Conditionally compact semitopological oneparameter inverse semigroups of partial isometries
Author:
M. O. Bertman
Journal:
Trans. Amer. Math. Soc. 240 (1978), 263275
MSC:
Primary 22A20; Secondary 47D05
MathSciNet review:
0476906
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Abstract: The algebraic structure of oneparameter inverse semigroups has been completely described. Furthermore, if B is the bicyclic semigroup and if B is contained in any semitopological semigroup, the relative topology on B is discrete. We show that if F is an inverse semigroup generated by an element and its inverse, and F is contained in a compact semitopological semigroup, then the relative topology is discrete; in fact, if F is any oneparameter inverse semigroup contained in a compact semitopological semigroup, then the multiplication on F is jointly continuous if and only if the inversion is continuous on F, and we describe in that case. We also show that if is a oneparameter semigroup of bounded linear operators on a (separable) Hilbert space, then generates a oneparameter inverse semigroup T with if and only if is a oneparameter semigroup of partial isometries, and we describe the weak operator closure of T in that case.
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 G. Brown and W. Moran, Idempotents of compact monothetic semigroups, Proc. London Math. Soc. 22 (1971), 203216. MR 0288210 (44:5408)
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 C. Eberhart and J. Selden, On the closure of the bicyclic semigroup, Trans. Amer. Math. Soc. 144 (1969), 115126. MR 0252547 (40:5767)
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 , Oneparameter inverse semigroups, Trans. Amer. Math. Soc. 168 (1972), 5366. MR 0296197 (45:5258)
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 P. R. Halmos, A Hilbert space problem book, Van Nostrand, Princeton, N. J., 1967. MR 0208368 (34:8178)
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 K. DeLeeuw and I. Glicksberg, Applications of almost periodic compactifications, Acta Math. 105 (1961), 6398. MR 0131784 (24:A1632)
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 D. B. McAllister, A homomorphism theorem for semigroups, J. London Math. Soc. 43 (1968), 355366. MR 0224730 (37:329)
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 L. J. Wallen, Decomposition of semigroups of partial isometries, Indiana Univ. Math. J. 20 (1970), 207212. MR 0268720 (42:3617)
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 T. T. West, Weakly compact monothetic semigroups of operators in Banach spaces, Proc. Roy. Irish Acad. Sect. A 67 (1968), 2737. MR 0239467 (39:824)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197804769067
PII:
S 00029947(1978)04769067
Keywords:
Partial isometries,
oneparameter semigroups,
inverse semigroups,
compact semitopological semigroups
Article copyright:
© Copyright 1978
American Mathematical Society
