## $K$-theory and multipliers of stable $C^ \ast$-algebras

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- by J. A. Mingo PDF
- Trans. Amer. Math. Soc.
**299**(1987), 397-411 Request permission

## Abstract:

The main theorem is that if $A$ is a $C^{\ast }$-algebra with a countable approximate identity consisting of projections, then the unitary group of $M(A \otimes K)$ is contractible. This gives a realization, via the index map, of ${K_0}(A)$ as components in the set of Fredholm operators on ${H_A}$.## References

- Huzihiro Araki, Mi-soo Bae Smith, and Larry Smith,
*On the homotopical significance of the type of von Neumann algebra factors*, Comm. Math. Phys.**22**(1971), 71–88. MR**288587**, DOI 10.1007/BF01651585 - M. F. Atiyah,
*$K$-theory*, W. A. Benjamin, Inc., New York-Amsterdam, 1967. Lecture notes by D. W. Anderson. MR**0224083** - Bruce Blackadar and David Handelman,
*Dimension functions and traces on $C^{\ast }$-algebras*, J. Functional Analysis**45**(1982), no. 3, 297–340. MR**650185**, DOI 10.1016/0022-1236(82)90009-X - M. Breuer,
*On the homotopy type of the group of regular elements of semifinite von Neumann algebras*, Math. Ann.**185**(1970), 61–74. MR**264408**, DOI 10.1007/BF01350761 - Jochen Brüning and Wolfgang Willgerodt,
*Eine Verallgemeinerung eines Satzes von N. Kuiper*, Math. Ann.**220**(1976), no. 1, 47–58. MR**405483**, DOI 10.1007/BF01354528 - Robert C. Busby,
*Double centralizers and extensions of $C^{\ast }$-algebras*, Trans. Amer. Math. Soc.**132**(1968), 79–99. MR**225175**, DOI 10.1090/S0002-9947-1968-0225175-5 - A. Connes,
*An analogue of the Thom isomorphism for crossed products of a $C^{\ast }$-algebra by an action of $\textbf {R}$*, Adv. in Math.**39**(1981), no. 1, 31–55. MR**605351**, DOI 10.1016/0001-8708(81)90056-6 - Edward G. Effros and E. Christopher Lance,
*Tensor products of operator algebras*, Adv. Math.**25**(1977), no. 1, 1–34. MR**448092**, DOI 10.1016/0001-8708(77)90085-8 - George A. Elliott,
*Derivations of matroid $C^{\ast }$-algebras. II*, Ann. of Math. (2)**100**(1974), 407–422. MR**352999**, DOI 10.2307/1971079 - K. R. Goodearl and D. E. Handelman,
*Stenosis in dimension groups and AF $C^{\ast }$-algebras*, J. Reine Angew. Math.**332**(1982), 1–98. MR**656856**, DOI 10.1515/crll.1982.332.1 - G. G. Kasparov,
*Hilbert $C^{\ast }$-modules: theorems of Stinespring and Voiculescu*, J. Operator Theory**4**(1980), no. 1, 133–150. MR**587371** - G. G. Kasparov,
*The operator $K$-functor and extensions of $C^{\ast }$-algebras*, Izv. Akad. Nauk SSSR Ser. Mat.**44**(1980), no. 3, 571–636, 719 (Russian). MR**582160** - Nicolaas H. Kuiper,
*The homotopy type of the unitary group of Hilbert space*, Topology**3**(1965), 19–30. MR**179792**, DOI 10.1016/0040-9383(65)90067-4 - James A. Mingo,
*On the contractibility of the unitary group of the Hilbert space over a $C^{\ast }$-algebra*, Integral Equations Operator Theory**5**(1982), no. 6, 888–891. MR**682305**, DOI 10.1007/BF01694068 - A. S. Miščenko and A. T. Fomenko,
*The index of elliptic operators over $C^{\ast }$-algebras*, Izv. Akad. Nauk SSSR Ser. Mat.**43**(1979), no. 4, 831–859, 967 (Russian). MR**548506** - Gert K. Pedersen,
*$C^{\ast }$-algebras and their automorphism groups*, London Mathematical Society Monographs, vol. 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979. MR**548006** - M. Pimsner, S. Popa, and D. Voiculescu,
*Homogeneous $C^{\ast }$-extensions of $C(X)\otimes K(H)$. I*, J. Operator Theory**1**(1979), no. 1, 55–108. MR**526291** - Graeme Segal,
*Equivariant contractibility of the general linear group of Hilbert space*, Bull. London Math. Soc.**1**(1969), 329–331. MR**248877**, DOI 10.1112/blms/1.3.329 - J. L. Taylor,
*Banach algebras and topology*, Algebras in analysis (Proc. Instructional Conf. and NATO Advanced Study Inst., Birmingham, 1973) Academic Press, London, 1975, pp. 118–186. MR**0417789**

## Additional Information

- © Copyright 1987 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**299**(1987), 397-411 - MSC: Primary 46L80; Secondary 18F25, 19K56, 46M20
- DOI: https://doi.org/10.1090/S0002-9947-1987-0869419-7
- MathSciNet review: 869419