A Plancherel formula for idyllic nilpotent Lie groups
Eloise
Carlton
1-42
Abstract: A procedure is developed which can be used to compute the Plancherel measure for a certain class of nilpotent Lie groups, including the Heisenberg groups, free groups, two-and three-step groups, the nilpotent part of an Iwasawa decomposition of the R-split form of the classical simple groups $ {A_l},{C_l},{G_2}$. Let G be a connected, simply connected nilpotent Lie group. The Plancherel formula for G can be expressed in terms of Plancherel measure of a normal subgroup N and projective Plancherel measures of certain subgroups of $G/N$. To get an explicit measure for G, we need an explicit formula for (1) the disintegration of Plancherel measure of N under the action of G on N, and (2) projective Plancherel measures of ${G_\gamma }/N$, where ${G_\gamma }$ is the stability subgroup at $ \gamma$ in N. When both N and $ {G_\gamma }/N$ are abelian, the measures (1) and (2) are obtained as special cases of more general problems. These measures combine into Plancherel measure for G.
Parametric perturbation problems in ordinary differential equations
Thomas G.
Hallam
43-59
Abstract: The asymptotic behavior of solutions of a nonlinear differential equation that arises through a nonlinear parametric perturbation of a linear system of differential equations is discussed. Fundamental hypotheses include the admissibility of a pair of Banach spaces for the linear system. Conclusions about the behavior of the perturbed system evolve through the behavior of certain manifolds of solutions of the unperturbed linear system. Asymptotic representations are found on a semi-infinite axis ${R_ + }$ and on the real line R. The bifurcation condition, which is shown to be trivial on $ {R_ + }$, plays an essential role for the perturbation problem on R. Illustrations and examples, primarily on the space $ {{\text{L}}^\infty }$, of the theoretical results are presented.
Nonregular ultrafilters and large cardinals
Jussi
Ketonen
61-73
Abstract: The relationship between the existence of nonregular ultrafilters and large cardinals in the constructible universe is studied.
Lie algebras of type $BC\sb{1}$
B. N.
Allison
75-86
Abstract: Let L be a central simple Lie algebra of type $B{C_1}$ with highest root space of dimension greater than one over a field of characteristic zero. It is shown that either L is isomorphic to the simple Lie algebra associated with a skew hermitian form of index one or L can be constructed from the tensor product of two composition algebras. This result is obtained by completing the description (begun in [3]) of the corresponding class of ternary algebras.
Inner product modules arising from compact automorphism groups of von Neumann algebras
William L.
Paschke
87-102
Abstract: Let M be a von Neumann algebra of operators on a separable Hilbert space H, and G a compact, strong-operator continuous group of $^\ast$-automorphisms of M. The action of G on M gives rise to a faithful, ultraweakly continuous conditional expectation of M on the subalgebra $N = \{ A \in M:g(A) = A\forall g \in G\}$, which in turn makes M into an inner product module over N. The inner product module M may be ``completed'' to yield a self-dual inner product module $\bar M$ over N; our most general result states that the ${W^\ast}$-algebra $A(\bar M)$ of bounded N-module maps of $ \bar M$ into itself is isomorphic to a restriction of the crossed product $M \times G$ of M by G. When G is compact abelian, we give conditions for $ A(\bar M)$ and $M \times G$ to be isomorphic and show, among other things, that if G acts faithfully on M, then $M \times G$ is a factor if and only if N is a factor. As an example, we discuss certain compact abelian automorphism groups of group von Neumann algebras.
Maximal chains of prime ideals in integral extension domains. I
L. J.
Ratliff;
S.
McAdam
103-116
Abstract: Let (R, M) be a local domain, let k be a positive integer, and let Q be a prime ideal in ${R_k} = R[{X_1}, \ldots ,{X_k}]$ such that $M{R_k} \subset Q$. Then the following statements are equivalent: (1) There exists an integral extension domain of R which has a maximal chain of prime ideals of length n. (2) There exists a minimal prime ideal z in the completion of R such that depth $z = n$. (3) There exists a minimal prime ideal w in the completion of $ {({R_k})_Q}$ such that depth $w = n + k - {\text{depth}}\;Q$. (4) There exists an integral extension domain of ${({R_k})_Q}$ which has a maximal chain of prime ideals of length $n + k - {\text{depth}}\;Q$. (5) There exists a maximal chain of prime ideals of length $n + k - {\text{depth}}\;Q$ in ${({R_k})_Q}$. (6) There exists a maximal chain of prime ideals of length $n + 1$ in $ R{[{X_1}]_{(M,{X_1})}}$.
Maximal chains of prime ideals in integral extension domains. II
L. J.
Ratliff
117-141
Abstract: Four related subjects are investigated: (1) If (L, N) is a locality over a local domain (R, M) such that $N \cap R = M$, and if there exists an integral extension domain of L which has a maximal chain of prime ideals of length n (for short, a mcpil n), then there exists an integral extension domain of R which has a mcpil $n - {\text{trd}}\;L/R + {\text{trd}}(L/N)/(R/M)$. A refinement of the altitude inequality follows from this. (2) A condition for the converse of (1) to hold is given. (3) The class of local domains R such that there exists an integral extension domain of R which has a mcpil n if and only if there exists a mcpil n in R is studied. (4) Two new equivalences for the existence of mcpil n in an integral extension domain of a local domain are given.
The spectral theory of posets and its applications to $C\sp*$-algebras
A. H.
Dooley
143-155
Abstract: This paper uses methods from the spectral theory of partially ordered sets to clarify and extend some recent results concerning approximately finite-dimensional $ {C^\ast}$-algebras. An extremely explicit description is obtained of the Jacobson topology on the primitive ideal space, and it is shown that this topology has a basis of quasi-compact open sets. In addition, the main results of [4] are proved using only elementary means.
Local Fatou theorem and area theorem for symmetric spaces of rank one
A.
Korányi;
R. B.
Putz
157-168
Abstract: The classical results for the unit disc mentioned in the title are extended to harmonic functions on symmetric spaces of rank one.
$Z\sb{p}$actions on symplectic manifolds
R. J.
Rowlett
169-177
Abstract: A bordism classification is studied for periodic maps of prime period p preserving a symplectic structure on a smooth manifold. In sharp contrast to the corresponding oriented bordism, this theory contains nontrivial ptorsion even when p is odd. Calculation gives an upper limit on the size of this p-torsion.
Partially ordered linear algebras with multiplicative diagonal map
Taen Yu
Dai;
Ralph
DeMarr
179-187
Abstract: The diagonal of the product of two triangular matrices is the product of the diagonals of each matrix. This idea is used to characterize partially ordered linear algebras which have order properties similar to an algebra of real triangular matrices.
The primitive lifting problem in the equivalence problem for transitive pseudogroup structures: a counterexample
Pierre
Molino
189-192
Abstract: A transitive Lie pseudogroup ${\Gamma _M}$ on M is a primitive extension of $ {\Gamma _N}$ if ${\Gamma _N}$ is the quotient of ${\Gamma _M}$ by an invariant fibration $\pi :M \to N$ and if the pseudogroup induced by ${\Gamma _M}$ on the fiber of $\pi$ is primitive. In the present paper an example of this situation is given with the following property (counterexample to the primitive lifting property): the equivalence theorem is true for almost- ${\Gamma _N}$-structures but false for almost- ${\Gamma _M}$-structures.