Compact groups operating on Banach algebras

Björk, Jan-Erik

doi:10.1007/bf01362698

Cited by 12 publications

(16 citation statements)

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“…G· x, it cannot be the case that h(x) = y. Equivalently, Because the maps z --+ z-I and z --+ (zO', T) are holomorphic on GL(n), we conclude in either case there is a holomorphic function F on GL(n) such that F(zo) > 1, while IF(z)1 :::; 1 for each z E U(n). It now follows from a result of Bjork [4,Theorem 6.1J that every continuous complex-valued function on U(n) can be uniformly approximated by restrictions to U(n) ofholomorphic functions on GL(n).…”

Section: (T+p-2) (T+ Q -2) T-2 T-2mentioning

confidence: 99%

“…In the dimensions in which we work, this means that we shall be able to avoid umbilic points entirely. The second remark is that the property of being nonumbilic depends only on the jet of the function tp of order 4 The plan of the proof is to first construct a domain on which G acts by automorphisms, and then to find a strictly pseudoconvex subdomain Do which is G-invariant. By using a transversality argument, we then find a perturbation D of Do with the property that points of bD which belong to distinct orbits of G have different Burns-Shnider-Wells curvature invariants.…”

Section: Letmentioning

confidence: 99%

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The isometry groups of manifolds and the automorphism groups of domains

Saerens¹,

Zame²

1987

Trans. Amer. Math. Soc.

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Section: (T+p-2) (T+ Q -2) T-2 T-2mentioning

confidence: 99%

Section: Letmentioning

confidence: 99%

The isometry groups of manifolds and the automorphism groups of domains

Saerens¹,

Zame²

1987

Trans. Amer. Math. Soc.

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“…In [2], Björk found a typical situation where analytic annuli appear in the maximal ideal space M A of a commutative Banach algebra A which admits a nontrivial action of T by automorphisms: this happens if T -invariant functions on M A do not separate distinct T-orbits. In [7], it was noted that analytic strips and/or annuli appear in Gv if the stable subgroup of v in G C does not coincide with the complexification of the stable subgroup of v in G. Proof.…”

Section: Finite Extensions Of T That Keep a T C -Orbitmentioning

confidence: 99%

Orbits of tori extended by finite groups and their polynomial hulls: the case of connected complex orbits

Gichev¹

2008

Complex Analysis and Dynamical Systems III

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Let V be a complex linear space, G ⊂ GL(V ) be a compact group. We consider the problem of description of polynomial hulls c Gv for orbits Gv, v ∈ V , assuming that the identity component of G is a torus T . The paper contains a universal construction for orbits which satisfy the inclusion Gv ⊂ T C v and a characterization of pairs (G, V ) such that it is true for a generic v ∈ V . The hull of a finite union of T -orbits in T C v can be distinguished in clos T C v by a finite collection of inequalities of the type |z 1 | s 1 . . . |zn| sn ≤ c. In particular, this is true for Gv. If powers in the monomials are independent of v, Gv ⊂ T C v for a generic v, and either the center of G is finite or T C has an open orbit, then the space V and the group G are products of standard ones; the latter means that G = SnT , where Sn is the group of all permutations of coordinates and T is either T n or SU(n) ∩ T n , where T n is the torus of all diagonal matrices in U(n). The paper also contains a description of polynomial hulls for orbits of isotropy groups of bounded symmetric domains. This result is already known, but we formulate it in a different form and supply with a shorter proof.1991 Mathematics Subject Classification. Primary 32E20; Secondary 32M15, 32M05.

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“…Hence by the Local Maximum Modulus Principle for uniform algebras, for each y in N we can find x in f~\3D) with whence by (2) we have (3) \g exρ(-P(/))(y)| = 1.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…Questions related to the result just proved are studied by J. E. Bjόrk in [3], 4. Invariant disks in C 2 .…”

Section: ζLφ(ζ)e H(x)mentioning

confidence: 99%