We introduce the notion of a totally ($K$-) bounded element of a W*-probability space $(M, \varphi)$ and, borrowing ideas of Kadison, give an intrinsic characterization of the $^*$-subalgebra $M_{tb}$ of ā¦
We introduce the notion of a totally ($K$-) bounded element of a W*-probability space $(M, \varphi)$ and, borrowing ideas of Kadison, give an intrinsic characterization of the $^*$-subalgebra $M_{tb}$ of totally bounded elements. Namely, we show that $M_{tb}$ is the unique strongly dense $^*$-subalgebra $M_0$ of totally bounded elements of $M$ for which the collection of totally $1$-bounded elements of $M_0$ is complete with respect to the $\|\cdot\|_\varphi^\#$-norm and for which $M_0$ is closed under all operators $h_a(\log(\Delta))$ for $a \in \mathbb{N}$, where $\Delta$ is the modular operator and $h_a(t):=1/\cosh(t-a)$ (see Theorem 4.3). As an application, we combine this characterization with Rieffel and Van Daele's bounded approach to modular theory to arrive at a new language and axiomatization of W*-probability spaces as metric structures. Previous work of Dabrowski had axiomatized W*-probability spaces using a smeared version of multiplication, but the subalgebra $M_{tb}$ allows us to give an axiomatization in terms of the original algebra operations. Finally, we prove the (non-)axiomatizability of several classes of W*-probability spaces.
Abstract We show that the universal theory of the hyperfinite II $_1$ factor is not computable. The proof uses the recent result that MIP*=RE. Combined with an earlier observation of ā¦
Abstract We show that the universal theory of the hyperfinite II $_1$ factor is not computable. The proof uses the recent result that MIP*=RE. Combined with an earlier observation of the authors, this yields a proof that the Connes Embedding Problem has a negative solution that avoids the equivalences with Kirchbergās QWEP Conjecture and Tsirelsonās Problem.
We survey the developments in the model theory of tracial von Neumann algebras that have taken place in the last 15 years. We discuss the appropriate first-order language for axiomatizing ā¦
We survey the developments in the model theory of tracial von Neumann algebras that have taken place in the last 15 years. We discuss the appropriate first-order language for axiomatizing this class as well as the subclass of II1 factors and how modeltheoretic ideas were used to settle a variety of questions around isomorphism of ultrapowers of tracial von Neumann algebras with respect to different ultrafilters. We move on to more model-theoretic concerns, such as theories of II1 factors and existentially closed II1 factors, and conclude with two recent applications of model-theoretic ideas to questions around relative commutants.
We present an introduction to modern continuous model theory with an emphasis on its interactions with topics covered in this volume such as Cā-algebras and von Neumann algebras. The role ā¦
We present an introduction to modern continuous model theory with an emphasis on its interactions with topics covered in this volume such as Cā-algebras and von Neumann algebras. The role of ultraproducts is highlighted and expositions of definable sets, imaginaries, quantifier elimination, and separable categoricity are included.
We introduce the notion of a Tsirelson pair of C*-algebras, which is a pair of C*-algebras for which the space of quantum strategies obtained by using states on the minimal ā¦
We introduce the notion of a Tsirelson pair of C*-algebras, which is a pair of C*-algebras for which the space of quantum strategies obtained by using states on the minimal tensor product of the pair is dense in the space of quantum strategies obtained by using states on the maximal tensor product. We exhibit a number of examples of such pairs that are ānontrivialā in the sense that the minimal tensor product and the maximal tensor product of the pair are not isomorphic. For example, we prove that any pair containing a C*-algebra with Kirchbergās QWEP property is a Tsirelson pair. We then introduce the notion of a C*-algebra with the Tsirelson property (TP) and establish a number of closure properties for this class. We also show that the class of C*-algebras with the TP forms an elementary class (in the sense of model theory), but that this class does not admit an effective axiomatization.
The Kirchberg Embedding Problem (KEP) asks if every C*-algebra embeds into an ultrapower of the Cuntz algebra $\mathcal{O}_2$. In an effort to provide a negative solution to the KEP and ā¦
The Kirchberg Embedding Problem (KEP) asks if every C*-algebra embeds into an ultrapower of the Cuntz algebra $\mathcal{O}_2$. In an effort to provide a negative solution to the KEP and motivated by the recent refutation of the Connes Embedding Problem, we establish two computability-theoretic consequences of a positive solution to KEP. Both of our results follow from the a priori weaker assumption that there exists a locally universal C*-algebra with a computable presentation.
We present an introduction to modern continuous model theory with an emphasis on its interactions with topics covered in this volume such as $C^*$-algebras and von Neumann algebras. The role ā¦
We present an introduction to modern continuous model theory with an emphasis on its interactions with topics covered in this volume such as $C^*$-algebras and von Neumann algebras. The role of ultraproducts is highlighted and expositions of definable sets, imaginaries, quantifier elimination and separable categoricity are included.
We show that neither the class of C*-algebras with Kirchberg's QWEP property nor the class of W*-probability spaces with the QWEP property are effectively axiomatizable (in the appropriate languages). The ā¦
We show that neither the class of C*-algebras with Kirchberg's QWEP property nor the class of W*-probability spaces with the QWEP property are effectively axiomatizable (in the appropriate languages). The latter result follows from a more general result, namely that the hyperfinite III$_1$ factor does not have a computable universal theory in the language of W*-probability spaces. We also prove that the Powers' factors $\mathcal{R}_\lambda$, for $0<\lambda<1$, when equipped with their canonical Powers' states, do not have computable universal theory.
An open question in quantum complexity theory is whether or not the class $\operatorname{MIP}^{co}$, consisting of languages that can be efficiently verified using interacting provers sharing quantum resources according to ā¦
An open question in quantum complexity theory is whether or not the class $\operatorname{MIP}^{co}$, consisting of languages that can be efficiently verified using interacting provers sharing quantum resources according to the quantum commuting model, coincides with the class $coRE$ of languages with recursively enumerable complement. We introduce the notion of a qc-modulus, which encodes approximations to quantum commuting correlations, and show that the existence of a computable qc-modulus gives a negative answer to a natural variant of the aforementioned question.
We introduce the notion of a Tsirelson pair of C*-algebras, which is a pair of C*-algebras for which the space of quantum strategies obtained by using states on the minimal ā¦
We introduce the notion of a Tsirelson pair of C*-algebras, which is a pair of C*-algebras for which the space of quantum strategies obtained by using states on the minimal tensor product of the pair and the space of quantum strategies obtained by using states on the maximal tensor product of the pair coincide. We exhibit a number of examples of such pairs that are "nontrivial" in the sense that the minimal tensor product and the maximal tensor product of the pair are not isomorphic. For example, we prove that any pair containing a C*-algebra with Kirchberg's QWEP property is a Tsirelson pair. We then introduce the notion of a C*-algebra with the Tsirelson property (TP) and establish a number of closure properties for this class. We also show that the class of C*-algebras with the TP form an axiomatizable class (in the sense of model theory), but that this class admits no "effective" axiomatization.
We survey the developments in the model theory of tracial von Neumann algebras that have taken place in the last fifteen years. We discuss the appropriate first-order language for axiomatizing ā¦
We survey the developments in the model theory of tracial von Neumann algebras that have taken place in the last fifteen years. We discuss the appropriate first-order language for axiomatizing this class as well as the subclass of II$_1$ factors. We discuss how model-theoretic ideas were used to settle a variety of questions around isomorphism of ultrapowers of tracial von Neumann algebras with respect to different ultrafilters before moving on to more model-theoretic concerns, such as theories of II$_1$ factors and existentially closed II$_1$ factors. We conclude with two recent applications of model-theoretic ideas to questions around relative commutants.
We establish an approximate zero-one law for sentences of continuous logic over finite metric spaces of diameter at most $1$. More precisely, we axiomatize a complete metric theory $T_{\mathrm{as}}$ such ā¦
We establish an approximate zero-one law for sentences of continuous logic over finite metric spaces of diameter at most $1$. More precisely, we axiomatize a complete metric theory $T_{\mathrm{as}}$ such that, given any sentence $\sigma$ in the language of pure metric spaces and any $\epsilon>0$, the probability that the difference of the value of $\sigma$ in a random metric space of size $n$ and the value of $\sigma$ in any model of $T_{\mathrm{as}}$ is less than $\epsilon$ approaches $1$ as $n$ approaches infinity. We also establish some model-theoretic properties of the theory $T_{\mathrm{as}}$.
The Kirchberg Embedding Problem (KEP) asks if every C*-algebra embeds into an ultrapower of the Cuntz algebra $\cal O_2$. Motivated by the recent refutation of the Connes Embedding Problem using ā¦
The Kirchberg Embedding Problem (KEP) asks if every C*-algebra embeds into an ultrapower of the Cuntz algebra $\cal O_2$. Motivated by the recent refutation of the Connes Embedding Problem using the quantum complexity result MIP*=RE, we establish two quantum complexity consequences of a positive solution to KEP. Both results involve almost-commuting strategies to nonlocal games.
We show that any II 1 factor that has the same 4-quantifier theory as the hyperfinite II 1 factor ā satisfies the conclusion of the Popa Factorial Commutant Embedding Problem ā¦
We show that any II 1 factor that has the same 4-quantifier theory as the hyperfinite II 1 factor ā satisfies the conclusion of the Popa Factorial Commutant Embedding Problem (FCEP) and has the Brown property. These results improve recent results proving the same conclusions under the stronger assumption that the factor is actually elementarily equivalent to ā. In the same spirit, we improve a recent result of the first-named author, who showed that if (1) the amalgamated free product of embeddable factors over a property (T) base is once again embeddable, and (2) ā is an infinitely generic embeddable factor, then the FCEP is true of all property (T) factors. In this paper, it is shown that item (2) can be weakened to assume that ā has the same 3-quantifier theory as an infinitely generic embeddable factor.
The Kirchberg Embedding Problem (KEP) asks if every C*-algebra embeds into an ultrapower of the Cuntz algebra $\cal O_2$. Motivated by the recent refutation of the Connes Embedding Problem using ā¦
The Kirchberg Embedding Problem (KEP) asks if every C*-algebra embeds into an ultrapower of the Cuntz algebra $\cal O_2$. Motivated by the recent refutation of the Connes Embedding Problem using the quantum complexity result MIP*=RE, we establish two quantum complexity consequences of a positive solution to KEP. Both results involve almost-commuting strategies to nonlocal games.
Abstract We show that the following operator algebras have hyperarithmetic theory: the hyperfinite II$_1$ factor $\mathcal R$, $L(\varGamma )$ for $\varGamma $ a finitely generated group with solvable word problem, ā¦
Abstract We show that the following operator algebras have hyperarithmetic theory: the hyperfinite II$_1$ factor $\mathcal R$, $L(\varGamma )$ for $\varGamma $ a finitely generated group with solvable word problem, $C^*(\varGamma )$ for $\varGamma $ a finitely presented group, $C^*_\lambda (\varGamma )$ for $\varGamma $ a finitely generated group with solvable word problem, $C(2^\omega )$ and $C(\mathbb P)$ (where $\mathbb P$ is the pseudoarc). We also show that the Cuntz algebra $\mathcal O_2$ has a hyperarithmetic theory provided that the Kirchberg embedding problems have affirmative answers. Finally, we prove that if there is an existentially closed (e.c.) II$_1$ factor (resp. $\textrm{C}^*$-algebra) that does not have hyperarithmetic theory, then there are continuum many theories of e.c. II$_1$ factors (resp. e.c. $\textrm{C}^*$-algebras).
We show that any II$_1$ factor that has the same 4-quantifier theory as the hyperfinite II$_1$ factor $\mathcal{R}$ satisfies the conclusion of the Popa Factorial Commutant Embedding Problem (FCEP) and ā¦
We show that any II$_1$ factor that has the same 4-quantifier theory as the hyperfinite II$_1$ factor $\mathcal{R}$ satisfies the conclusion of the Popa Factorial Commutant Embedding Problem (FCEP) and has the Brown property. These results improve recent results proving the same conclusions under the stronger assumption that the factor is actually elementarily equivalent to $\mathcal{R}$. In the same spirit, we improve a recent result of the first-named author, who showed that if (1) the amalgamated free product of embeddable factors over a property (T) base is once again embeddable, and (2) $\mathcal{R}$ is an infinitely generic embeddable factor, then the FCEP is true of all property (T) factors. In this paper, it is shown that item (2) can be weakened to assume that $\mathcal{R}$ has the same 3-quantifier theory as an infinitely generic embeddable factor.
We show that the universal theory of the hyperfinite II$_1$ factor is not computable. The proof uses the recent result that MIP*=RE. Combined with an earlier observation of the authors, ā¦
We show that the universal theory of the hyperfinite II$_1$ factor is not computable. The proof uses the recent result that MIP*=RE. Combined with an earlier observation of the authors, this yields a proof that the Connes Embedding Problem has a negative solution that avoids the equivalences with Kirchberg's QWEP Conjecture and Tsirelson's Problem.+
We show that the following operator algebras have hyperarithmetic theory: the hyperfinite II$_1$ factor $\mathcal R$, $L(\Gamma)$ for $\Gamma$ a finitely generated group with solvable word problem, $C^*(\Gamma)$ for $\Gamma$ ā¦
We show that the following operator algebras have hyperarithmetic theory: the hyperfinite II$_1$ factor $\mathcal R$, $L(\Gamma)$ for $\Gamma$ a finitely generated group with solvable word problem, $C^*(\Gamma)$ for $\Gamma$ a finitely presented group, $C^*_\lambda(\Gamma)$ for $\Gamma$ a finitely generated group with solvable word problem, $C(2^\omega)$, and $C(\mathbb P)$ (where $\mathbb P$ is the pseudoarc). We also show that the Cuntz algebra $\mathcal O_2$ has a hyperarithmetic theory provided that the Kirchberg embedding problem has an affirmative answer. Finally, we prove that if there is an existentially closed (e.c.) II$_1$ factor (resp. C$^*$-algebra) that does not have hyperarithmetic theory, then there are continuum many theories of e.c. II$_1$ factors (resp. e.c. C$^*$-algebras).
We show that the following operator algebras have hyperarithmetic theory: the hyperfinite II$_1$ factor $\mathcal R$, $L(\Gamma)$ for $\Gamma$ a finitely generated group with solvable word problem, $C^*(\Gamma)$ for $\Gamma$ ā¦
We show that the following operator algebras have hyperarithmetic theory: the hyperfinite II$_1$ factor $\mathcal R$, $L(\Gamma)$ for $\Gamma$ a finitely generated group with solvable word problem, $C^*(\Gamma)$ for $\Gamma$ a finitely presented group, $C^*_\lambda(\Gamma)$ for $\Gamma$ a finitely generated group with solvable word problem, $C(2^\omega)$, and $C(\mathbb P)$ (where $\mathbb P$ is the pseudoarc). We also show that the Cuntz algebra $\mathcal O_2$ has a hyperarithmetic theory provided that the Kirchberg embedding problem has an affirmative answer. Finally, we prove that if there is an existentially closed (e.c.) II$_1$ factor (resp. C$^*$-algebra) that does not have hyperarithmetic theory, then there are continuum many theories of e.c. II$_1$ factors (resp. e.c. C$^*$-algebras).
We show that any II$_1$ factor that has the same 4-quantifier theory as the hyperfinite II$_1$ factor $\mathcal{R}$ satisfies the conclusion of the Popa Factorial Commutant Embedding Problem (FCEP) and ā¦
We show that any II$_1$ factor that has the same 4-quantifier theory as the hyperfinite II$_1$ factor $\mathcal{R}$ satisfies the conclusion of the Popa Factorial Commutant Embedding Problem (FCEP) and has the Brown property. These results improve recent results proving the same conclusions under the stronger assumption that the factor is actually elementarily equivalent to $\mathcal{R}$. In the same spirit, we improve a recent result of the first-named author, who showed that if (1) the amalgamated free product of embeddable factors over a property (T) base is once again embeddable, and (2) $\mathcal{R}$ is an infinitely generic embeddable factor, then the FCEP is true of all property (T) factors. In this paper, it is shown that item (2) can be weakened to assume that $\mathcal{R}$ has the same 3-quantifier theory as an infinitely generic embeddable factor.
We show that the universal theory of the hyperfinite II$_1$ factor is not computable. The proof uses the recent result that MIP*=RE. Combined with an earlier observation of the authors, ā¦
We show that the universal theory of the hyperfinite II$_1$ factor is not computable. The proof uses the recent result that MIP*=RE. Combined with an earlier observation of the authors, this yields a proof that the Connes Embedding Problem has a negative solution that avoids the equivalences with Kirchberg's QWEP Conjecture and Tsirelson's Problem.+
We introduce a metric space $\mathbb{AS}$ and show that its (continuous) theory is the almost-sure theory of finite metric spaces of diameter at most $1$.
We introduce a metric space $\mathbb{AS}$ and show that its (continuous) theory is the almost-sure theory of finite metric spaces of diameter at most $1$.
We introduce the notion of strong $p$-semi-regularity and show that if $p$ is a regular type which is not locally modular then any $p$-semi-regular type is strongly $p$-semi-regular. Moreover, for ā¦
We introduce the notion of strong $p$-semi-regularity and show that if $p$ is a regular type which is not locally modular then any $p$-semi-regular type is strongly $p$-semi-regular. Moreover, for any such $p$-semi-regular type, "domination implies isolation" which allows us to prove the following: Suppose that $T$ is countable, classifiable and $M$ is any model. If $p\in S(M)$ is regular but not locally modular and $b$ is any realization of $p$ then every model $N$ containing $M$ that is dominated by $b$ over $M$ is both constructible and minimal over $Mb$.
We study correspondences of tracial von Neumann algebras from the model-theoretic point of view. We introduce and study an ultraproduct of correspondences and use this ultraproduct to prove, for a ā¦
We study correspondences of tracial von Neumann algebras from the model-theoretic point of view. We introduce and study an ultraproduct of correspondences and use this ultraproduct to prove, for a fixed pair of tracial von Neumann algebras M and N, that the class of M-N correspondences forms an elementary class. We prove that the corresponding theory is classifiable, all of its completions are stable, that these completions have quantifier elimination in an appropriate language, and that one of these completions is in fact the model companion. We also show that the class of triples (M, H, N), where M and N are tracial von Neumann algebras and H is an M-N correspondence, form an elementary class. As an application of our framework, we show that a II_1 factor M has property (T) precisely when the set of central vectors form a definable set relative to the theory of M-M correspondences. We then use our approach to give a simpler proof that the class of structures (M, Phi), where M is a sigma-finite von Neumann algebra and Phi is a faithful normal state, forms an elementary class. Finally, we initiate the study of a family of Connes-type ultraproducts on C*-algebras.
We give an algebraic description of the structure of the analytic universal cover of a complex abelian variety which suffices to determine the structure up to isomorphism. More generally, we ā¦
We give an algebraic description of the structure of the analytic universal cover of a complex abelian variety which suffices to determine the structure up to isomorphism. More generally, we classify the models of theories of āuniversal coversā of rigid divisible commutative finite Morley rank groups.
We study correspondences of tracial von Neumann algebras from the model-theoretic point of view. We introduce and study an ultraproduct of correspondences and use this ultraproduct to prove, for a ā¦
We study correspondences of tracial von Neumann algebras from the model-theoretic point of view. We introduce and study an ultraproduct of correspondences and use this ultraproduct to prove, for a fixed pair of tracial von Neumann algebras M and N, that the class of M-N correspondences forms an elementary class. We prove that the corresponding theory is classifiable, all of its completions are stable, that these completions have quantifier elimination in an appropriate language, and that one of these completions is in fact the model companion. We also show that the class of triples (M, H, N), where M and N are tracial von Neumann algebras and H is an M-N correspondence, form an elementary class. As an application of our framework, we show that a II_1 factor M has property (T) precisely when the set of central vectors form a definable set relative to the theory of M-M correspondences. We then use our approach to give a simpler proof that the class of structures (M, Phi), where M is a sigma-finite von Neumann algebra and Phi is a faithful normal state, forms an elementary class. Finally, we initiate the study of a family of Connes-type ultraproducts on C*-algebras.
Recently, Boutonnet, Chifan, and Ioana proved that McDuff's examples of continuum many pairwise non-isomorphic separable II$_1$ factors are in fact pairwise non-elementarily equivalent. Their proof proceeded by showing that any ā¦
Recently, Boutonnet, Chifan, and Ioana proved that McDuff's examples of continuum many pairwise non-isomorphic separable II$_1$ factors are in fact pairwise non-elementarily equivalent. Their proof proceeded by showing that any ultrapowers of any two distinct McDuff examples are not isomorphic. In a paper by the first two authors of this paper, Ehrenfeucht-Fra\isse games were used to find an upper bound on the quantifier complexity of sentences distinguishing the McDuff examples, leaving it as an open question to find concrete sentences distinguishing the McDuff factors. In this paper, we answer this question by providing such concrete sentences.
We show that the Urysohn sphere is pseudofinite. As a consequence, we derive an approximate $0$-$1$ law for finite metric spaces of diameter at most $1$.
We show that the Urysohn sphere is pseudofinite. As a consequence, we derive an approximate $0$-$1$ law for finite metric spaces of diameter at most $1$.
Recently, Boutonnet, Chifan, and Ioana proved that McDuff's examples of continuum many pairwise non-isomorphic separable II$_1$ factors are in fact pairwise non-elementarily equivalent. Their proof proceeded by showing that any ā¦
Recently, Boutonnet, Chifan, and Ioana proved that McDuff's examples of continuum many pairwise non-isomorphic separable II$_1$ factors are in fact pairwise non-elementarily equivalent. Their proof proceeded by showing that any ultrapowers of any two distinct McDuff examples are not isomorphic. In a paper by the first two authors of this paper, Ehrenfeucht-Fra\"isse games were used to find an upper bound on the quantifier complexity of sentences distinguishing the McDuff examples, leaving it as an open question to find concrete sentences distinguishing the McDuff factors. In this paper, we answer this question by providing such concrete sentences.
Recently, Boutonnet, Chifan, and Ioana proved that McDuffās family of continuum many pairwise non-isomorphic separable II|$_1$| factors are in fact pairwise non-elementarily equivalent by proving that any ultrapowers of two ā¦
Recently, Boutonnet, Chifan, and Ioana proved that McDuffās family of continuum many pairwise non-isomorphic separable II|$_1$| factors are in fact pairwise non-elementarily equivalent by proving that any ultrapowers of two distinct members of the family are non-isomorphic. We use EhrenfeuchtāFraisse games to provide an upper bound on the quantifier-depth of sentences which distinguish these theories.
Abstract The Connes Embedding Problem (CEP) asks whether every separable II 1 factor embeds into an ultrapower of the hyperfinite II 1 factor. We show that the CEP is equivalent ā¦
Abstract The Connes Embedding Problem (CEP) asks whether every separable II 1 factor embeds into an ultrapower of the hyperfinite II 1 factor. We show that the CEP is equivalent to the statement that every type II 1 tracial von Neumann algebra has a computable universal theory.
We begin the model theoretic study of nuclear $\mathrm{C}^*$-algebras using the tools of continuous logic.
We begin the model theoretic study of nuclear $\mathrm{C}^*$-algebras using the tools of continuous logic.
Recently, Boutonnet, Chifan, and Ioana proved that McDuff's family of continuum many pairwise nonisomorphic separable II$_1$ factors are in fact pairwise non-elementarily equivalent by proving that any ultrapowers of two ā¦
Recently, Boutonnet, Chifan, and Ioana proved that McDuff's family of continuum many pairwise nonisomorphic separable II$_1$ factors are in fact pairwise non-elementarily equivalent by proving that any ultrapowers of two distinct members of the family are nonsiomorphic. We use Ehrenfeucht-Fraisse games to provide an upper bound on the quantifier-depth of sentences which distinguish these theories.
We begin the systematic model theoretic study of $\mathrm{C}^*$-algebras using the tools of continuous logic.
We begin the systematic model theoretic study of $\mathrm{C}^*$-algebras using the tools of continuous logic.
We begin the study of categorical logic for continuous model theory. In particular, we 1. introduce the notions of metric logical categories and functors as categorical equivalents of a metric ā¦
We begin the study of categorical logic for continuous model theory. In particular, we 1. introduce the notions of metric logical categories and functors as categorical equivalents of a metric theory and interpretations, 2. prove a continuous version of conceptual completeness showing that $T^\eq$ is the maximal conservative expansion of $T$, and 3. define the concept of a metric pre-topos.
Recently, Boutonnet, Chifan, and Ioana proved that McDuff's family of continuum many pairwise nonisomorphic separable II$_1$ factors are in fact pairwise non-elementarily equivalent by proving that any ultrapowers of two ā¦
Recently, Boutonnet, Chifan, and Ioana proved that McDuff's family of continuum many pairwise nonisomorphic separable II$_1$ factors are in fact pairwise non-elementarily equivalent by proving that any ultrapowers of two distinct members of the family are nonsiomorphic. We use Ehrenfeucht-Fraisse games to provide an upper bound on the quantifier-depth of sentences which distinguish these theories.
The relative commutant $A'\cap A^{\mathcal{U}}$ of a strongly self-absorbing algebra $A$ is indistinguishable from its ultrapower $A^{\mathcal{U}}$. This applies both to the case when $A$ is the hyperfinite II$_1$ factor ā¦
The relative commutant $A'\cap A^{\mathcal{U}}$ of a strongly self-absorbing algebra $A$ is indistinguishable from its ultrapower $A^{\mathcal{U}}$. This applies both to the case when $A$ is the hyperfinite II$_1$ factor and to the case when it is a strongly self-absorbing C*-algebra. In the latter case we prove analogous results for $\ell_\infty(A)/c_0(A)$ and reduced powers corresponding to other filters on $\bf N$. Examples of algebras with approximately inner flip and approximately inner half-flip are provided, showing the optimality of our results. We also prove that strongly self-absorbing algebras are smoothly classifiable, unlike the algebras with approximately inner half-flip.
We examine the properties of existentially closed ($\mathcal {R}^\omega $-embeddable) ${\rm II}_1$ factors. In particular, we use the fact that every automorphism of an existentially closed ($\mathcal {R}^\omega $-embeddable) ${\rm ā¦
We examine the properties of existentially closed ($\mathcal {R}^\omega $-embeddable) ${\rm II}_1$ factors. In particular, we use the fact that every automorphism of an existentially closed ($\mathcal {R}^\omega $-embeddable) ${\rm II}_1$ factor is approx
The relative commutant $A'\cap A^{\mathcal{U}}$ of a strongly self-absorbing algebra $A$ is indistinguishable from its ultrapower $A^{\mathcal{U}}$. This applies both to the case when $A$ is the hyperfinite II$_1$ factor ā¦
The relative commutant $A'\cap A^{\mathcal{U}}$ of a strongly self-absorbing algebra $A$ is indistinguishable from its ultrapower $A^{\mathcal{U}}$. This applies both to the case when $A$ is the hyperfinite II$_1$ factor and to the case when it is a strongly self-absorbing C*-algebra. In the latter case we prove analogous results for $\ell_\infty(A)/c_0(A)$ and reduced powers corresponding to other filters on $\bf N$. Examples of algebras with approximately inner flip and approximately inner half-flip are provided, showing the optimality of our results. We also prove that strongly self-absorbing algebras are smoothly classifiable, unlike the algebras with approximately inner half-flip.
We use continuous model theory to obtain several results concerning isomorphisms and embeddings between II_1 factors and their ultrapowers. Among other things, we show that for any II_1 factor M, ā¦
We use continuous model theory to obtain several results concerning isomorphisms and embeddings between II_1 factors and their ultrapowers. Among other things, we show that for any II_1 factor M, there are continuum many nonisomorphic separable II_1 factors that have an ultrapower isomorphic to an ultrapower of M. We also give a poor man's resolution of the Connes Embedding Problem: there exists a separable II_1 factor such that all II_1 factors embed into one of its ultrapowers.
We give an algebraic description of the structure of the analytic cover of a complex abelian variety which suffices to determine the structure up to isomorphism. More generally, we classify ā¦
We give an algebraic description of the structure of the analytic cover of a complex abelian variety which suffices to determine the structure up to isomorphism. More generally, we classify the models of theories of universal covers of rigid divisible commutative finite Morley rank groups.
We give an algebraic description of the structure of the analytic universal cover of a complex abelian variety which suffices to determine the structure up to isomorphism. More generally, we ā¦
We give an algebraic description of the structure of the analytic universal cover of a complex abelian variety which suffices to determine the structure up to isomorphism. More generally, we classify the models of theories of "universal covers" of rigid divisible commutative finite Morley rank groups.
A metric structure is a many-sorted structure in which each sort is a complete metric space of finite diameter. Additionally, the structure consists of some distinguished elements as well as ā¦
A metric structure is a many-sorted structure in which each sort is a complete metric space of finite diameter. Additionally, the structure consists of some distinguished elements as well as some functions (of several variables) (a) between sorts and (b) from sorts to bounded subsets of ā, and these functions are all required to be uniformly continuous. Examples arise throughout mathematics, especially in analysis and geometry. They include metric spaces themselves, measure algebras, asymptotic cones of finitely generated groups, and structures based on Banach spaces (where one takes the sorts to be balls), including Banach lattices, C*-algebras, etc.
We use continuous model theory to obtain several results concerning isomorphisms and embeddings between II_1 factors and their ultrapowers. Among other things, we show that for any II_1 factor M, ā¦
We use continuous model theory to obtain several results concerning isomorphisms and embeddings between II_1 factors and their ultrapowers. Among other things, we show that for any II_1 factor M, there are continuum many nonisomorphic separable II_1 factors that have an ultrapower isomorphic to an ultrapower of M. We also give a poor man's resolution of the Connes Embedding Problem: there exists a separable II_1 factor such that all II_1 factors embed into one of its ultrapowers.
Several authors have considered whether the ultrapower and the relative commutant of a C*-algebra or II1 factor depend on the choice of the ultrafilter. We settle each of these questions, ā¦
Several authors have considered whether the ultrapower and the relative commutant of a C*-algebra or II1 factor depend on the choice of the ultrafilter. We settle each of these questions, extending results of GeāHadwin and the first author.
On considere la possibilite que la conjecture de Morley peut eventuellement etre demontree en donnant plus ou moins explicitement toutes les fonctions spectre possibles K(ā„Īŗ 1 )āI(T,K) avec chaque possibilite ā¦
On considere la possibilite que la conjecture de Morley peut eventuellement etre demontree en donnant plus ou moins explicitement toutes les fonctions spectre possibles K(ā„Īŗ 1 )āI(T,K) avec chaque possibilite se conformant a la condition de Morley
The primary purpose of this article is to show that a certain natural set of axioms yields a completeness result for continuous first-order logic. In particular, we show that in ā¦
The primary purpose of this article is to show that a certain natural set of axioms yields a completeness result for continuous first-order logic. In particular, we show that in continuous first-order logic a set of formulae is (completely) satisfiable if (and only if) it is consistent. From this result it follows that continuous first-order logic also satisfies an \emph{approximated} form of strong completeness, whereby $\Sigma\vDash\varphi$ (if and) only if $\Sigma\vdash\varphi\dotminus 2^{-n}$ for all $n<\omega$. This approximated form of strong completeness asserts that if $\Sigma\vDash\varphi$, then proofs from $\Sigma$, being finite, can provide arbitrary better approximations of the truth of $\varphi$.
We prove that there exist uncountably many separable II$_1$ factors whose ultrapowers (with respect to arbitrary ultrafilters) are non-isomorphic. In fact, we prove that the families of non-isomorphic II$_1$ factors ā¦
We prove that there exist uncountably many separable II$_1$ factors whose ultrapowers (with respect to arbitrary ultrafilters) are non-isomorphic. In fact, we prove that the families of non-isomorphic II$_1$ factors originally introduced by McDuff \cite{MD69a,MD69b} are such examples. This entails the existence of a continuum of non-elementarily equivalent II$_1$ factors, thus settling a well-known open problem in the continuous model theory of operator algebras.
We establish the Borel computability of various C * -algebra invariants, including the Elliott invariant and the Cuntz semigroup.As applications we deduce that AF algebras are classifiable by countable structures, ā¦
We establish the Borel computability of various C * -algebra invariants, including the Elliott invariant and the Cuntz semigroup.As applications we deduce that AF algebras are classifiable by countable structures, and that a conjecture of Winter and the second author for nuclear separable simple C * -algebras cannot be disproved by appealing to known standard Borel structures on these algebras. IntroductionThe classification theory of nuclear separable C * -algebras via K-theoretic and tracial invariants was initiated by G. A. Elliott c. 1990.An ideal result in this theory is of the following type:Let C 1 be a category of C * -algebras, C 2 a category of invariants, and F : C 1 ā C 2 a functor.We say that (F , C 2 ) classifies C 1 if for any isomorphism Ļ : F (A) ā F (B) there is an isomorphism Ī¦ : A ā B such that F (Ī¦) = Ļ, and if, moreover, the range of F can be identified.Given A, B ā C 1 , one wants to decide whether A and B are isomorphic.With a theorem as above in hand (and there are plenty such-see ?or ?for an overview), this reduces to deciding whether F (A) and F (B) are isomorphic; in particular, one must compute F (A) and F (B).What does it mean for an invariant to be computable?The broadest definition is available when the objects of C 1 and C 2 admit natural parameterizations as standard Borel spaces, for the computability of F (ā¢) then reduces to the question "Is F a Borel map?"The
Say that a separable, unital $C^*$-algebra $\mathcal {D} \ncong \mathbb {C}$ is strongly self-absorbing if there exists an isomorphism $\varphi : \mathcal {D} \to \mathcal {D} \otimes \mathcal {D}$ such ā¦
Say that a separable, unital $C^*$-algebra $\mathcal {D} \ncong \mathbb {C}$ is strongly self-absorbing if there exists an isomorphism $\varphi : \mathcal {D} \to \mathcal {D} \otimes \mathcal {D}$ such that $\varphi$ and $\mathrm {id}_{\mathcal {D}} \otimes \mathbf {1}_{\mathcal {D}}$ are approximately unitarily equivalent $*$-homomorphisms. We study this class of algebras, which includes the Cuntz algebras $\mathcal {O}_2$, $\mathcal {O}_{\infty }$, the UHF algebras of infinite type, the JiangāSu algebra $\mathcal {Z}$ and tensor products of $\mathcal {O}_{\infty }$ with UHF algebras of infinite type. Given a strongly self-absorbing $C^{*}$-algebra $\mathcal {D}$ we characterise when a separable $C^*$-algebra absorbs $\mathcal {D}$ tensorially (i.e., is $\mathcal {D}$-stable), and prove closure properties for the class of separable $\mathcal {D}$-stable $C^*$-algebras. Finally, we compute the possible $K$-groups and prove a number of classification results which suggest that the examples listed above are the only strongly self-absorbing $C^*$-algebras.
We present unified proofs of several properties of the corona of $Ļ$-unital C*-algebras such as AA-CRISP, SAW*, being sub-$Ļ$-Stonean in the sense of Kirchberg, and the conclusion of Kasparov's Technical ā¦
We present unified proofs of several properties of the corona of $Ļ$-unital C*-algebras such as AA-CRISP, SAW*, being sub-$Ļ$-Stonean in the sense of Kirchberg, and the conclusion of Kasparov's Technical Theorem. Although our results were obtained by considering C*-algebras as models of the logic for metric structures, the reader is not required to have any knowledge of model theory of metric structures (or model theory, or logic in general). The proofs involve analysis of the extent of model-theoretic saturation of corona algebras.
We examine the properties of existentially closed ($\mathcal {R}^\omega $-embeddable) ${\rm II}_1$ factors. In particular, we use the fact that every automorphism of an existentially closed ($\mathcal {R}^\omega $-embeddable) ${\rm ā¦
We examine the properties of existentially closed ($\mathcal {R}^\omega $-embeddable) ${\rm II}_1$ factors. In particular, we use the fact that every automorphism of an existentially closed ($\mathcal {R}^\omega $-embeddable) ${\rm II}_1$ factor is approx
We study countable saturation of the metric reduced products and introduce continuous fields of metric models indexed by locally compact, separable, completely metrizable spaces. Saturation of the reduced product depends ā¦
We study countable saturation of the metric reduced products and introduce continuous fields of metric models indexed by locally compact, separable, completely metrizable spaces. Saturation of the reduced product depends both on the underlying index space and the model. By using the Gelfand--Naimark duality we conclude that the assertion that the \vCech--Stone remainder of the half-line has only trivial automorphisms is independent from ZFC. The consistency of this statement follows from Proper Forcing Axiom and this is the first known example of a connected space with this property.
The most recent wave of applications of logic to operator algebras is a young and rapidly developing field. This is a snapshot of the current state of the art.
The most recent wave of applications of logic to operator algebras is a young and rapidly developing field. This is a snapshot of the current state of the art.
Introduction. A theory, 1, (formalized in the first order predicate calculus) is categorical in power K if it has exactly one type of models of power K. This notion was ā¦
Introduction. A theory, 1, (formalized in the first order predicate calculus) is categorical in power K if it has exactly one type of models of power K. This notion was introduced by Los [ 9] and Vaught [ 16] in 1954. At that time they pointed out that a theory (e.g., the theory of dense linearly ordered sets without end points) may be categorical in power N0 and fail to be categorical in any higher power. Conversely, a theory may be categorical in every uncountable power and fail to be categorical in power N0 (e.g., the theory of algebraically closed fields of characteristic 0). Los' then raised the following question. Is a theory categorical in one uncountable power necessarily categorical in every uncountable power? The principal result of this paper is an affirmative answer to that question. We actually prove a stronger result, namely: If a theory is categorical in some uncountable power then every uncountable model of that theory is saturated. (Terminology used in the Introduction will be defined in the body of the paper; roughly speaking, a model is saturated, or universalhomogeneous, if it contains an element of every possible elementary type relative to its subsystems of strictly smaller power.) It is known(2) that a theory can have (up to isomorphism) at most one saturated model in each power. It is interesting to note that our results depend essentially on an analogue of the usual analysis of topological spaces in terms of their derived spaces and the Cantor-Bendixson theorem. The paper is divided into five sections. In ?1 terminology and some meta-mathematical results are summarized. In particular, for each theory, 1, there is described a theory, *, which has essentially the same models as z but is neater to work with. In ?2 is defined a topological space, S(A), corresponding to each subsystem, A, of a model of a theory, 1; the points of S(A) being the isomorphism types of elements with respect to A. With each monomorphism (= isomorphic imbedding), f: A -+ B, is associated a dual continuous map, f*: S(B) -+ S(A). Then there is defined for each S(A) a decreasing sequence ISa(A) I of subspaces which is analogous to (but different from)
We prove that it is relatively consistent with the usual axioms of mathematics that all automorphisms of the Calkin algebra are inner.Together with a 2006 Phillips-Weaver construction of an outer ā¦
We prove that it is relatively consistent with the usual axioms of mathematics that all automorphisms of the Calkin algebra are inner.Together with a 2006 Phillips-Weaver construction of an outer automorphism using the Continuum Hypothesis, this gives a complete solution to a 1977 problem of Brown-Douglas-Fillmore.We also give a simpler and self-contained proof of the Phillips-Weaver result.
We develop continuous first order logic, a variant of the logic described in \cite{Chang-Keisler:ContinuousModelTheory}. We show that this logic has the same power of expression as the framework of open ā¦
We develop continuous first order logic, a variant of the logic described in \cite{Chang-Keisler:ContinuousModelTheory}. We show that this logic has the same power of expression as the framework of open Hausdorff cats, and as such extends Henson's logic for Banach space structures. We conclude with the development of local stability, for which this logic is particularly well-suited.
In functional analysis, approximative properties of an object become precise in its ultrapower. We discuss this idea and its consequences for automorphisms of II1 factors. Here are some sample results: ā¦
In functional analysis, approximative properties of an object become precise in its ultrapower. We discuss this idea and its consequences for automorphisms of II1 factors. Here are some sample results: (1) an automorphism is approximately inner if and only if its ultrapower is ×0-locally inner; (2) the ultrapower of an outer automorphism is always outer; (3) for unital *-homomorphisms from a separable nuclear C*-algebra into an ultrapower of a II1 factor, equality of the induced traces implies unitary equivalence. All statements are proved using operator algebraic techniques, but in the last section of the paper we indicate how the underlying principle is related to theorems of Hensonās positive bounded logic.
We give an example of a simple separable C*-algebra that is not isomorphic to its opposite algebra. Our example is nonnuclear and stably finite, has real rank zero and stable ā¦
We give an example of a simple separable C*-algebra that is not isomorphic to its opposite algebra. Our example is nonnuclear and stably finite, has real rank zero and stable rank one, and has a unique tracial state. It has trivial <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K 1"> <mml:semantics> <mml:msub> <mml:mi>K</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">K_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and its <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K 0"> <mml:semantics> <mml:msub> <mml:mi>K</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">K_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-group is order isomorphic to a countable subgroup of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">R</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathbf R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.
We construct a unital separable C *-algebra Z as an analog of the hyperfinite type II 1 factor. Besides being nuclear, simple, projectionless, and infinite-dimensional, Z has a unique tracial ā¦
We construct a unital separable C *-algebra Z as an analog of the hyperfinite type II 1 factor. Besides being nuclear, simple, projectionless, and infinite-dimensional, Z has a unique tracial state, and is KK-equivalent to [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /], the algebra of complex numbers. It is shown that unital endomorphisms on Z are approximately inner, and that Z is isomorphic to the infinite tensor product of its replicas. It is also shown that A ā Z ā A for certain interesting classes of unital simple nuclear C *-algebras A of real rank zero.
By a variety, we mean a class of structures in some language containing only function symbols which is equationally defined or equivalently is closed under homomorphisms, submodels and products. If ā¦
By a variety, we mean a class of structures in some language containing only function symbols which is equationally defined or equivalently is closed under homomorphisms, submodels and products. If K is a class of -structures then I ( K , Ī») denotes the number of nonisomorphic models in K of cardinality Ī». When we say that K has few models, we mean that I ( K ,Ī») < 2 Ī» for some Ī» > ā£ ā£. If I ( K ,Ī») = 2 Ī» for all Ī» > ā£ ā£, then we say K has many models. In [9] and [10], Shelah has shown that for an elementary class K , having few models is a strong structural condition.
We prove a strong dichotomy for the number of ultrapowers of a given model of cardinality ā¤ 2 āµ 0 associated with nonprincipal ultrafilters on ā. They are either all ā¦
We prove a strong dichotomy for the number of ultrapowers of a given model of cardinality ā¤ 2 āµ 0 associated with nonprincipal ultrafilters on ā. They are either all isomorphic, or else there are 2 2 āµ 0 many nonisomorphic ultrapowers. We prove the analogous result for metric structures, including C*-algebras and II 1 factors, as well as their relative commutants and include several applications. We also show that the CAF001-algebra [Formula: see text] always has nonisomorphic relative commutants in its ultrapowers associated with nonprincipal ultrafilters on ā.
It is shown that a unital C*-algebra A has the Dixmier property if and only if it is weakly central and satisfies certain tracial conditions. This generalises the Haagerup-Zsido theorem ā¦
It is shown that a unital C*-algebra A has the Dixmier property if and only if it is weakly central and satisfies certain tracial conditions. This generalises the Haagerup-Zsido theorem for simple C*-algebras. We also study a uniform version of the Dixmier property, as satisfied for example by von Neumann algebras and the reduced C*-algebras of Powers groups, but not by all C*-algebras with the Dixmier property, and we obtain necessary and sufficient conditions for a simple unital C*-algebra with unique tracial state to have this uniform property. We give further examples of C*-algebras with the uniform Dixmier property, namely all C*-algebras with the Dixmier property and finite radius of comparison-by-traces. Finally, we determine the distance between two Dixmier sets, in an arbitrary unital C*-algebra, by a formula involving tracial data and algebraic numerical ranges.
We report on recent progress in the program to classify separable amenable C<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Superscript asterisk"><mml:semantics><mml:msup><mml:mi /><mml:mo>ā<!-- ā --></mml:mo></mml:msup><mml:annotation encoding="application/x-tex">^*</mml:annotation></mml:semantics></mml:math></inline-formula>-algebras. Our emphasis is on the newly apparent role of ā¦
We report on recent progress in the program to classify separable amenable C<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Superscript asterisk"><mml:semantics><mml:msup><mml:mi /><mml:mo>ā<!-- ā --></mml:mo></mml:msup><mml:annotation encoding="application/x-tex">^*</mml:annotation></mml:semantics></mml:math></inline-formula>-algebras. Our emphasis is on the newly apparent role of regularity properties such as finite decomposition rank, strict comparison of positive elements, and<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper Z"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">Z</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathcal {Z}</mml:annotation></mml:semantics></mml:math></inline-formula>-stability, and on the importance of the Cuntz semigroup. We include a brief history of the programās successes since 1989, a more detailed look at the Villadsen-type algebras which have so dramatically changed the landscape, and a collection of announcements on the structure and properties of the Cuntz semigroup.
Abstract This is an introductory survey of the emerging theory of two new classes of (discrete, countable) groups, called hyperlinear and sofic groups. They can be characterized as subgroups of ā¦
Abstract This is an introductory survey of the emerging theory of two new classes of (discrete, countable) groups, called hyperlinear and sofic groups. They can be characterized as subgroups of metric ultraproducts of families of, respectively, unitary groups U(n) and symmetric groups S n , n ā ā. Hyperlinear groups come from theory of operator algebras (Connes' Embedding Problem), while sofic groups, introduced by Gromov, are motivated by a problem of symbolic dynamics (Gottschalk's Surjunctivity Conjecture). Open questions are numerous, in particular it is still unknown if every group is hyperlinear and/or sofic.
Abstract In this paper, we solve a question of Simon Wassermann, whether the Calkin algebra can be written as a C*-tensor product of two infinite dimensional C*-algebras. More generally, we ā¦
Abstract In this paper, we solve a question of Simon Wassermann, whether the Calkin algebra can be written as a C*-tensor product of two infinite dimensional C*-algebras. More generally, we show that there is no surjective *-homomorphism from a SAW *-algebra onto C*-tensor product of two infinite dimensional C*-algebras.
We investigate C*-algebras whose central sequence algebra has no characters, and we raise the question if such C*-algebras necessarily must absorb the JiangāSu algebra (provided that they also are separable). ā¦
We investigate C*-algebras whose central sequence algebra has no characters, and we raise the question if such C*-algebras necessarily must absorb the JiangāSu algebra (provided that they also are separable). We relate this question to a question of Dadarlat and Toms if the JiangāSu algebra always embeds into the infinite tensor power of any unital C*-algebra without characters. We show that absence of characters of the central sequence algebra implies that the C*-algebra has the so-called strong Corona Factorization Property, and we use this result to exhibit simple nuclear separable unital C*-algebras whose central sequence algebra does admit a character. We show how stronger divisibility properties on the central sequence algebra imply stronger regularity properties of the underlying C*-algebra.
By a variety, we mean a class of structures in some language containing only function symbols which is equationally defined or equivalently is closed under homomorphisms, submodels and products. If ā¦
By a variety, we mean a class of structures in some language containing only function symbols which is equationally defined or equivalently is closed under homomorphisms, submodels and products. If K is a class of -structures then I(K, Ī») denotes the number of nonisomorphic models in K of cardinality Ī». When we say that K has few models, we mean that I(K,Ī») < 2Ī» for some Ī» > ā£ā£. If I(K,Ī») = 2Ī» for all Ī» > ā£ā£, then we say K has many models. In [9] and [10], Shelah has shown that for an elementary class K, having few models is a strong structural condition.
We initiate a general study of ultraproducts of C*-algebras, including topics on representations, homomorphisms, isomorphisms, positive linear maps and their ultraproducts. We partially settle a question of D. McDuff by ā¦
We initiate a general study of ultraproducts of C*-algebras, including topics on representations, homomorphisms, isomorphisms, positive linear maps and their ultraproducts. We partially settle a question of D. McDuff by proving, for a finite von Neumann algebra with a separable predual, that the continuum hypothesis implies the isomorphism of all of its tracial ultrapowers with respect to different free ultrafilters on the natural numbers. The analog for C*-ultrapowers of separable C*-algebras is equivalent to the continuum hypothesis. We also prove a finite local reflexivity theorem for operator spaces that implies that the second dual of a C*-algebra can be embedded in some ultra-power of the algebra using a completely positive completely isometric map.
In functional analysis, approximative properties of an object become precise in its ultrapower. We discuss this idea and its consequences for automorphisms of II_1 factors. Here are some sample results: ā¦
In functional analysis, approximative properties of an object become precise in its ultrapower. We discuss this idea and its consequences for automorphisms of II_1 factors. Here are some sample results: (1) an automorphism is approximately inner if and only if its ultrapower is aleph_0-locally inner; (2) the ultrapower of an outer automorphism is always outer; (3) for unital *-homomorphisms from a separable nuclear C*-algebra into an ultrapower of a II_1 factor, equality of the induced traces implies unitary equivalence. All statements are proved using operator algebraic techniques, but in the last section of the paper we indicate how the underlying principle is related to theorems of Henson's positive bounded logic.
This is a detailed survey on the QWEP conjecture and Connes' embedding problem. Most of contents are taken from Kirchberg's paper [Invent. Math. 112 (1993)].
This is a detailed survey on the QWEP conjecture and Connes' embedding problem. Most of contents are taken from Kirchberg's paper [Invent. Math. 112 (1993)].