Compact Transformation Groups Research Articles

Enveloppes holomorphiquement convexes invariantes

A result of A. Sadullaev [6], used in the potential theory, says that the polynomial hull of a circled compact set in n coincides with its convex hull with respect to holomorphic homogeneous polynomials. We give a natural generalization of this result to invariant compact sets in complex spaces under the action of a compact Lie group of holomorphic transformations.

An Equivariant Brauer Group and Actions of Groups onC*-Algebras

Suppose that (G, T) is a second countable locally compact transformation group given by a homomorphism ℓ:G→Homeo(T), and thatAis a separable continuous-traceC*-algebra with spectrumT. An actionα:G→Aut(A) is said to cover ℓ if the induced action ofGonTcoincides with the original one. We prove that the set BrG(T) of Morita equivalence classes of such systems forms a group with multiplication given by the balanced tensor product: [A, α][B, β]=[A⊗C0(T)B, α⊗β], and we refer to BrG(T) as the Equivariant Brauer Group. We give a detailed analysis of the structure of BrG(T) in terms of the Moore cohomology of the groupGand the integral cohomology of the spaceT. Using this, we can characterize the stable continuous-traceC*-algebras with spectrumTwhich admit actions covering ℓ. In particular, we prove that ifG=R, then every stable continuous-traceC*-algebra admits an (essentially unique) action covering ℓ, thereby substantially improving results of Raeburn and Rosenberg.

The equivariant Brauer groups of commuting free and proper actions are isomorphic

If X is a locally compact space which admits commuting free and proper actions of locally compact groups G and H, then the Brauer groups BrH(G\X) and BrG(X/H) are naturally isomorphic. Rieffel's formulation of Mackey's Imprimitivity Theorem asserts that if H is a closed subgroup of a locally compact group G, then the group C ∗ -algebra C ∗ (H) is Morita equivalent to the crossed product C0(G/H) ⋊ G. Subsequently, Rieffel found a symmetric version, involving two subgroups of G, and Green proved the following Symmetric Imprimitivity Theorem: if two locally compact groups act freely and properly on a locally compact space X, G on the left and H on the right, then the crossed products C0(G\X) ⋊ H and C0(X/H) ⋊ G are Morita equivalent. (For a discussion and proofs of these results, see (15).) Here we shall show that in this situation there is an isomorphism BrH(G\X) ∼ = BrG(X/H) of the equivariant Brauer groups introduced in (2). Suppose (G, X) is a second countable locally compact transformation group. The objects in the underlying set BrG(X) of the equivariant Brauer group BrG(X) are dynamical systems (A, G, α), in which A is a separable continuous-trace C ∗ -algebra with spectrum X, and α : G → Aut(A) is a strongly continuous action of G on A inducing the given action of G on X. The equivalence relation on such systems is the equivariant Morita equivalence studied in (1), (3). The group operation is given by (A, α) � (B, β) = (A ⊗C(X) B, α ⊗ β), the inverse of (A, α) is the conjugate system (A, α), and the identity is represented by (C0(X), τ), where τs(f)(x) = f(s −1 � x). Notation. Suppose that H is a locally compact group, that X is a free and proper right H-space, and that (B, H, β) a dynamical system. Then Ind X(B, β) will be the

Compact differentiable transformation groups on exotic spheres

0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 1 Statement of the main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 2 Higher dimensional subgroups of O(n + 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 3 Higher dimensional transformation groups on homology spheres, geometric weight systems and orbit structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 4 Higher dimensional transformation groups on homotopy spheres and the proof of Theorem B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 5 Degree of symmetry of exotic spheres revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 6 Large transformation groups on exotic spheres in bP.+ 1 . . . . . . . . . . . . . . . . . . . . . . . . . 384

On transformation group 𝐶*-algebras with continuous trace

In this paper we answer some questions posed by Dana Williams in [19] concerning the problem under which conditions the transformation group C ∗ {C^ \ast } -algebra C ∗ ( G , Ω ) {C^\ast }(G,\Omega ) of a locally compact transformation group (G, Ω \Omega ) has continuous trace. One consequence will be, for compact G, that C ∗ ( G , Ω ) {C^\ast }(G,\Omega ) has continuous trace if and only if the stabilizer map is continuous. We also give a complete solution to the problem if G is discrete.

Uniformly quasi-isometric foliations

AbstractConsider the natural action of 1-jets of the holonomy pseudogroup H on the transverse tangent bundle of a C1 compact foliated manifold (M, ℱ). If these 1-jets act in an equicontinuous way then it is possible to use a C1 ‘Ellis semi-group’ technique, as applied to a neighborhood of the identity in H, to produce a sheaf of compact local transformation groups whose orbits are the topological closures of leaves of ℱ. This action on the transverse manifold to ℱ then decomposes M into a union of minimal sets. These minimal sets we show to be C1 embedded submanifolds of M and the action of H on them is locally transitive.

Absolute continuity of measures on compact transformation groups

Absolute continuity of measures on compact transformation groups

An F. and M. Riesz theorem on locally compact transformation groups

An F. and M. Riesz theorem on locally compact transformation groups

A discretization ofp-adic quantum mechanics

We show that some compact subgroups (ℋn,m) of thep-adic Heisenberg group act irreducibly on corresponding finite dimensional spaces of test-functions (Sm,n). Under certain conditions, a compact group (Am+n) of linear canonical transformations, isomorphic toSL(2,Zp), can be represented unitarily onSm,n as a group of automorphisms of ℋn,m. The restriction toSm,n can be considered as a discretization because an invariant subgroup (In,m) ofAm+n is represented trivially. It is possible to take a limit whereIm,n becomes an arbitrarily small neighborhood of the identity, while the dimension ofSm,n becomes arbitrarily large. This is a possible definition of the “continuum limit” that we relate to other projective limits appearing naturally in the present context.

On probabilistic invariance

The invariance principle is one important design principle in low-level pattern recognition. In the first part of this paper we investigate some theoretical aspects of this principle. In the second part we show how an adaptive filter system reacts to pattern sets that involve some kind of invariance. In its most general, group-theoretical form the invariance principle can be formulated as follows: We assume that we have a class of patterns of interest { p i : i ϵ I} and a given pattern q. The problem is to decide if q was a pattern of interest or not, i.e., if q = p i 0 for some i 0 ϵ I. If the patterns p i are generated from a common prototype pattern p 0 with the help of transformations g i and if these transformations g i form a group G then this problem can be solved by finding maximum correlation filter sets. The relation between these filter sets and the prototype pattern p 0 and the transformation group G is completely known. The construction of the filter functions is, however, based on the assumption that all transformations g i ϵ G (and therefore also all patterns p i ) appear equally often. The purpose of this paper is the generalization of the filter construction process to pattern classes whose elements obey some probability distribution. In the special cases of 2-D rotation invariance and scale invariance, we will show how the maximum correlation filters depend on the prototype pattern, the transformation group, and the probability distribution. We will also sketch the theory for the case of arbitrary compact transformation groups and what type of problems turn up in this case. In the last part of the paper we describe an adaptive filter system that can find the maximum correlation filter functions automatically from a given set of patterns of interest. This filter system also uses a kind of maximum separation principle so that the resulting feature vectors can be easily used by classifiers. We demonstrate how such a filter system stabilizes in states that correspond to the theoretically derived filter functions and how this filter system reacts to invariance properties in the trainings set.

An almost classification of compact Lie groups with Borsuk-Ulam properties

We say that a compact Lie group G has the Borsuk-Ulam property in the weak sense if for every orthogonal representation V of G and every G-equivariant map /: S{V) -> S(V), VG = {0}, of the unit sphere we have deg/ Φ 0 . We say that G has the Borsuk-Ulam property in the strong sense if for any two orthogonal representations V, W of G with dim W = dim V and WG = VG = {0} and every G-equivariant map /: S(V) —> S{W) of the unit spheres we have deg/ Φ 0. In this paper a complete classification, up to isomorphism, of group with the weak Borsuk-Ulam property is given. A classification of groups with the strong Borsuk-Ulam property does not cover nonabelian p-groups with all elements of the order p . In fact we deal with a more general definition admitting a nonempty fixed point set of G on the sphere S(V). 1. The main theorems. In order to formulate our main results we introduce the following notation. Let G be a compact Lie group. We denote by Go the component of identity of and by Γ the quotient group G/GQ . We use standard notation of the theory of compact transformation groups (see for instance [4] or [5]). In particular, for every subgroup H c G, the fixed point set of H on a G-space X is denoted by XH. Also, for a G-equivariant map f:X—>Y between two G-spaces, we denote by fH its restriction to the space XH . The symbol (n, m) stands for the greatest common divisor of the integers n, m with the notation (0, 1) = 0 and \G for the rank of the (finite) group G. We will work with the following definition of the BorsukUlam property (cf. [11]). DEFINITION I. (A) We say that G has the Borsuk-Ulam property in the weak sense A if for every orthogonal representation V of G and every G-equivariant map /: S(V) -> S(V) if (deg/ G, |Γ|) = 1 then

Manifolds on Which Only Tori Can Act

A list of various types of connected, closed oriented manifolds are given. Each of the manifolds support some of the well-known compact transformation group properties enjoyed by aspherical manifolds. We list and describe these classes and their transformation group properties in increasing generality. We show by various examples that these implications can never be reversed. This establishes a hierarchy in terms of spaces in one direction and the properties they enjoy in the opposite direction.

Compact 16-Dimensional Projective Planes with Large Collineation Groups. IV

Let be a topological projective plane with compact point set P of finite (covering) dimension. In the compact-open topology (of uniform convergence), the group Σ of continuous collineations of is a locally compact transformation group of P.THEOREM. If dim Σ > 40, thenis isomorphic to the Moufang plane 6 over the real octonions (and dim Σ = 78).By [3] the translation planes with dim Σ = 40 form a one-parameter family and have Lenz type V. Presumably, there are no other planes with dim Σ = 40, cp. [17].

Manifolds on which only tori can act

A list of various types of connected, closed oriented manifolds are given. Each of the manifolds support some of the well-known compact transformation group properties enjoyed by aspherical manifolds. We list and describe these classes and their transformation group properties in increasing generality. We show by various examples that these implications can never be reversed. This establishes a hierarchy in terms of spaces in one direction and the properties they enjoy in the opposite direction.

Automorphism groups of compact projective planes

Compact connected projective planes have been investigated extensively in the last 30 years, mostly by studying their automorphism groups. It is our aim here to remove the connectedness assumption in some general results of Salzmann [31] and Hahl [14] on automorphism groups of compact projective planes. We show that the continuous collineations of every compact projective plane form a locally compact transformation group (Theorem 1), and that the continuous collineations fixing a quadrangle in a compact translation plane form a compact group (Corollary to Theorem 3). Furthermore we construct a metric for the topology of a quasifield belonging to a compact projective translation plane, using the modular function of its additive group (Theorem 2).

A generalization of correlation functions and the Wiener-Khinchin theorem

Correlation functions and energy spectra are closely related to translation invariance properties. This can be generalized for more complex transformations, such as translations and discrete rotations or even to rotations of a sphere for functions defined on its surface (applicable to the earth's magnetic field). The concepts of correlation function and energy spectrum are extended to functions on a set M under a compact transformation group G with respect to G-invariant integrals on M and G. Two definitions of correlation are possible, which coincide in special cases. They both have Wiener-Khinchin-like relations to suitably defined energy spectra, connected with the irreducible representations of G.

The geometry of spontaneous symmetry breaking

The problem of classifying the theoretically allowed patterns of spontaneous symmetry breading, in theories where the ground state is determined as a minimum of a G-invariant potential ( G a compact group of transformations), is analyzed. A detailed, complete, and rigorous justification of a recently proposed approach to the determination of the minima of G-invariant potentials (M. Abud and G. Sartori, Phys. Lett. B 104 (1981), 147) is presented. The results are obtained through an analysis of the geometry of the finite-dimensional representations of G, which leads to a complete characterization of the structure of orbit space and its partition in subsets (strata) formed by orbits with the same symmetry under G-transformations (orbit type), and to a new theorem stating that the gradients of complex analytic G-invariant functions annihilate on one-dimensional strata. Polynomial potentials in particular are studied. Conditions for instability of the residual symmetry (second-order phase transitions) are determined.

Supplement to: ``Compact transformation groups on $Z\sb2$-cohomology spheres with orbit of codimension 1'' [Hiroshima Math. J. {11} (1981), no. 3, 571--616; MR0635040 (83b:57021)

Supplement to: ``Compact transformation groups on $Z\sb2$-cohomology spheres with orbit of codimension 1'' [Hiroshima Math. J. {11} (1981), no. 3, 571--616; MR0635040 (83b:57021)

On the triangulation of stratified sets and singular varieties

We show that every compact stratified set in the sense of Thom can be triangulated as a simplicial complex. The proof uses that author’s description of a stratified set as the geometric realisation of a certain type of diagram of smooth fibre bundles and smooth imbeddings, and the triangulability of smooth fibre bundles. As a consequence, we obtain proofs of the classical triangulation theorems for analytic and subanalytic sets, and a correct proof of Yang’s theorem that the orbit space of a smooth compact transformation group is triangulable.

Some Group Actions on Homotopy Spheres of Dimension Seven and Fifteen

0. Introduction. This paper is based on the simple observation that every 4-plane bundle over S4 admits a natural action of 50(3) by bundle maps and that every 8-plane bundle over 58 admits a natural action of the compact Lie group G2 by bundle maps. (These actions are easy to see once one remembers that SO (3) is the group of automorphisms of the quaternions and that G2 is the group of automorphisms of the Cayley numbers.) The study of these actions is closely connected to several well-known phenomena in differential topology and compact transformation groups. The most obvious connection is to Milnor's original construction of exotic 7-spheres as 3-sphere bundles over 54, [26]. Milnor proved that if the Euler class of such a bundle is a generator of H4(S4; Z), then its total space is homeomorphic to 57. He also defined a numerical invariant of the diffeomorphism type and used it to detect an exotic differential structure on some of these sphere bundles. Subsequently, Eells and Kuiper [12] introduced a refinement of this invariant, called the it-invariant. Using the ,u-invariant, they proved that the sphere bundle M7 (Milnor's notation) is a generator for the group of homotopy 7-spheres. These constructions also work for 7-sphere bundles over 58, [32], and the manifold M15 is a generator for bP16, the group of homotopy 15-spheres which bound wN-manifolds. It is a routine matter to check that Milnor's arguments and their subsequent refinements work G-equivariantly where G 5SO(3) or G2, and we shall do this in Sections 2 and 3. In particular, each sphere bundle with the correct Euler class is G-homeomorphic to an orthogonal action on S2,,+1, where 2n + 1 = 7 or 15. Moreover, distinct sphere bundles have distinct oriented G-diffeomorphism types. The proof of the second fact uses an equivariant version of the it-invariant. As originally defined this invariant takes values in Q/Z. However, it is well-known that in the presence of a G-action, with S C G, its value in Q is well-defined. Using