203 research outputs found

### Dual Bialgebroids for Depth Two Ring Extensions

We introduce a general notion of depth two for ring homomorphism N --> M, and
derive Morita equivalence of the step one and three centralizers, R = C_M(N)
and C = End_{N-M}(M \o_N M), via dual bimodules and step two centralizers A =
End_NM_N and B = (M \o_N M)^N, in a Jones tower above N --> M. Lu's
bialgebroids End_k A' and A' \o_k {A'}^op over a k-algebra A' are generalized
to left and right bialgebroids A and B with B the R-dual bialgebroid of A. We
introduce Galois-type actions of A on M and B on End_NM when M_N is a balanced
module. In the case of Frobenius extensions M | N, we prove an endomorphism
ring theorem for depth two. Further in the case of irreducible extensions, we
extend previous results on Hopf algebra and weak Hopf algebra actions in
subfactor theory [Szymanski, Nikshych-Vainerman] and its generalizations
[Kadison-Nikshych: RA/0107064, RA/0102010] by methods other than nondegenerate
pairing. As a result, we have concrete expressions for the Hopf or weak Hopf
algebra structures on the step two centralizers. Semisimplicity of B is
equivalent to separability of the extension M | N. In the presence of depth
two, we show that biseparable extensions are QF.Comment: 2 new sections added, 37 page

### Diagonalizing matrices over AW*-algebras

Every commuting set of normal matrices with entries in an AW*-algebra can be
simultaneously diagonalized. To establish this, a dimension theory for properly
infinite projections in AW*-algebras is developed. As a consequence, passing to
matrix rings is a functor on the category of AW*-algebras.Comment: 24 pages. Comments very welcome

### A Topos Foundation for Theories of Physics: II. Daseinisation and the Liberation of Quantum Theory

This paper is the second in a series whose goal is to develop a fundamentally
new way of constructing theories of physics. The motivation comes from a desire
to address certain deep issues that arise when contemplating quantum theories
of space and time. Our basic contention is that constructing a theory of
physics is equivalent to finding a representation in a topos of a certain
formal language that is attached to the system. Classical physics arises when
the topos is the category of sets. Other types of theory employ a different
topos. In this paper, we study in depth the topos representation of the
propositional language, PL(S), for the case of quantum theory. In doing so, we
make a direct link with, and clarify, the earlier work on applying topos theory
to quantum physics. The key step is a process we term `daseinisation' by which
a projection operator is mapped to a sub-object of the spectral presheaf--the
topos quantum analogue of a classical state space. In the second part of the
paper we change gear with the introduction of the more sophisticated local
language L(S). From this point forward, throughout the rest of the series of
papers, our attention will be devoted almost entirely to this language. In the
present paper, we use L(S) to study `truth objects' in the topos. These are
objects in the topos that play the role of states: a necessary development as
the spectral presheaf has no global elements, and hence there are no
microstates in the sense of classical physics. Truth objects therefore play a
crucial role in our formalism.Comment: 34 pages, no figure

### Leibniz Seminorms and Best Approximation from C*-subalgebras

We show that if B is a C*-subalgebra of a C*-algebra A such that B contains a
bounded approximate identity for A, and if L is the pull-back to A of the
quotient norm on A/B, then L is strongly Leibniz. In connection with this
situation we study certain aspects of best approximation of elements of a
unital C*-algebra by elements of a unital C*-subalgebra.Comment: 24 pages. Intended for the proceedings of the conference "Operator
Algebras and Related Topics". v2: added a corollary to the main theorem, plus
several minor improvements v3: much simplified proof of a key lemma,
corollary to main theorem added v4: Many minor improvements. Section numbers
increased by

### Reduction of Lie-Jordan Banach algebras and quantum states

A theory of reduction of Lie-Jordan Banach algebras with respect to either a
Jordan ideal or a Lie-Jordan subalgebra is presented. This theory is compared
with the standard reduction of C*-algebras of observables of a quantum system
in the presence of quantum constraints. It is shown that the later corresponds
to the particular instance of the reduction of Lie-Jordan Banach algebras with
respect to a Lie-Jordan subalgebra as described in this paper. The space of
states of the reduced Lie-Jordan Banach algebras is described in terms of
equivalence classes of extensions to the full algebra and their GNS
representations are characterized in the same way. A few simple examples are
discussed that illustrates some of the main results

### Applications of Automata and Graphs: Labeling-Operators in Hilbert Space I

We show that certain representations of graphs by operators on Hilbert space
have uses in signal processing and in symbolic dynamics. Our main result is
that graphs built on automata have fractal characteristics. We make this
precise with the use of Representation Theory and of Spectral Theory of a
certain family of Hecke operators. Let G be a directed graph. We begin by
building the graph groupoid G induced by G, and representations of G. Our main
application is to the groupoids defined from automata. By assigning weights to
the edges of a fixed graph G, we give conditions for G to acquire fractal-like
properties, and hence we can have fractaloids or G-fractals. Our standing
assumption on G is that it is locally finite and connected, and our labeling of
G is determined by the "out-degrees of vertices". From our labeling, we arrive
at a family of Hecke-type operators whose spectrum is computed. As
applications, we are able to build representations by operators on Hilbert
spaces (including the Hecke operators); and we further show that automata built
on a finite alphabet generate fractaloids. Our Hecke-type operators, or
labeling operators, come from an amalgamated free probability construction, and
we compute the corresponding amalgamated free moments. We show that the free
moments are completely determined by certain scalar-valued functions.Comment: 69 page

### Free energy density for mean field perturbation of states of a one-dimensional spin chain

Motivated by recent developments on large deviations in states of the spin
chain, we reconsider the work of Petz, Raggio and Verbeure in 1989 on the
variational expression of free energy density in the presence of a mean field
type perturbation. We extend their results from the product state case to the
Gibbs state case in the setting of translation-invariant interactions of finite
range. In the special case of a locally faithful quantum Markov state, we
clarify the relation between two different kinds of free energy densities (or
pressure functions).Comment: 29 pages, Section 5 added, to appear in Rev. Math. Phy

### Entropy and the variational principle for actions of sofic groups

Recently Lewis Bowen introduced a notion of entropy for measure-preserving
actions of a countable sofic group on a standard probability space admitting a
generating partition with finite entropy. By applying an operator algebra
perspective we develop a more general approach to sofic entropy which produces
both measure and topological dynamical invariants, and we establish the
variational principle in this context. In the case of residually finite groups
we use the variational principle to compute the topological entropy of
principal algebraic actions whose defining group ring element is invertible in
the full group C*-algebra.Comment: 44 pages; minor changes; to appear in Invent. Mat

### A quantum subgroup depth

The Green ring of the half quantum group is computed in [9]. The tensor product formulas between indecomposables may be used for a generalized subgroup depth computation in the setting of quantum groups-to compute the depth of the Hopf subalgebra H in its Drinfeld double D(H). In this paper the Hopf subalgebra quotient module Q (a generalization of the permutation module of cosets for a group extension) is computed and, as H-modules, Q and its second tensor power are decomposed into a direct sum of indecomposables. We note that the least power n, referred to as depth, for which has the same indecomposable constituents as is , since contains all H-module indecomposables, which determines the minimum even depth

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