Maximal Abelian Subalgebra Research Articles

Maximal Abelian subgroups of the isometry and conformal groups of Euclidean and Minkowski spaces

The maximal Abelian subalgebras of the Euclidean e(p,0) and pseudoeuclidean e(p,1)Lie algebras are classified into conjugacy classes under the action of the corresponding Lie groups E(p,0) and E(p,1), and also under the conformal groups O(p+1,1) and O(p+1,2), respectively. The results are presented in terms of decomposition theorems. For e(p,0) orthogonally indecomposable MASAs exist only for p=1 and p=2. For e(p,1), on the other hand, orthogonally indecomposable MASAs exist for all values of p. The results are used to construct new coordinate systems in which wave equations and Hamilton-Jacobi equations allow the separation of variables.

Applications of free entropy to finite von Neumann algebras

We apply Voiculescu's free entropy to show that free group factors do not possess a simple maximal abelian self-adjoint subalgebra, hence answering a longstanding question of Ambrose and Singer.

Spectral stochastic processes arising in quantum mechanical models with a non-L 2 ground state

A functional integral representation is given for a large class of quantum mechanical models with a non-L 2 ground state. As a prototype, the particle in a periodic potential is discussed: a unique ground state is shown to exist as a state on the Weyl algebra, and a functional measure (spectral stochastic process) is constructed on trajectories taking values in the spectrum of the maximal Abelian subalgebra of the Weyl algebra isomorphic to the algebra of almost periodic functions. The thermodynamical limit of the finite-volume functional integrals for such models is discussed, and the superselection sectors associated to an ‘observable’ subalgebra of the Weyl algebra are described in terms of boundary conditions and/or topological terms in the finite-volume measures.

Harmonic analysis on spannors

The spannor representation of the universal covering group of the conformal group of Minkowski space–time is studied herein. It is an example of an indecomposable representation. Various parallelizations for spannors are described. Normalized K̃-finite basis fields for the spannors are introduced, and a computation of the actions of the scale generator and other generators of the conformal group on these basis fields is given. The irreducibility of all irreducible composition factors is established, and it is decided which ones are infinitesimally unitary. Two sets of generators for maximal Abelian subalgebras of the enveloping algebra of the conformal group are introduced. The action of one of the sets on K̃-finite basis fields in the spannor representation is studied; this is important for understanding of the problem of degeneracy in the spannor representation, since K̃ types occur with multiplicity 2. Many calculations make use of the lowest (highest) weight module structures of the unitarizable and positive (negative) energy, irreducible composition factors; and, thus, many of our results and techniques can be used to initiate a study of the quantum deformations of the associated Lie algebra representations. Results on the Poincaré content of the unitarizable, irreducible composition factors are stated, and unitary equivalences between certain of these factors are established.

Modular algebras in geometric quantization

Presented here is a discussion on the connection between geometric quantization and algebraic quantization. The former procedure relies on a construction of unitary irreducible representations that starts from co-adjoint orbits and uses polarizations, while the latter depends on the purely algebraic characterization of unitary irreducible representations, which is based on central decompositions of von Neumann algebras in involutive duality, and their decompositions in terms of maximal Abelian subalgebras. Intermediate stages of these two quantization methods turn out to be complementary, leading thus to a new characterization of the so-called discrete series representations.

Maximal Abelian subalgebras of complex Euclidean Lie algebras

Maximal Abelian subalgebras (MASAs) of the complex Euclidean Lie algebra [Formula: see text] are classified into conjugacy classes under the action of the Lie group [Formula: see text] Use is made of an earlier classification of MASAs of the orthogonal Lie algebra [Formula: see text] These are then extended to nonsplitting MASAs of [Formula: see text] in which maximal Abelian nilpotent subalgebras of [Formula: see text] are coupled with translations in a nontrivial manner. The methods presented are applicable to the classification of MASAs of any affine Lie algebra.

Hecke algebra on homogeneous trees and relations with Toeplitz and Hankel operators

We consider the Hecke algebra on homogeneous trees. We prove that it is a maximal abelian subalgebra of some operator algebras if the degree of the tree is greater than 2. There we show the influence of geometry of the tree on that fact. If the degree is 2 (for example, in the case of integers) then we show that operators which commute with the Hecke algebra can be uniquely represented as a sum of Hankel and Toeplitz matrices.

Diagonalization in Compact Lie Algebras and a New Proof of a Theorem of Kostant

We exhibit a simple algorithmic procedure to show that any element of a compact Lie algebra is conjugate to an element of a fixed maximal abelian subalgebra. An estimate of the convergence of the algorithm is obtained. As an application, we provide a new proof of Kostant’s theorem on the projection of orbits onto a maximal abelian subalgebra.

Diagonalization in compact Lie algebras and a new proof of a theorem of Kostant

We exhibit a simple algorithmic procedure to show that any element of a compact Lie algebra is conjugate to an element of a fixed maximal abelian subalgebra. An estimate of the convergence of the algorithm is obtained. As an application, we provide a new proof of Kostant’s theorem on the projection of orbits onto a maximal abelian subalgebra.

Maximal abelian subalgebras of pseudoorthogonal lie algebras

The classification of maximal abelian subalgebras (MASAs) of o( p, q) is reduced to the construction of orthogonally indecomposable (OID) MASAs. Five types of OID MASAs exist, and their classification amounts to an analysis of maximal abelian nilpotent subalgebras (MANSs) of o( p,q), su( p',q'), sl( n, R ), or sl( n, C ). Theorems are provided making possible the classification of all MANSs of o( p, q), and low-dimensional examples are analyzed in detail.

Singularity of radial subalgebras in II 1 factors associated with free products of groups

Let G be the free product of N groups each having order k ⩽ N and let A be the maximal abelian subalgebra of the group von Neumann algebra L ( G), called the radial algebra of G. The Pukánszky invariant of the abelian algebra A =( A ∨ JAJ)″ is computed in this case. If N ⩾ 3, A is isomorphic to A ⊕ ( A ⊗ A) and A is singular. If N = k = 2, A is isomorphic to A ⊕ A and A is a Cartan subalgebra.

Singularity of the radial subalgebra of ℒ(FN) and the Pukánszky invariant

Let ℒ(F N ) be the von Neumann algebra of the free group with N generators x 1 ,⋯,x N , N≥2 and let A be the abelian von Neumann subalgebra generated by x 1 +x 1 -1 +⋯+x N +x N -1 acting as a left convolutor on l 2 (F N ). The radial algebra A appeared in the harmonic analysis of the free group as a maximal abelian subalgebra of ℒ(F N ), the von Neumann algebra of the free group. The aim of this paper is to prove that A is singular (which means that there are no unitaries u in ℒ(F N ) except those coming from A such that u * Au⊆A). This is done by showing that the Pukanszky invariant of A is infinite, where the Pukanszky invariant of A is the type of the commutant of the algebra A in B(l 2 (F N )) generated by A and x 1 +x 1 -1 +⋯+x N +x N -1 regarded also as a right convolutor on l 2 (F N )

Maximal abelian subalgebras of complex orthogonal lie algebras

The study of maximal abelian subalgebras (MASAs) of o(n,C) is reduced to the study of orthogonally indecomposable MASAs. Their classification is in turn reduced to that of maximal abelian nilpotent subalgebras (MANSs) of sl(n/2,C) (for n even) and of o(n,C) itself. General results and guidelines are given, making it possible to construct all MANSs, and hence all MASAs, for any given n. Representatives of all O(n,C) conjugacy classes of MASAs of o(n,C) are constructed for 2⩽n⩽7, as an application of the general techniques.

Maximal Abelian subalgebras of pseudounitary lie algebras

The task of classifying and constructing all maximal Abelian subalgebras of su( p, q) ( p⩾ q⩾1) is reduced to that of classifying orthogonally indecomposable MASAs. These are either maximal Abelian nilpotent subalgebras (represented by nilpotent matrices in any finite-dimensional representation), or for p= q they can be (nonorthogonally) decomposable and their study can be reduced to a construction of MANSs of sl( p, C). Two types of MANSs of su( p, q) are shown to exist (“one-rowed” and “non-one-rowed”). Numerous classification theorems are proven and applied to obtain all MASAs of su( p,1), su( p,2), and su( p, q) with p+ q⩽6. Physical applications are discussed.

Some abelian Banach algebras of operators on the matricial 𝐼𝐼₁ factor

We show that, if G G is an amenable discrete group, then the set of completely bounded [completely positive] multiplication operators on V N ( G ) VN(G) is maximal abelian and norm- 1 1 complemented in various sets of bounded [positive] operators on V N ( G ) VN(G) . Since there are many different amenable discrete groups G G which generate the matricial I I 1 II_{1} factor R R , this shows the set of completely bounded normal operators on R R contains uncountably many non-isomorphic maximal abelian subalgebras each of which is complemented by a positivity-preserving projection of norm one. Our results are closely related to [16], which considers the case of group C ∗ {C^ * } -algebras of general locally compact groups.

Maximal Abelian Subalgebras with Simple Normalizer

Maximal Abelian Subalgebras with Simple Normalizer

Continuous subgroups of the generalized Schrödinger groups

Some general results on the subalgebras of the Lie algebra ASch(n) of the generalized Schrödinger group Sch(n) and on the subalgebras of the Lie algebra ASch(n) of the generalized extended Schrödinger group Sch(n) have been obtained. The subalgebra structure of the algebras ASch(n) and ASch(n) are studied with respect to inner automorphisms of the groups Sch(n) and Sch(n), respectively. The maximal Abelian subalgebras and the one-dimensional subalgebras of the algebras ASch(n) and ASch(n) have been explicitly found. The full classification of the subalgebras of the algebras ASch(3), ASch(3), which are nonconjugate to the subalgebras of ASch(2), ASch(2), respectively, has been carried out.

Matrix hierarchies and vector bundles over Riemann surfaces

To a compact Riemann surface, a point on the surface, a local coordinate around this point, and a holomorphic transfer function of degree m is associated a point in the Grassmann manifold of l2(Z)m. This leads to solutions of the nonlinear hierarchies corresponding to maximal Abelian subalgebras of End(Cm).

Maximal abelian subalgebras of function algebras associated with free products

Let Γ be the free product of n groups each having order k. The characteristic function χ 1 of the set of elements of length one in Γ generates subalgebras, called radial algebras, of the regular C ∗- and von Neumann algebras of Γ. If n < k, a nonzero, nonradial element of C r ∗(Γ) which commutes with χ 1 is demonstrated, while for n ⩾ k, any element of L 2( Γ) which commutes with χ 1 is radial. Hence the radial C ∗- and von Neumann algebras are maximal abelian if and only n ⩾ k.

The algebra formed by the invariant charges of the Nambu-Goto theory: Identification of a maximal Abelian subalgebra

Continuing the analysis of the algebraic structure of the invariant charges of the Nambu-Goto theory, we identify a complete set of commuting observables for the bosonic strings.