces and Lipschitz maps where $i_\...
3
votes
2answers
393 views

References for the Keisler Order

Are there any good/modern references on the Keisler order. I have been reading Keisler's original paper, "Ultraproducts which are not Saturated", which introduces the order. However it is somewhat ...
6
votes
1answer
383 views

Ultraproduct of Forcing Extensions & Forcing Extension of Ultraproduct

Notation: $M[{\mathbb{P}}:G]$ denotes the forcing extension of $M$ by $\mathbb{P}$-generic filter $G$. $\prod_{\mathcal{F}}\langle M_i~|~i\in I\rangle$ denotes the ultraproduct of models using the ...
10
votes
1answer
381 views

K-theory of ultrapowers

It may well be a trivial question but I was wondering if there is any relation between $K$-groups and ultrapowers of $C^*$-algebras. For instance, if $A$ is a $C^*$-algebra does $K_0(A^U)$ depend on ...
6
votes
0answers
166 views

Ultrapowers and complemented subspaces

Let $Y$ be a closed subspace of a Banach space $X$, and let $\mathcal{U}$ be a nontrivial ultrafilter on the set $\mathbb{N}$ of all integer numbers. It is not difficult to see that if $Y$ is ...
5
votes
1answer
195 views

Complemented subspaces of ultrapowers

It's a famous result of Maurey that a Banach space $E$ is finitely representable in $X$ if and only if it is a subspace of some ultrapower of $X$. Is there an analogous result for complemented ...
2
votes
1answer
230 views

How many elementary equivalent models are unifiable by ultrapower?

Definition. A class $\mathcal{C}$ of pairwise elementary equivalent $\mathcal{L}$-structures is unifiable by ultrapower if there is an index set $I$ and an ultrafilter $F$ on it such that $\forall M,N\...
13
votes
4answers
1k views

Is it ever a good idea to use Keisler-Shelah to show elementary equivalence?

The most useful way I know to show that two structures are elementarily equivalent is Ehrenfeucht-Fraisse games. These are quite nice and intuitive, and even when I can't use them to solve my problem ...
1
vote
0answers
228 views

Ultrapowers of ultrapowers

Suppose that you have some structure $S$, and you want to construct an ultrapower of cardinality $\kappa$ to obtain $S^*_\kappa$. Then, say you want to construct a new ultrapower from $S^*_\kappa$, ...
11
votes
3answers
670 views

Surreal numbers, ultrapowers of $\Bbb R$, ordinal-valued functions and the slow-growing hierarchy

Philip Ehrlich's paper “The Absolute Arithmetic Continuum and the Unification of All Numbers Great and Small”, The Bulletin of Symbolic Logic 18 (1) 2012, pp. 1-45. claims as a theorem that, in NBG, $\...
2
votes
1answer
225 views

Can we flex the rigid models by enough power?

Definition (1): ‎An ‎‎$‎‎‎\mathcal{L}‎$ -‎ ‎structure ‎‎$‎‎‎\mathcal{M}‎$ ‎called ‎"‎‎rigid" ‎iff ‎‎there ‎is ‎no ‎non-trivial automorphism on ‎$‎\mathcal{M}‎$.‎ ‎‎ Definition (2): ‎An ‎‎$‎‎‎\mathcal{...
5
votes
1answer
179 views

Is there a truth approximation on a‎ cumulative hierarchy‎‎‎‎?

‎‎‎Note ‎to ‎the ‎following ‎well known theorem:‎ Theorem (1): ‎If ‎‎$‎‎\kappa‎‎$ ‎be a ‎‎"measurable" ‎cardinal ‎and ‎‎$‎‎‎\mathcal{F}‎$ be a‎ ‎"non-principal ‎$‎‎‎\kappa‎$-complete normal‎" ‎...
3
votes
0answers
119 views

What is known about the krull dimension of an ultrapower ring?

Let $R$ be a ring, $F$ a free ultrafilter on a set $X$ which is not countably complete, and $R_F$ the ultrapower of $R$ with $R \not\cong R_F$. The following two results are from a masters thesis ...
16
votes
2answers
697 views

Why does CH imply that there is a unique ultrapower of $\mathbb{N}$?

I've read these words: "How many ultra products $∏_Uℕ$ exist up to isomorphism, where $U$ is a non-principal ultrafilter over $ℕ$? If continuum hypothesis(CH) holds, then obviously just one ..." i ...
4
votes
1answer
128 views

Galois action on ultrapowers

Let $K$ be a char $0$ field with algebraic closure $\bar K$ and absolute Galois group $G$. Let $\mathcal U$ be an ultrafilter on $\mathbb N$ and $F=\bar K^\mathbb N/\mathcal U$ be the ultrapower of $\...
1
vote
1answer
238 views

When can we “displace” an ultrafilter limit with another limit?

Let $\cal A$ be a Banach algebra, $\cal U$ be a free ultrafilter, and $\phi$ be a character. Let ${(w_{\alpha})}_{\alpha}$ be a net in $(\cal A)_{\cal U}$, and suppose that for every $(a_i)\in (\cal A)...
5
votes
0answers
273 views

Double ultrapower of the hyperfinite $II_1$-factor

Let $\omega$ be a free ultrafilter on the natural numbers and $R$ be the hyperfinite $II_1$-factor (the definition of $R$ is recalled in the comments). Question: Does there exist another free ...
1
vote
1answer
293 views

an elementary substructure of a natural numbers ultrapower

Hi I'm looking for an elementary substructure of a natural numbers ultrapower with a free ultrafilter over a numerable set also must not be isomorphism between the elementary substructure and any ...
4
votes
2answers
417 views

Set forcing and ultrapowers

The following is a result of Woodin (the proof is found after Theorem 5 of "Generalizations of the Kunen Inconsistency" by J.D.Hamkins, G.Kirmayer and N.L.Perlmutter): (Woodin) Let $V[G]$ be a set-...
6
votes
2answers
379 views

Ultrapowers of operators

Can we prove that for each infinite dimensional Banach space $X$ and any free ultrafilter (possibly over uncountable set of indices) $\mathcal{U}$ the obvious embedding $$({\mathcal{L}(X)})_{\mathcal{...
3
votes
1answer
312 views

Is an ultrapower of a faithful Banach algebra always faithful?

Let $A$ be an infinite dimensional faithful Banach algebra and let $\mathcal U$ be a free ultrafilter. Is the ultrapower $(A)_{\mathcal U}$ faithful?
2
votes
1answer
163 views

Whether the result that an ultraproduct which satisfies ACCP is automatically a field generalizes to ultrafilters on larger indexing sets

Let $F$ be an ultrafilter on some set $X$, $R$ an integral domain and $R_F$ the resulting ultraproduct ring. For an element $(a)$ in the product ring of $R$ indexed by $X$, denote its equivalence ...
11
votes
1answer
414 views

Do ultrapowers of classical Banach spaces have unconditional bases?

I am trying to imagine (to some extent, of course) the geometry of ultrapowers of certain 'easy-to-handle' Banach spaces. Let me start with $X = \ell_p$, $p\in (1,\infty)$ or $X=c_0$. Since the ...
6
votes
1answer
375 views

iterating ultrapowers of C*-algebras: the Calkin algebra

Elsewhere I asked about ultrapowers of the C*-algebra $A$ of compact operators on separable infinite-dimensional Hilbert space. My question was whether the process of taking ultrapowers of ultrapowers ...
5
votes
2answers
416 views

iterating ultrapowers of C*-algebras

Let $A$ be something interesting like the C*-algebra of compact operators on a separable infinite-dimensional Hilbert space and let $A^1$ be an ultrapower of $A$. Then $A^1$ is a primitive C*-algebra ...
6
votes
2answers
600 views

An Extender is a Generalization of an Ultrafilter?

I'm not sure whether this belongs here or on math stackexchange, but I'll give a try here. I've heard that an extender is a generalization of an ultrafilter. This is not to say that an ultrafilter ...
5
votes
2answers
354 views

Do (Banach) ultrapowers carry some sort of 'elementary equivalence'?

The (model-theoretic) ultrapowers had been used for studying elementary equivalnce of first-order structures. Then, they have been adapted to Banach spaces, which are, let me say, second-order ...
9
votes
1answer
511 views

Homomorphic images of a Cartesian product of finite groups

What can be said about the class of groups which can be represented as a homomorphic image of an (infinite) Cartesian product (=unrestricted direct product) of finite groups? What would be simple ...
8
votes
2answers
486 views

Is Every field Extension of an Ultrafield an Ultrafield?

Let $K=lim(K_{i})$ be an ultrafield (over a non-principal ultrafilter), and let $K\hookrightarrow K'$ be a field extension of $K$. When the field $K'$ is finite over $K$ it is also an ultrafield by ...
16
votes
0answers
569 views

Dual of the Ultraproduct of a Banach Space

Suppose we have a Banach space ultraproduct $(E_i)_U$. I can think of three natural objects which are "dual-like": $(E_i^*)_U$, the ultraproduct of the duals of the ground spaces. The space made up ...
4
votes
1answer
230 views

cardinality of local bases in the non-standard reals

Given a index set $S$ together with a ultrafilter $\mu$ on $S$ (such that no set of cardinality $< |S|$ has measure $1$). Let the ordered field $\mathbb{R}(S,\mu)$ denote the ultrapower of $\mathbb{...
9
votes
4answers
863 views

ultrapowers and higher order logic

One of the reasons I think ultrapowers are interesting is the following corollary of Łoś's theorem: Let $V$ be a relational structure and $^*V$ an ultrapower of $V$. Then a first order statement ...
5
votes
1answer
314 views

What is known about the ultra-inverse limit?

Given a nonprincipal ultrafilter $\mu$ on $\mathbb{N}$ and a sequence of groups $G_i$, one can define its ultraproduct as: $$ ^*\prod_{i\in \mathbb{N}}G_i:=\{(x_i)_{i \in \mathbb{N}}| x_i\in G_i\}/\...