Inductive limits of -convex algebras

Author:
Allan C. Cochran

Journal:
Proc. Amer. Math. Soc. **37** (1973), 489-496

MSC:
Primary 46H20

MathSciNet review:
0310639

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The ideas of inductive limits and bornological spaces are extended to *A*-convex algebras. The usual locally convex linear space characterization of bornological spaces has an appropriate analog for *A*-convex algebras. The results here have a strong connection to locally *m*-convex algebras (*A*-convexity generalizes *m*-convexity) and some additional information relating to *m*-convex algebras is given. Comparisons of various inductive limits are obtained including results which insure that these limits coincide. Finally, *A*-bornological algebras are introduced and studied. It is shown that the property of being *A*-bornological is preserved with respect to several general constructions.

**[1]**G. R. Allan,*A spectral theory for locally convex algebras*, Proc. London Math. Soc. (3)**15**(1965), 399–421. MR**0176344****[2]**Richard Arens,*A generalization of normed rings*, Pacific J. Math.**2**(1952), 455–471. MR**0051445****[3]**R. Creighton Buck,*Bounded continuous functions on a locally compact space*, Michigan Math. J.**5**(1958), 95–104. MR**0105611****[4]**A. C. Cochran, R. Keown, and C. R. Williams,*On a class of topological algebras*, Pacific J. Math.**34**(1970), 17–25. MR**0273399****[5]**Allan C. Cochran,*Weak 𝐴-convex algebras*, Proc. Amer. Math. Soc.**26**(1970), 73–77. MR**0262830**, 10.1090/S0002-9939-1970-0262830-X**[6]**Allan C. Cochran,*Topological algebras and Mackey topologies*, Proc. Amer. Math. Soc.**30**(1971), 115–119. MR**0291807**, 10.1090/S0002-9939-1971-0291807-4**[7]**Ernest A. Michael,*Locally multiplicatively-convex topological algebras*, Mem. Amer. Math. Soc.,**No. 11**(1952), 79. MR**0051444****[8]**Ju-kwei Wang,*Multipliers of commutative Banach algebras*, Pacific J. Math.**11**(1961), 1131–1149. MR**0138014****[9]**Seth Warner,*Weak locally multiplicatively-convex algebras*, Pacific J. Math.**5**(1955), 1025–1032. MR**0076295****[10]**Seth Warner,*Inductive limits of normed algebras*, Trans. Amer. Math. Soc.**82**(1956), 190–216. MR**0079226**, 10.1090/S0002-9947-1956-0079226-3**[11]**Seth Warner,*Bornological structures*, Illinois J. Math.**4**(1960), 231–245. MR**0110939**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
46H20

Retrieve articles in all journals with MSC: 46H20

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1973-0310639-3

Keywords:
*A*-convex algebra,
*m*-convex algebra,
bornological space,
normed algebra,
inductive limit of algebras,
Mackey topology

Article copyright:
© Copyright 1973
American Mathematical Society