Uniform algebras generated by holomorphic and pluriharmonic functions

Author:
Alexander J. Izzo

Journal:
Trans. Amer. Math. Soc. **339** (1993), 835-847

MSC:
Primary 46J15; Secondary 32E25, 46E15

DOI:
https://doi.org/10.1090/S0002-9947-1993-1139494-6

MathSciNet review:
1139494

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that if are pluriharmonic on (the open unit ball in and on , and the matrix is invertible at every point of , then the norm-closed algebra generated by the ball algebra and is equal to . Extensions of this result to more general strictly pseudoconvex domains are also presented.

**[A-S]**S. Axler and A. Shields,*Algebras generated by analytic and harmonic functions*, Indiana Univ. Math. J.**36**(1987), 631-638. MR**905614 (88h:46102)****[B]**A. Browder,*Cohomology of maximal ideal spaces*, Bull. Amer. Math. Soc.**67**(1961), 515-516. MR**0130580 (24:A440)****[Č]**E. M. Čirka,*Approximation by holomorphic functions on smooth manifolds in*, Mat. Sb.**78**(1969), 101-123; English transl., Math. USSR Sb.**7**(1969), 95-114. MR**0239121 (39:480)****[G]**T. W. Gamelin,*Uniform algebras*, 2nd ed., Chelsea, New York, 1984.**[H]**L. Hörmander,*An introduction to complex analysis in several variables*, 2nd ed., North-Holland, 1979.**[I]**A. J. Izzo,*Uniform algebras generated by holomorphic and harmonic functions of one and several complex variables*, Ph.D. Thesis, Univ. of California at Berkeley, 1989.**[R-S]**R. M. Range and Y. T. Sui,*Uniform estimates for the*-*equation on domains with piecewise smooth strictly pseudoconvex boundaries*, Math. Ann.**206**(1973), 325-354. MR**0338450 (49:3214)****[R]**W. Rudin,*Function theory in the unit ball of*, Springer-Verlag, New York, 1980. MR**601594 (82i:32002)****[S]**E. L. Stout,*The theory of uniform algebras*, Bogden & Quigley, New York, 1971. MR**0423083 (54:11066)****[Wei1]**B. M. Weinstock,*Zero-sets of continuous holomorphic functions on the boundary of a strongly pseudoconvex domain*, J. London Math. Soc.**18**(1978), 484-488. MR**518233 (80e:32010)****[Wei2]**B. M. Weinstock,*Uniform approximation on the graph of a smooth map in*, Canad. J. Math.**17**(1980), 1390-1396. MR**604694 (82e:32027)****[Wer]**J. Wermer,*Polynomially convex disks*, Math. Ann.**158**(1965), 6-10. MR**0174968 (30:5158)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
46J15,
32E25,
46E15

Retrieve articles in all journals with MSC: 46J15, 32E25, 46E15

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1993-1139494-6

Article copyright:
© Copyright 1993
American Mathematical Society