The bar construction BG of a topological group G has a subcomplex B_{com}G ⊂ BG assembled from spaces of commuting elements in G. If G = U;O (the infinite unitary / orthogonal groups) then B_{com}U and B_{com}O are E_{∞}ring spaces. The corresponding cohomology theory is called commutative Ktheory. In this work we study properties of the spaces B_{com}G and of infinite loop spaces built from them, with an emphasis on the cases G = U,O. The content of this thesis is organised as follows: In Chapter 1 we consider a family of selfmaps of B_{com}G and apply these to study the question when the inclusion map B_{com}G ⊂ BG admits a section up to homotopy. In Chapter 2 we show that B_{com}U is a model for the E_{∞}ring space underlying the kugroup ring of ℂP^{∞}. Thus we provide a complete description of complex commutative Ktheory. We also study the space B_{com}O. Our results include a computation of the torsionfree part of the homotopy groups of B_{com}O and a long exact sequence relating real commutative Ktheory to singular mod2 homology. Chapter 3 is selfcontained. We prove a result about the acyclicity of the "comparison map" M_{∞} → ΩBM in the groupcompletion theorem and apply this to compare the infinite loop space associated to a commutative 𝕀monoid with the Quillen plusconstruction. Chapter 4 is concerned with a previously known filtration of Ω_{0}^{∞}S^{∞} by certain infinite loop spaces {hocolim_{𝕀}B(q, Σ_)}_{q≥2}. For each term in this filtration we construct another filtration on the spectrum level, whose subquotients we describe. Our setup is more general, but the space hocolim_{𝕀}B(q, Σ_) will serve as our main example. Appendix A is an excerpt from the author's Oxford transfer thesis. There we gave a construction of an infinite loop space associated to certain subspaces B(q, Γ_{g,1}) ⊂ BΓ_{g;1}, where Γ_{g;1} is the mapping class group of a genus g surface with one boundary component.
