Inductive limits of -convex algebras

Author:
Allan C. Cochran

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

MSC:
Primary 46H20

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

MathSciNet review:
0310639

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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**31**#619. MR**0176344 (31:619)****[2]**R. F. Arens,*A generalization of normed rings*, Pacific J. Math.**2**(1952), 455-471. MR**14,**482. MR**0051445 (14:482b)****[3]**R. C. Buck,*Bounded continuous functions on a locally compact space*, Michigan Math. J.**5**(1968), 95-104. MR**0105611 (21:4350)****[4]**A. C. Cochran, C. R. Williams and R. Keown,*On a class of topological algebras*, Pacific J. Math.**34**(1970), 17-25. MR**42**#8278. MR**0273399 (42:8278)****[5]**A. C. Cochran,*Weak-A-convex algebras*, Proc. Amer. Math. Soc.**26**(1970), 73-77. MR**41**#7435. MR**0262830 (41:7435)****[6]**-,*Topological algebras and Mackey topologies*, Proc. Amer. Math. Soc.**30**(1971), 115-119. MR**0291807 (45:897)****[7]**E. A. Michael,*Locally multiplicative-convex topological algebras*, Mem. Amer. Math. Soc. No. 11 (1952). MR**14,**482. MR**0051444 (14:482a)****[8]**J. Wang,*Multipliers of commutative Banach algebras*, Pacific J. Math.**11**(1961), 1131-1149. MR**25**#1462. MR**0138014 (25:1462)****[9]**S. Warner,*Weak locally multiplicatively-convex algebras*, Pacific J. Math.**5**(1955) 1025-1032. MR**17,**876. MR**0076295 (17:876e)****[10]**-,*Inductive limits of normed algebras*, Trans. Amer. Math. Soc.**82**(1956), 190-216. MR**18**, 52. MR**0079226 (18:52e)****[11]**-,*Bornological structures*, Illinois J. Math.**4**(1960), 231-245. MR**22**#1807. MR**0110939 (22:1807)**

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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