nLab pro-C-star-algebra

A pro-C-star-algebra (also called locally C-star algebra) is very much like a pro-object in the category of C-star-algebras, but the inverse limits are in fact taken in an ambient category of topological *-algebras. $\sigma$-$C^*$-algebras are those pro-$C^*$-algebras which may be obtained as countable inverse limits (there are also intrinsic characterizations in terms of seminorms).

• N. C. Phillips, Inverse limits of $C^\ast$-algebras, J. Operator Theory 19 (1988) 159–195, pdf; Inverse limits of $C^\ast$-algebras, in: Operator Algebras and Applications vol. 1, Structure Theory K-theory, Geometry and Topology (London Math. Soc. Lecture Note Series), D. E. Evans, M. Takesaki, eds.; Representable K-theory for $\sigma$-$C^*$-algebras, K-Theory 3, 441–478 (1989)
• Massoud Amini, Locally compact pro-$C^\ast$-algebras, math.OA/0205253
• Maria Joița, Hilbert modules over locally $C^*$-algebras, thesis, Editura Universitatii Bucuresti 2006 pdf; Crossed products of locally $C^*$-algebras, Rocky Mountain J. Math. 37:5 (2007) 1623-1644 jstor; A new look at the crossed products of pro-$C^*$-algebras; Annals of Functional Analysis, 2015; On representations associated with completely n-positive linear maps on pro-$C^*$-algebras, Chinese Annals of Mathematics, Series B, 2008; Pro-$C^*$-algebras associated to tensor products of pro-$C^*$-correspondences, J. Math. Anal. Appl. 455: 2 (2017) 1822-1834 doi; Crossed Products of pro-$C^*$-algebras and Hilbert pro-$C^*$-modules_, M. Bull. Malays. Math. Sci. Soc. (2015) 38:1053 doi
• Maria Joiţa, Ioannis Zarakas, Crossed products by Hilbert pro-$C^*$-bimodules, Studia Mathematica 215 (2013), 139-156 doi; A construction of pro-$C^*$-algebras from pro-$C^*$-correspondences, J. Operator Theory 74:1 (2015) 195-211, doi
• Rachid El Harti, N. Christopher Phillips, Paulo R. Pinto, Profinite pro-$C^\ast$-algebras and pro-$C^\ast$-algebras of profinite groups, arxiv/1110.3411
• Snigdhayan Mahanta, Noncommutative correspondence categories, simplicial sets and pro-$C^\ast$-algebras, arxiv/0906.5400
• Ilan Barnea, Michael Joachimand Snigdhayan Mahanta, Model structure on projective systems of $C^*$-algebras and bivariant homology theories, New York J. Math. 23 (2017) 383–439 pdf
• Atiyah-Jänich Theorem for σ-C-algebras_ 2017, Applied Categorical Structures 25:5 (2017) 893–905 doi

There is a functor $(-)_b:pro-C^\ast\to C^\ast$ which is the dual of Stone-Čech compactification:

• Rachid El Harti, Gábor Lukács, Bounded and unitary elements in pro-$C^\ast$-algebras, Applied Categorical Structures 14:2, 2006, 151–164 doi math.CT/0511068

A class of topological algebras which correspond to pro-$C^\ast$-algebras can be characterized in terms of properties of the topological algebra:

• S.J. Bhatt, D.J. Karia, An intrinsic characterization of pro-$C^\ast$-algebras and its applications, J. Math. Anal. Appl. 175:1 (1993) 68–80, doi