Celebratio Mathematica — Baum — Complete Bibliography

Complete Bibliography

Works connected to Jonathan Micah Rosenberg

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P. Baum: “\( K \)-homology and D-branes,” pp. 81–94 in Superstrings, geometry, topology, and \( C^* \)-algebras (Fort Worth, TX, 18–22 May 2009). Edited by R. S. Doran, G. Friedman, and J. Rosenberg. Proceedings of Symposia in Pure Mathematics 81. American Mathematical Society (Providence, RI), 2010. MR 2681759 Zbl 1210.81079 incollection

\( K \)-homology is the dual theory to \( K \)-theory. In algebraic geometry [14], [7], the \( K \)-homology of a (possibly singular) projective variety \( X \) is the Grothendieck group of coherent algebraic sheaves on \( X \). In topology there are three ways to define \( K \)-homology. First \( K \)-homology is the homology theory determined by the Bott spectrum. Second, \( K \)-homology is the group of geometric \( K \)-cycles introduced by Baum–Douglas [6]. Third, using funtional analysis, \( K \)-homology is the group of abstract elliptic operators as in the work of M. .F. Atiyah [1], Brown–Douglas–Filmore [15], and G. Kasparov [21].

The D-branes of string theory [31] are twisted geometric \( K \)-cycles which are endowed with some additional structure. The charge of a D-brane is the element in the twisted \( K \)-homology of spacetime determined by the underlying twisted \( K \)-cycle of the D-brane. Essentially, the Baum–Douglas theory [6] was rediscovered in terms of constraints on open strings. The aim of this expository note is to briefly describe this development.

Greg Brian Friedman	Related
Robert Stuart Doran	Related
Jonathan Micah Rosenberg	Related

P. F. Baum and P. M. Hajac: “Local proof of algebraic characterization of free actions” in Special issue on noncommutative geometry and quantum groups in honor of Marc A. Rieffel, published as SIGMA 10. Issue edited by G. Elliott, P. M. Hajac, H. Li, and J. Rosenberg. 2014. paper no. 060. MR 3226990 Zbl 1295.22010 ArXiv 1402.3024 incollection

Let \( G \) be a compact Hausdorff topological group acting on a compact Hausdorff topological space \( X \). Within the \( C^* \)-algebra \( C(X) \) of all continuous complex-valued functions on \( X \), there is the Peter–Weyl algebra \( \mathcal{P}_G(X) \) which is the (purely algebraic) direct sum of the isotypical components for the action of \( G \) on \( C(X) \). We prove that the action of \( G \) on \( X \) is free if and only if the canonical map \[ \mathcal{P}_G(X)\otimes^{}_{C(X/G)}\mathcal{P}_G(X)\to\mathcal{P}_G(X)\otimes\mathcal{O}(G) \] is bijective. Here both tensor products are purely algebraic, and \( \mathcal{O}(G) \) denotes the Hopf algebra of “polynomial” functions on \( G \).

George Arthur Elliott	Related
Jonathan Micah Rosenberg	Related
Marc A. Rieffel	Related
Piotr Mieczysław Hajac	Related
Hanfeng Li	Related