14,502 research outputs found
Logic and operator algebras
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.Comment: A minor chang
Quantifier elimination in C*-algebras
The only C*-algebras that admit elimination of quantifiers in continuous
logic are , Cantor space and
. We also prove that the theory of C*-algebras does not have
model companion and show that the theory of is not
-axiomatizable for any .Comment: More improvements and bug fixes. To appear in IMR
Model theory of operator algebras III: Elementary equivalence and II_1 factors
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.Comment: 16 page
Notions of Infinity in Quantum Physics
In this article we will review some notions of infiniteness that appear in
Hilbert space operators and operator algebras. These include proper
infiniteness, Murray von Neumann's classification into type I and type III
factors and the class of F{/o} lner C*-algebras that capture some aspects of
amenability. We will also mention how these notions reappear in the description
of certain mathematical aspects of quantum mechanics, quantum field theory and
the theory of superselection sectors. We also show that the algebra of the
canonical anti-commutation relations (CAR-algebra) is in the class of F{/o}
lner C*-algebras.Comment: 11 page
Quantum logic is undecidable
We investigate the first-order theory of closed subspaces of complex Hilbert
spaces in the signature , where `' is the
orthogonality relation. Our main result is that already its quasi-identities
are undecidable: there is no algorithm to decide whether an implication between
equations and orthogonality relations implies another equation. This is a
corollary of a recent result of Slofstra in combinatorial group theory. It
follows upon reinterpreting that result in terms of the hypergraph approach to
quantum contextuality, for which it constitutes a proof of the inverse sandwich
conjecture. It can also be interpreted as stating that a certain quantum
satisfiability problem is undecidable.Comment: 11 pages. v3: improved exposition. v4: minor clarification
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