# Article

Full entry | PDF   (0.1 MB)
Keywords:
compact Hausdorff space \$X\$; the sup-norm algebra \$C(X)\$ of all complex-valued continuous functions on \$X\$; its closed subalgebras (called function algebras); pervasive algebras; the algebra \$A(X)\$ of all functions continuous on \$\Bbb S\$ and holomorphic on \$\Bbb S\smallsetminus X\$
Summary:
We characterize compact sets \$X\$ in the Riemann sphere \$\Bbb S\$ not separating \$\Bbb S\$ for which the algebra \$A(X)\$ of all functions continuous on \$\Bbb S\$ and holomorphic on \$\Bbb S\smallsetminus X\$, restricted to the set \$X\$, is pervasive on \$X\$.
References:
[1] Hoffman K., Singer I.M.: Maximal algebras of continuous functions. Acta Math. 103 (1960), 217-241. MR 0117540 | Zbl 0195.13903
[2] Gamelin T.W.: Uniform Algebras. Prentice Hall, Inc., Englewood Cliffs, N.J., 1969. MR 0410387 | Zbl 1118.47014
[3] Fuka J.: A remark to maximality of several function algebras (in Russian). Čas. Pěst. Mat. 93 (1968), 346-348. MR 0251539
[4] Saks S., Zygmund A.: Analytic Functions. Polskie Towarzystwo Matematyczne, Warszawa, 1952. MR 0055432 | Zbl 0136.37301
[5] Urysohn P.S.: Sur une fonction analytique partout continue. Fund. Math. 4 (1922), 144-150.
[6] McKissick R.: A nontrivial normal sup norm algebra. Bull. Amer. Math. Soc. 69 (1963), 391-395. MR 0146646 | Zbl 0113.31502

Partner of